**The Computational Unified Field Theory (CUFT) – Revising Quantum & Relativistic Models**

Jehonathan Bentwich

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59175

#### **1. Introduction**

#### **1.1. The Computational Unified Field Theory (CUFT)**

Over the past three years, a new hypothetical 'Computational Unified Field Theory' (CUFT) has been discovered which sets to unify between Quantum Mechanics and Relativity Theory (e.g., whose current theoretical contradiction is considered to be most likely the greatest unresolved enigma in modern Science). Indeed, several previous articles have demonstrated that this new hypothetical CUFT is capable of resolving the principle quantum-relativistic theoretical inconsistencies, replicating all of their key empirical phenomena, and was able to identify (three) "differential-critical" predictions differentiating it from both quantum and relativistic models of physical reality; Indeed, before proceeding to describe a (recent) empirical validation of one of these three CUFT 'differential-critical' predictions it may be helpful to delineate the key theoretical postulates underlying the CUFT as well as its associated "'Cinematic-Film Metaphor";

#### *1.1.1. The 'Duality Principle'*

The first theoretical postulate underlying the CUFT is the computational **'Duality Principle'** [4] which identified a basic "computational flaw" associated with both Quantum and Rela‐ tivistic computational systems; The Duality Principle demonstrates that both quantum and relativistic computational systems comprise a 'Self-Referential Ontological Computational System' (SROCS) which assumes that it is possible to determine the value of any given 'y' (e.g., subatomic 'target' or relativistic 'space-time' or 'energy-mass' entity) strictly based on its direct (or indirect) physical interaction with another (exhaustive) 'x' factor/s (e.g., subatomic 'probe' or relativistic observer). It proves that such SROCS computational structure inevitably leads

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to both "logical inconsistency" and "computational indeterminacy" which are contradicted by robust empirical evidence indicating the capacity of both quantum and relativistic to determine the particular value of any given subatomic 'target' element or relativistic 'space-time' or 'energy-mass' phenomenon. Hence, the 'Duality Principle' negates the assumed SROCS computational structure underlying both quantum and relativistic computational systems, instead pointing at the existence of a (singular) higher-ordered 'Universal Computational Principle' (UCP) which computes the "simultaneous co-occurrence" of all exhaustive subato‐ mic 'probe-target' or relativistic 'observer- (space-time or energy-mass) phenomenon' physical interactions (at any given point in time).

Indeed, the identification of such a singular **'Universal Computational Principle' (UCP)** responsible for the computation of all quantum and relativistic (exhaustive) physical interac‐ tions which is also postulated to possess three **'Computational Dimensions'** constitutes the second theoretical postulate of the CUFT; In order to perhaps better sense these three *'Computational Dimensions'* of the UCP let us examine a closely related **"Cinematic-Film Metaphor"** which may be used to explain these three 'Computational Dimensions' (as well as some other features and aspects of the CUFT):

Imagine yourself sitting in a cinema film presentation (e.g., seeing a film for the first time – unaware of the 'mechanics' of a film being presented to you)... In this case you could measure (for instance) the "velocity" (or energy) of a jet-plane zooming through the screen, the "time" it took this jet-plane to get from point 'A' to point 'B' (on the screen), the "spatial" length of the plane etc. – being unaware that (in truth) all of these 'spatial', 'temporal', 'energy' (and 'mass') "physical" features are produced based on the 'higher-ordered' computation of the degree of "displacement" or "lack of displacement" occurring across the series of cinematicfilm frames!? Thus, for instance, the plane's "energy" (or velocity) is computed based on the number of 'pixels' that plane has been displaced across a given series of frames... Conversely, the plane's "spatial" measure is give based on the computation of the number of 'spatial pixels' that remain constant across a series of cinematic film frames (e.g., resulting in the fact that the plane's length doesn't "increase" or "decrease" across these frames)... Likewise, the "tempo‐ ral" length of the plane's flight is computed based on the number of changes that occur in- or around- the plane (across a given number of film frames): imagine for instance what would happen to that plane's flight temporal value if the frames were projected more slowly (e.g., in "slow-motion" where there is a smaller number of changes taking place in the plane's flight, giving rise to a "dilated time" measure) or in a case in which precisely the same frame was presented over and over again for say one minute – time would "stand-still"... Similarly, we can devise a special 'cinematic-film' operation in which any given object is projected at "belowthreshold" intensity at any given single frame such that only the presentation of the same object (in the same spatial configuration) across multiple number of frames may produce a visible object and that its apparent "mass" value will be computed as a function of the number of frames in which that object appeared 'spatially-consistent'... So, we can see that at least in the "cinematic-film metaphor", 'energy', 'space'; 'time' or 'mass' – are all produced as secondary computational measures being computed by a higher-ordered (singular) computation relating to the degree of 'changes'- or 'lack of changes'- of a given object across the frame, or as measured in the object itself (across a given series of cinematic film-frames)...

Quite similarly, the CUFT posits that the four basic physical features of 'space', 'time', 'energy' and 'mass' are produced through the computation of a singular (higher-ordered) 'Universal Computational Principle' (represented by the Hebrew letter "yud") – of the degree of 'consis‐ tency' or 'inconsistency' across a series of extremely rapid (c2 /h) 'Universal Simultaneous Computational Frames' (USCF's): According to the CUFT, this Universal Computational Principle (UCP) employs two 'Computational Dimensions' to compute these four (secondary computational) physical features of 'space', 'time', 'energy' and 'mass' which are: 'Consistency' ('consistent' vs. 'inconsistent') and 'Framework' ('frame' vs. 'object'), and an additional Computational Dimension of 'Locus' ('global' vs. 'local') which accounts for relativistic phenomena.

#### *1.1.2. The UCP's computational dimensions*

Hence, the CUFT hypothesizes that the above mentioned 'Universal Computational Principle' (UCP) possesses three 'Computational Dimensions': The *'Framework'* Dimension relates to certain 'computational features' that are computed at the 'object' level, or at the 'frame' (USCF's) level; The 'Consistency' Dimension relates to the UCP's computation of the degree of 'consistency' or 'inconsistency' of an object across a series of USCF's frames (e.g., regarding its above mentioned 'object' or 'frame' measures and also relating to the below mentioned 'Locus' Dimension computation); and the 'Locus Dimension' relates to the UCP's computation of any 'Framework-Consistency' combination from computational perspective of the 'frame' (termed: 'global') or from the 'object's' computational perspective (termed: 'local'); The fascinating facet of these UCP's three Computational Dimensions is that they produce the four physical features of 'space', 'energy', 'mass' and 'time' – i.e., as secondary computational combinations of the 'Framework' and 'Consistency' Computational Dimensions: The CUFT posits that 'space' and 'energy' emerge as a result of the UCP's computation of the degree of 'consistent' or 'inconsistent' measure of an 'object' (e.g., comprising one of the computational levels of the 'Framework' Dimension) the 'Framework' Dimension; Likewise, the basic physical features of 'mass' and 'time' arise as secondary computational features associated with the degree of 'consistent' or 'inconsistent' measure of an object relative to the 'frame' (also comprising the 'Framework' Dimension)!

Hence, the (new) computational definitions of 'space', 'energy', 'mass' and 'energy' are given by:

#### **S:(fi{x,y,z}[USCF(***i***)]+… f***j***{x,y,z}[USCF(***n***)]) / h x n{USCF's}**

such that:

to both "logical inconsistency" and "computational indeterminacy" which are contradicted by robust empirical evidence indicating the capacity of both quantum and relativistic to determine the particular value of any given subatomic 'target' element or relativistic 'space-time' or 'energy-mass' phenomenon. Hence, the 'Duality Principle' negates the assumed SROCS computational structure underlying both quantum and relativistic computational systems, instead pointing at the existence of a (singular) higher-ordered 'Universal Computational Principle' (UCP) which computes the "simultaneous co-occurrence" of all exhaustive subato‐ mic 'probe-target' or relativistic 'observer- (space-time or energy-mass) phenomenon' physical

Indeed, the identification of such a singular **'Universal Computational Principle' (UCP)** responsible for the computation of all quantum and relativistic (exhaustive) physical interac‐ tions which is also postulated to possess three **'Computational Dimensions'** constitutes the second theoretical postulate of the CUFT; In order to perhaps better sense these three *'Computational Dimensions'* of the UCP let us examine a closely related **"Cinematic-Film Metaphor"** which may be used to explain these three 'Computational Dimensions' (as well as

Imagine yourself sitting in a cinema film presentation (e.g., seeing a film for the first time – unaware of the 'mechanics' of a film being presented to you)... In this case you could measure (for instance) the "velocity" (or energy) of a jet-plane zooming through the screen, the "time" it took this jet-plane to get from point 'A' to point 'B' (on the screen), the "spatial" length of the plane etc. – being unaware that (in truth) all of these 'spatial', 'temporal', 'energy' (and 'mass') "physical" features are produced based on the 'higher-ordered' computation of the degree of "displacement" or "lack of displacement" occurring across the series of cinematicfilm frames!? Thus, for instance, the plane's "energy" (or velocity) is computed based on the number of 'pixels' that plane has been displaced across a given series of frames... Conversely, the plane's "spatial" measure is give based on the computation of the number of 'spatial pixels' that remain constant across a series of cinematic film frames (e.g., resulting in the fact that the plane's length doesn't "increase" or "decrease" across these frames)... Likewise, the "tempo‐ ral" length of the plane's flight is computed based on the number of changes that occur in- or around- the plane (across a given number of film frames): imagine for instance what would happen to that plane's flight temporal value if the frames were projected more slowly (e.g., in "slow-motion" where there is a smaller number of changes taking place in the plane's flight, giving rise to a "dilated time" measure) or in a case in which precisely the same frame was presented over and over again for say one minute – time would "stand-still"... Similarly, we can devise a special 'cinematic-film' operation in which any given object is projected at "belowthreshold" intensity at any given single frame such that only the presentation of the same object (in the same spatial configuration) across multiple number of frames may produce a visible object and that its apparent "mass" value will be computed as a function of the number of frames in which that object appeared 'spatially-consistent'... So, we can see that at least in the "cinematic-film metaphor", 'energy', 'space'; 'time' or 'mass' – are all produced as secondary computational measures being computed by a higher-ordered (singular) computation relating to the degree of 'changes'- or 'lack of changes'- of a given object across the frame, or as

measured in the object itself (across a given series of cinematic film-frames)...

interactions (at any given point in time).

176 Selected Topics in Applications of Quantum Mechanics

some other features and aspects of the CUFT):

#### *fj{x,y,z}[USCF(i)]) ≤ fi{x+(hxn),y+(hxn),z+(hxn)}[USCF(i…n)]*

where the 'space' measure of a given object (or event) is computed based on a *frame consis‐ tent* computation that adds the specific USCF's (x,y,z) localization across a series of USCF's [1...n] – which nevertheless do *not exceed* the threshold of Planck's constant per each ('n') number of frames (e.g., thereby providing the CUFT's definition of "space" as 'frameconsistent' USCF's measure).

Conversely, the 'energy' of an object (e.g., whether it is the spatial dimensions of an object or event or whether it relates to the spatial location of an object) is computed based on the *frame's differences* of a given object's location/s or size/s across a series of USCF's, divided by the speed of light 'c' multiplied by the number of USCF's across which the object's energy value has been measured:

#### **E:(f***j***{x,y,z}[USCF(n)])–f***i***{(x+n),(y+n),(z+n)} [USCF(***i***...***n***)] ) /c x n{USCF's}**

such that:

#### *fj{x,y,z}[USCF(n)])>fi{x+(hxn),y+(hxn),z+(hxn) [USCF(i…n)])*

wherein the energetic value of a given object, event etc. is computed based on the subtraction of that object's "universal pixels" location/s across a series of USCF's, divided by the speed of light multiplied by the number of USCF's.

In contrast, the of 'mass' of an object is computed based on a measure of the number of times an *'object'* is presented *'consistently'* across a series of USCF's, divided by Planck's constant (e.g., representing the minimal degree of inter-frame's changes):

#### **M: Σ[o***j***{x,y,z} [USCF(***n***)] = o(***i***…***j-1***) {(x),(y),(z)} {***USCF***(***i***...***n***)} / h x n{USCF's} {***USCF***(***1***...***n***)} / h x n{USCF's}**

*where the measure of 'mass' is computed based on a comparison of the number of instances in which an object's (or event's) 'universal-pixels' measures* (e.g., along the three axes 'x', y' and 'z') is identical across a series of USCF's (e.g., Σoi{x,y,z} [USCF(n)] = oj{(x+m),(y+m),(z+m)} [USCF(1...n)]), divided by Planck's constant.

Again, the measure of 'mass' represents an *object-consistent* computational measure – e.g., regardless of any changes in that object's spatial (frame) position across these frames.

Finally, the *'time'* measure is computed based on an *'object-inconsistent'* computation of the number of instances in which an 'object' (i.e., corresponding to only a particular segment of the entire USCF) changes across two subsequent USCF's (e.g., Σ o*i*{x,y,z} [USCF(*n*)] ≠ o*j*{(x+m), (y+m),(z+m)}[USCF(*1...n*)]), ivided by 'c':

#### **T : Σ o***j***{x,y,z} [USCF(***n***)] ≠ o(***i…j-1***){(x),(y),(z)} [USCF(***1...n***)] /c x n{USCF's}**

such that:

#### T:Σo*i*{x,y,z}[USCF(*n*)]-*j*{(x+m),(y+m),(z+m)} [USCF(*1...n*)] ≤ c x n{USCF's}

Hence, the measure of 'time' represents a computational measure of the number of *'objectinconsistent'* presentations any given object (or event) possesses across subsequent USCF' (e.g., once again- regardless of any changes in that object's 'frame's' spatial position across this series of USCF's). Finally, the combination of the 'Locus' Dimension together with the 'Framework-Consistency' Dimensions, e.g., producing the four physical features of 'space', 'energy', 'mass', and 'time' – produces all known relativistic effects and phenomenon, e.g., such as 'timedilation', 'energy-mass' equivalence and even the curvature of 'space-time'!

#### *1.1.3. The Computational Invariance Principle*

Conversely, the 'energy' of an object (e.g., whether it is the spatial dimensions of an object or event or whether it relates to the spatial location of an object) is computed based on the *frame's differences* of a given object's location/s or size/s across a series of USCF's, divided by the speed of light 'c' multiplied by the number of USCF's across which the object's energy value has been

wherein the energetic value of a given object, event etc. is computed based on the subtraction of that object's "universal pixels" location/s across a series of USCF's, divided by the speed of

In contrast, the of 'mass' of an object is computed based on a measure of the number of times an *'object'* is presented *'consistently'* across a series of USCF's, divided by Planck's constant

**M: Σ[o***j***{x,y,z} [USCF(***n***)] = o(***i***…***j-1***) {(x),(y),(z)} {***USCF***(***i***...***n***)} / h x n{USCF's} {***USCF***(***1***...***n***)} / h**

*where the measure of 'mass' is computed based on a comparison of the number of instances in which an object's (or event's) 'universal-pixels' measures* (e.g., along the three axes 'x', y' and 'z') is identical across a series of USCF's (e.g., Σoi{x,y,z} [USCF(n)] = oj{(x+m),(y+m),(z+m)} [USCF(1...n)]),

Again, the measure of 'mass' represents an *object-consistent* computational measure – e.g.,

Finally, the *'time'* measure is computed based on an *'object-inconsistent'* computation of the number of instances in which an 'object' (i.e., corresponding to only a particular segment of the entire USCF) changes across two subsequent USCF's (e.g., Σ o*i*{x,y,z} [USCF(*n*)] ≠ o*j*{(x+m),

Hence, the measure of 'time' represents a computational measure of the number of *'objectinconsistent'* presentations any given object (or event) possesses across subsequent USCF' (e.g., once again- regardless of any changes in that object's 'frame's' spatial position across this series of USCF's). Finally, the combination of the 'Locus' Dimension together with the 'Framework-Consistency' Dimensions, e.g., producing the four physical features of 'space', 'energy', 'mass', and 'time' – produces all known relativistic effects and phenomenon, e.g., such as 'time-

regardless of any changes in that object's spatial (frame) position across these frames.

**T : Σ o***j***{x,y,z} [USCF(***n***)] ≠ o(***i…j-1***){(x),(y),(z)} [USCF(***1...n***)] /c x n{USCF's}**

T:Σo*i*{x,y,z}[USCF(*n*)]-*j*{(x+m),(y+m),(z+m)} [USCF(*1...n*)] ≤ c x n{USCF's}

dilation', 'energy-mass' equivalence and even the curvature of 'space-time'!

**E:(f***j***{x,y,z}[USCF(n)])–f***i***{(x+n),(y+n),(z+n)} [USCF(***i***...***n***)] ) /c x n{USCF's}**

*fj{x,y,z}[USCF(n)])>fi{x+(hxn),y+(hxn),z+(hxn) [USCF(i…n)])*

(e.g., representing the minimal degree of inter-frame's changes):

light multiplied by the number of USCF's.

178 Selected Topics in Applications of Quantum Mechanics

measured:

such that:

**x n{USCF's}**

such that:

divided by Planck's constant.

(y+m),(z+m)}[USCF(*1...n*)]), ivided by 'c':

Another key theoretical postulate comprising the CUFT is the 'Computational Invariance Principle' which identifies this 'Universal Computational Principle' as the sole 'computation‐ ally invariant' element which both produces all four 'computationally variant' physical features of 'space', 'time', 'energy' and 'mass' and also exists independently of these physical features "in-between" any two subsequent 'USCF's frames; As such, the 'Computational Invariance Principle' recognizes the Universal Computational Principle as the sole (and singular) 'invariant' reality underlying the production of the four secondary computational 'variant' physical properties of 'space', 'time', 'energy' and 'mass' (based in part on a wellknown scientific principle: "Ockham's Razor" which prefers the simplest most parsimonious theoretical account for complex phenomena) [1]...

#### *1.1.4. The Universal Computational Formula*

Finally, this recognition of the Universal Computational Principle as the sole and singular reality producing and sustaining all four (secondary computational) physical properties of space, time, energy and mass has also lead to the formulation of a singular 'Universal Com‐ putational Formula' which completely integrates these four secondary computational physical properties, as well as all known quantum and relativistic properties (e.g., as embedded within the higher-ordered Universal Computational Formula):

$$\text{Universal Componental Formula:} \left\{ \frac{\text{\textdegree}}{\text{h}} \right\} = \frac{\text{s}}{\text{t}} \times \frac{\text{e}}{\text{m}}$$

#### **2. The CUFT: Quantum-relativistic harmonization- embedding- & transcendence**

Hence, the next necessary step in validating the CUFT as a satisfactory TOE is to demon‐ strate that it's capable of harmonizing between quantum and relativistic models, embed both models within the CUFT's Universal Computational Formula, and providing certain "differential critical predictions" which transcend these quantum and relativistic models (e.g., and if validated empirically validate the CUFT as an satisfactory TOE!) First, we set to demonstrate that the CUFT is able to bridge the (above mentioned) key theoretical inconsistencies that seem to exist between quantum and relativistic models based on its reformulation- and embedding- of quantum and relativistic computation within the singular (higher-ordered) Universal Computational Principle (e.g., due to the Duality Principle's identification of a mutual computational flaw underlying both models, as shown above); Interestingly, based on the singularity of the Universal Computational Principle' computa‐ tion of both quantum and relativistic relationships (e.g., as embedded within an exhaus‐ tive USCF's frames' series) the CUFT is able to embrace both quantum's probabilistic and positivistic relativistic modeling. This is because the Universal Computational Principle's (rapid) production of all exhaustive spatial-pixels in the physical universe comprising each USCF frame – allows it to embed "single spatial-temporal" relativistic objects' (or subato‐ mic 'particles') measurements as well as "multi spatial-temporal" subatomic 'wave' measures! In fact, one of the elegant features of the CUFT is precisely the fact that it conceptualizes such 'single spatial-temporal' relativistic 'objects' or subatomic quantum 'particles' and 'multi spatial-temporal' quantum subatomic 'wave' measurements – within the exhaustive computational framework of the Universal Computational Principle's rapid production of the series of USCF's frames (e.g., comprising all such 'single spatial-tempo‐ ral' relativistic object or subatomic particle and 'multi spatial-temporal' quantum wave measurements...) Moreover, this exhaustive computational framework of the CUFT allows it to reconceptualize quantum's Uncertainty Principle's 'complimentary pairs' of 'space and energy' or 'time and mass' merely representing a computational constraint intrinsically embedded within the Universal Computational Principle's computation of the two 'Framework' and 'Consistency' Computational Dimensions – i.e., based on the fact that 'space' and 'energy' exhaustively comprise the Framework's Dimensions' 'frame' level, and likewise 'mass' and 'time' exhaustively comprising Framework's 'object' computational level... Hence, the CUFT is capable of embedding both 'single spatial-temporal' relativis‐ tic objects (and quantum 'particles'), and (apparently) 'probabilistic' 'multi spatial-tempo‐ ral' quantum wave measures within the broader and more exhaustive Universal Computational Principle's rapid computation of the series of USCF's frames (e.g., thereby also resolving the 'particle-wave duality' postulate of Quantum Mechanics!)

The CUFT's resolution of the second key quantum-relativistic theoretical inconsistency relating to quantum's instantaneous 'entanglement' phenomenon as opposed to Relativity's speed of light constraint set on the transmission of any signal across space is also anchored in the above mentioned Universal Computational Principle's rapid computation of these USCF's frames ; Since the CUFT posits that the Universal Computational Principle's (rapid) compu‐ tation of each of the Universal Simultaneous Computational Frame (USCF) simultaneously computes all of the spatial-pixels in the physical universe at a minimal time-point (e.g., c2 /h), then this computation extends the phenomenon of 'quantum entanglement' to all exhaustive spatial points in the universe (e.g., at any such minimal time-point! On the other hand, based on the above embedding of all 'single spatial-temporal' relativistic objects (or subatomic particles) as well as 'multi spatial-temporal' subatomic wave measures within the Universal Computational Principle's exhaustive USCF's computation – it allows for Relativity's apparent speed of light constraint imposed on any such 'single spatial-temporal' relativistic object (or subatomic 'particle') transmission!

The next step towards the validation of the CUFT as a satisfactory TOE involves an articulation of the embedding of quantum and relativistic models within the singular higher-ordered CUFT's Universal Computational Formula – i.e., which is shown to both maintain- and transcend- the (currently) known quantum and relativistic relationships! As can be seen from the two 'quantum' and 'relativistic' formats of the Universal Computational Principle (below), the highlighted portions of these formats conforms to the known mathematical relationships found in quantum and relativistic models, e.g., Relativity's energy and mass equivalence, and Quantum's 'complimentary pairs' of 'space and energy', 'mass and time' as constrained by the Uncertainty Principle's 'h' Planck's constant simultaneous measurement accuracy constraint:

I. Relativistic Format: e <sup>×</sup> <sup>s</sup> <sup>t</sup> =m <sup>×</sup> <sup>c</sup><sup>2</sup> h

positivistic relativistic modeling. This is because the Universal Computational Principle's (rapid) production of all exhaustive spatial-pixels in the physical universe comprising each USCF frame – allows it to embed "single spatial-temporal" relativistic objects' (or subato‐ mic 'particles') measurements as well as "multi spatial-temporal" subatomic 'wave' measures! In fact, one of the elegant features of the CUFT is precisely the fact that it conceptualizes such 'single spatial-temporal' relativistic 'objects' or subatomic quantum 'particles' and 'multi spatial-temporal' quantum subatomic 'wave' measurements – within the exhaustive computational framework of the Universal Computational Principle's rapid production of the series of USCF's frames (e.g., comprising all such 'single spatial-tempo‐ ral' relativistic object or subatomic particle and 'multi spatial-temporal' quantum wave measurements...) Moreover, this exhaustive computational framework of the CUFT allows it to reconceptualize quantum's Uncertainty Principle's 'complimentary pairs' of 'space and energy' or 'time and mass' merely representing a computational constraint intrinsically embedded within the Universal Computational Principle's computation of the two 'Framework' and 'Consistency' Computational Dimensions – i.e., based on the fact that 'space' and 'energy' exhaustively comprise the Framework's Dimensions' 'frame' level, and likewise 'mass' and 'time' exhaustively comprising Framework's 'object' computational level... Hence, the CUFT is capable of embedding both 'single spatial-temporal' relativis‐ tic objects (and quantum 'particles'), and (apparently) 'probabilistic' 'multi spatial-tempo‐ ral' quantum wave measures within the broader and more exhaustive Universal Computational Principle's rapid computation of the series of USCF's frames (e.g., thereby

also resolving the 'particle-wave duality' postulate of Quantum Mechanics!)

subatomic 'particle') transmission!

180 Selected Topics in Applications of Quantum Mechanics

The CUFT's resolution of the second key quantum-relativistic theoretical inconsistency relating to quantum's instantaneous 'entanglement' phenomenon as opposed to Relativity's speed of light constraint set on the transmission of any signal across space is also anchored in the above mentioned Universal Computational Principle's rapid computation of these USCF's frames ; Since the CUFT posits that the Universal Computational Principle's (rapid) compu‐ tation of each of the Universal Simultaneous Computational Frame (USCF) simultaneously computes all of the spatial-pixels in the physical universe at a minimal time-point (e.g., c2

then this computation extends the phenomenon of 'quantum entanglement' to all exhaustive spatial points in the universe (e.g., at any such minimal time-point! On the other hand, based on the above embedding of all 'single spatial-temporal' relativistic objects (or subatomic particles) as well as 'multi spatial-temporal' subatomic wave measures within the Universal Computational Principle's exhaustive USCF's computation – it allows for Relativity's apparent speed of light constraint imposed on any such 'single spatial-temporal' relativistic object (or

The next step towards the validation of the CUFT as a satisfactory TOE involves an articulation of the embedding of quantum and relativistic models within the singular higher-ordered CUFT's Universal Computational Formula – i.e., which is shown to both maintain- and transcend- the (currently) known quantum and relativistic relationships! As can be seen from the two 'quantum' and 'relativistic' formats of the Universal Computational Principle (below), the highlighted portions of these formats conforms to the known mathematical relationships

/h),

II. Quantum Format: t <sup>×</sup> <sup>m</sup> <sup>×</sup> <sup>c</sup><sup>2</sup> <sup>h</sup> =s <sup>×</sup> <sup>e</sup>

#### **3. The CUFT's "differential-critical predictions"**

However, it also becomes clear that the CUFT's Universal Computational Principle's embed‐ ding of those empirically validated quantum and relativistic relationships – also transcends and critically differs from these relationships! Indeed, these computational differences between the Universal Computational Formula's 'quantum' and 'relativistic' formats and the 'standard' relativistic 'E=Mc2 ' and quantum 'complimentary pairs' constitutes one (of three) "differentia-critical predictions" that differentiate the CUFT model from both quantum and relativistic predictions, e.g., thereby providing an empirically testable means for validating the CUFT as a satisfactory TOE...

Another key "differential-critical prediction" that differentiates the CUFT from both relativ‐ istic and quantum models' predictions are: the CUFT's prediction regarding the more consis‐ tent (spatial) presentation of more massive particles (or elements) – across a given series of USCF's frames, relative to less massive particles' appearance across the same series of USCF's frames. In fact, this 'differential critical prediction' regarding the more consistent spatial presentation of more massive particles (or elements) across a series of USCF's frames, relative to the spatial presentation of less massive particles (or elements) precisely replicates the empirical findings of the recently discovered 'Proton-Radius Puzzle', thereby providing a first empirical validation for the CUFT as a satisfactory TOE!

The third 'differential-critical prediction' differentiating the CUFT from both quantum and relativistic models involves a possible "reversal of the space-time spatial-electromagnetic pixels sequence" across a series of USCF's frames electromagnetic spatial-pixels' sequence of a given object or phenomenon; This may be achieved through a precise recording of that object (or phenomenon's) spatial-electromagnetic pixels values (across a given series of USCF's), and a manipulation of these electromagnetic-spatial pixels values (through precise electromagnetic stimulation) so as to produce the reverse sequence of the recorded spatial-electromagnetic values sequence! Interestingly, due to the fact that quantum theory precludes the possibility of the "un-collapse" of the probability wave function following a certain interaction between the any such probe particle and the target particle's wave function – this 'differential critical prediction' is ruled out as a possible prediction of Quantum Mechanics; Likewise, since Relativity sets the speed of light as a clear "unsurpassable" limit for the transference of any signals it also precludes the possibility of "reversing time"; In contrast, since the CUFT defines 'time' (e.g., alongside the other three physical features of 'space', 'energy' and 'mass') merely as a secondary computational property produced by the Universal Computational Principle's three Computational Dimensions' computation of the degree of an "object's-inconsistency" across a series of USCF's frames – then it should allow for the "reversal" of the 'space-time' sequence (e.g., of the particular spatial-electromagnetic pixels' values) across a series of USCF's frames!

### **4. Empirical validation of the CUFT as satisfactory 'TOE': The 'proton radius puzzle'**

Fortunately, the second (abovementioned) 'differential-critical prediction' of the CUFT regarding the more consistent spatial presentations of a more massive particle (or element), relative to the spatial-consistency of a less massive particle (or element) across a given series of USCF's frames – has now received initial empirical validation through the findings associ‐ ated with the 'Proton-Radius Puzzle'! This is because the 'Proton-Radius Puzzle' empirical findings indicate that the more massive 'Moun Hydrogen Proton' is measured (approximately) 200 times – smaller and more accurate than the standard Hydrogen (e.g., with the 200 times lighter electron particle instead of the Muon)... In order to fully understand how these 'Proton-Radius Puzzle' findings (Bernauer & Pohl, 2014) empirically confirm the differential-critical prediction of the CUFT, lets us return to the CUFT's computational definitions of "mass"; Mass is defined by the CUFT as a measure of the degree of "spatial-consistency" of a particle across a given series of USCF's frames! In mathematical terms, it is measured as the number of times that this particle was presented across the same spatial pixels (measured from within the object's frame of reference) across a series of USCF's frames... This computational definition of 'mass' implies at least two empirically measurable predictions:


Interestingly, the 'Proton-Radius Puzzle' precisely confirms the first of these two CUFT 'differential critical' predictions – i.e., indicating that the (200 times) more massive Muon particle (e.g., when embedded within the Hydrogen Proton) is measured as (200 times) 'smaller' and 'more accurate' than the (200 times) less massive electron (associated) Hydrogen Proton! Hence, these findings provide an initial empirical confirmation of the CUFT – as differing from the predictions of both quantum and relativistic models' predictions (e.g., which cannot account for these "Proton-Radius Puzzle" findings)!

*Efforts should be made to empirically validate the second (abovementioned) aspect of the CUFT's differential-critical prediction regarding the appearance of 'more massive' particles such as the Muon across a greater number of USCF's frames than the appearance of less massive particles (such as the electron)!*

### **5. The CUFT: Challenging quantum & relativistic "materialisticreductionistic" assumption**

'time' (e.g., alongside the other three physical features of 'space', 'energy' and 'mass') merely as a secondary computational property produced by the Universal Computational Principle's three Computational Dimensions' computation of the degree of an "object's-inconsistency" across a series of USCF's frames – then it should allow for the "reversal" of the 'space-time' sequence (e.g., of the particular spatial-electromagnetic pixels' values) across a series of USCF's

**4. Empirical validation of the CUFT as satisfactory 'TOE': The 'proton**

of 'mass' implies at least two empirically measurable predictions:

USCF's frames than the (lighter) electron!

cannot account for these "Proton-Radius Puzzle" findings)!

Fortunately, the second (abovementioned) 'differential-critical prediction' of the CUFT regarding the more consistent spatial presentations of a more massive particle (or element), relative to the spatial-consistency of a less massive particle (or element) across a given series of USCF's frames – has now received initial empirical validation through the findings associ‐ ated with the 'Proton-Radius Puzzle'! This is because the 'Proton-Radius Puzzle' empirical findings indicate that the more massive 'Moun Hydrogen Proton' is measured (approximately) 200 times – smaller and more accurate than the standard Hydrogen (e.g., with the 200 times lighter electron particle instead of the Muon)... In order to fully understand how these 'Proton-Radius Puzzle' findings (Bernauer & Pohl, 2014) empirically confirm the differential-critical prediction of the CUFT, lets us return to the CUFT's computational definitions of "mass"; Mass is defined by the CUFT as a measure of the degree of "spatial-consistency" of a particle across a given series of USCF's frames! In mathematical terms, it is measured as the number of times that this particle was presented across the same spatial pixels (measured from within the object's frame of reference) across a series of USCF's frames... This computational definition

**a.** That the more massive 'Muon' particle should be measured as more accurate- and as smaller- than the less massive electron particle; this is due to the fact that the more massive

**b.** That more massive particles (e.g., such as the Muon) should be measured across a greater number of USCF's frames, relative to less massive particles (such as the electron); In other words, we could expect to measure the (more massive) Muon across a greater number of

Interestingly, the 'Proton-Radius Puzzle' precisely confirms the first of these two CUFT 'differential critical' predictions – i.e., indicating that the (200 times) more massive Muon particle (e.g., when embedded within the Hydrogen Proton) is measured as (200 times) 'smaller' and 'more accurate' than the (200 times) less massive electron (associated) Hydrogen Proton! Hence, these findings provide an initial empirical confirmation of the CUFT – as differing from the predictions of both quantum and relativistic models' predictions (e.g., which

a particle is the greater its spatial-consistency across USCF's frames! And/or:

frames!

**radius puzzle'**

182 Selected Topics in Applications of Quantum Mechanics

Thus far, we've been able to demonstrate that the CUFT may be considered a satisfactory 'TOE' capable of resolving all major quantum-relativistic theoretical inconsistencies, replicating their primary empirical phenomena, identifying and empirically validating one of the CUFT's 'differential-critical' predictions differentiating it from both quantum and relativistic predic‐ tions... The primary aim of the current manuscript is to utilize this recognition of the CUFT as a satisfactory TOE, e.g., which also embeds both quantum and relativistic models within its broader more comprehensive (singular) 'Universal Computational Principle' theoretical framework *– towards recognizing the need to revise certain key theoretical aspects of both quantum and relativistic fields, i.e., based on the CUFT's singularity of the UCP sole production of the (extremely rapid: c*<sup>2</sup> /h) *series of the 'Universal Simultaneous Computational Frames' (USCF's); Specifically, the CUFT's emphasis on the singularity of the UCP (rapid) production of the USCF's series – forces us to revise both Quantum and Relativistic "materialistic-reductionistic" basic assumption whereby any (quantum or relativistic) physical relationship (or entities, value/s, phenomenon) can be determined solely based on an exhaustive probe-target (subatomic) interaction or observer-phenomenon (e.g., spacetime or energy-mass) relationship in such a manner as to point at the sole and singular production of all such quantum and relativistic entities, phenomena, relationship/s by the UCP's USCF's production...*

Indeed, if we revert back to the CUFT's (first) 'Duality Principle' theoretical postulate, we can see that both Quantum and Relativistic computational systems comprise a '*Self-Referential Ontological Computational Systems' (SROCS)* structure; this quantum and relativistic SROCS computational structure is synonymous with a **"materialistic-reductionistic" assumption**, wherein it is assumed that the determination of the "existence"/"non-existence" of any given 'y' entity (or value) is determined solely based on its direct (or indirect) physical interaction with another (exhaustive) 'x' factor/s... As we've seen, the Duality Principle in fact negates the validity of such assumed (quantum or relativistic) SROCS systems – instead, pointing at the existence of the singular higher-ordered (D2) 'Universal Computational Principle' (UCP) which alone computes the "simultaneous co-occurrence" of all (exhaustive) quantum and relativistic 'probe-target' and 'observer-phenomenon' pairs series (e.g., subsequently shown by the CUFT to comprise any minimal time-point 'Universal Simultaneous Computational Frame'). The CUFT further developed this 'Duality Principle' and 'UCP' (alongside its three 'Computational Dimensions') postulates towards the recognition of the *'Computational Invariance Principle'*: i.e., recognizing the fact that since only the UCP "exists" both 'during' each of the USCF's frames (in fact producing all of its exhaustive universal spatial pixels simultaneously at any such minimal time-point) as well as 'solely existing' "in-between" any two subsequent USCF frames (whereas the four secondary computational 'physical' features of 'space', 'time', 'energy' and 'mass' only exist "during" the UCP's production of the USCF's and its computation of these four secondary computational physical features), then we must conclude that only this singular UCP comprises an invariant "reality" (whereas these four secondary-computational 'physical' features may only be considered *'phenomenally' variant*)... Hence, the CUFT's 'Computational Invariance Principle' in fact points at the sole reality of the UCP (e.g., computationally invariant), as opposed to the "phenomenal" nature of the four secondary computational 'physical' features of 'space', 'time', 'energy' and 'mass' (e.g., computationally variant).

It is hereby suggested that a deeper analysis of these three particular theoretical postulates of the CUFT (e.g., the 'Duality Principle', 'Universal Computational Principle' and 'Computa‐ tional Invariance Principle') may negate the current (quantum and relativistic) *"materialisticreductionistic assumption"* (e.g., represented by the SROCS computational structure) based on the sole and singular reality of the 'Universal Computational Principle'; This is made partic‐ ularly clear based on (above mentioned) 'Computational Invariance' Principle's proof for the singular reality of the Universal Computational Principle – which is the only 'computationally invariant' element which "exists" both during its sole production the (rapid series of) USCF's frames, and "in-between" any two such (subsequent) USCF's frames! This is because based on this 'Computational Invariance Principle', the four physical features of 'space', 'time', 'energy' and 'mass' constitute computationally 'variant' properties and are therefore 'transient' (i.e., exist only "during" the Universal Computational Principle's production of the USCF's frames but ceases to exist "in-between" any two such USCF's frames)... Indeed, their "computational variant" composition makes them possess only "phenomenal" validity as opposed to the singular reality of the 'Universal Computational Principle' which exists permanently (and solely) – both as producing these 'phenomenal' (computationally variant) four 'physical' features "during" the USCF's frames and also "in-between" these USCF's frames... Therefore, the sole production- sustenance- and "transference" of any of these four "physical" features – during or across – any USCF frame/s is only made possible through the singular existence (and operation) of the 'Universal Computational Principle'! In other words, since these four 'phenomenal-physical' features "exist" only during each USCF frame, e.g., as produced by the singular 'Universal Computational Principle', but not "in-between" any two such subsequent USCF's frames – as opposed to the singularity of the UCP which solely produces these four physical features "during" the USCF's frames and also exists "in-between" the USCF's frames; then, we must conclude that the only means for the "production"- "sustenance"- or "trans‐ ference" of any given 'physical' feature across any two (subsequent) USCF's frames may only be done based on the UCP!

Hence, we must conclude that the basic assumption of "materialistic-reductionism", e.g., whereby it is possible to determine the "existence" or "non-existence" of any given 'physical' 'y' feature solely based on its direct or indirect physical interaction with another (exhaustive) 'x' factor/s – is negated (not only by the above mentioned Duality Principle) but even more explicitly through the recognition that it is not possible for any of these four (phenomenal) 'physical' features to be "transferred" across any two subsequent USCF's frames – except through the computation of the Universal Computational Principle, which constitutes the sole (computationally invariant) "reality" (which exists both "during" the USCF frames producing these four phenomena 'physical' features and solely exists "in-between" any two USCF's frames...) What this means is that the basic "materialistic-reductionistic" assumption under‐ lying both quantum and relativistic SROCS computational systems, i.e., which assumes that the "existence" (or "non-existence") of any given subatomic 'target' or relativistic (space-time or energy-mass) -'phenomenon' is determined solely based on their direct or indirect physical interaction/s with another (exhaustive) subatomic 'probe' element or relativistic 'observer' – is negated! Instead, the 'production'- 'sustenance'- or 'development'- of any 'physical' feature (relationship or phenomenon) – at the quantum or relativistic frameworks can only be computed through the singularity of the Universal Computational Principle!

#### **6. Revising physics: UCP a-causal computation**

of 'space', 'time', 'energy' and 'mass' only exist "during" the UCP's production of the USCF's and its computation of these four secondary computational physical features), then we must conclude that only this singular UCP comprises an invariant "reality" (whereas these four secondary-computational 'physical' features may only be considered *'phenomenally' variant*)... Hence, the CUFT's 'Computational Invariance Principle' in fact points at the sole reality of the UCP (e.g., computationally invariant), as opposed to the "phenomenal" nature of the four secondary computational 'physical' features of 'space', 'time', 'energy' and 'mass' (e.g.,

It is hereby suggested that a deeper analysis of these three particular theoretical postulates of the CUFT (e.g., the 'Duality Principle', 'Universal Computational Principle' and 'Computa‐ tional Invariance Principle') may negate the current (quantum and relativistic) *"materialisticreductionistic assumption"* (e.g., represented by the SROCS computational structure) based on the sole and singular reality of the 'Universal Computational Principle'; This is made partic‐ ularly clear based on (above mentioned) 'Computational Invariance' Principle's proof for the singular reality of the Universal Computational Principle – which is the only 'computationally invariant' element which "exists" both during its sole production the (rapid series of) USCF's frames, and "in-between" any two such (subsequent) USCF's frames! This is because based on this 'Computational Invariance Principle', the four physical features of 'space', 'time', 'energy' and 'mass' constitute computationally 'variant' properties and are therefore 'transient' (i.e., exist only "during" the Universal Computational Principle's production of the USCF's frames but ceases to exist "in-between" any two such USCF's frames)... Indeed, their "computational variant" composition makes them possess only "phenomenal" validity as opposed to the singular reality of the 'Universal Computational Principle' which exists permanently (and solely) – both as producing these 'phenomenal' (computationally variant) four 'physical' features "during" the USCF's frames and also "in-between" these USCF's frames... Therefore, the sole production- sustenance- and "transference" of any of these four "physical" features – during or across – any USCF frame/s is only made possible through the singular existence (and operation) of the 'Universal Computational Principle'! In other words, since these four 'phenomenal-physical' features "exist" only during each USCF frame, e.g., as produced by the singular 'Universal Computational Principle', but not "in-between" any two such subsequent USCF's frames – as opposed to the singularity of the UCP which solely produces these four physical features "during" the USCF's frames and also exists "in-between" the USCF's frames; then, we must conclude that the only means for the "production"- "sustenance"- or "trans‐ ference" of any given 'physical' feature across any two (subsequent) USCF's frames may only

Hence, we must conclude that the basic assumption of "materialistic-reductionism", e.g., whereby it is possible to determine the "existence" or "non-existence" of any given 'physical' 'y' feature solely based on its direct or indirect physical interaction with another (exhaustive) 'x' factor/s – is negated (not only by the above mentioned Duality Principle) but even more explicitly through the recognition that it is not possible for any of these four (phenomenal) 'physical' features to be "transferred" across any two subsequent USCF's frames – except through the computation of the Universal Computational Principle, which constitutes the sole

computationally variant).

184 Selected Topics in Applications of Quantum Mechanics

be done based on the UCP!

Hence, there seems to arise a necessity to revise both quantum and relativistic computational systems such that the 'existence' of any of the four (computationally variant phenomenal) physical features (of 'space', 'time', 'energy' or 'mass') in either quantum or relativistic theoretical frameworks be solely produced- sustained- or developed- solely based on the Universal Computational Principle's singular production of all spatial pixels in the universe at any minimal USCF frame/s time-point; Note, however, that this revision does not represent merely a 'philosophical' concept – i.e., in fact, it is suggested that this revision signifies a fundamental shift in Physics as it relies on the **UCP** singular (higher-ordered) **"A-Causal Computation"!**

In order to fully grasp the potential significance of this (novel) UCP 'A-Causal Computation', it may be helpful to specify the theoretical ramifications of recognizing the fact that in both Quantum Mechanics and Relativity Theory the sole production- sustenance- and develop‐ ment- of any of the four 'physical' (phenomenal) features can only be computed by the UCP; Given the fact that the UCP is postulated to compute *"simultaneous co-occurrence"* of all (exhaustive) quantum 'probe-target' and relativistic 'observer-phenomenon' relationships, this means that both within a *single USCF* frame and across a *series* of such *USCF's* frames – we cannot (any longer) rely on any 'materialistic-reductionistic' (SROCS) subatomic 'probetarget' or relativistic 'observer-phenomenon' interactions for determining any of the four (quantum or relativistic) 'physical' features... Instead, we must revise any such quantum or relativistic 'physical' feature based on the UCP computation of the "simultaneous co-occur‐ rence" of all (exhaustive) quantum and relativistic relationships comprising any (single or multiple) USCF frame/s! Indeed, this fundamental shift from the current "materialisticreductionistic" quantum or relativistic (SROCS) assumption towards a recognition of the sole computation of the UCP of all 'simultaneously co-occurring' quantum and relativistic ('probetarget' and 'observer-phenomenon') interactions is termed: the 'UCP A-Causal Computation' (i.e., of all 'simultaneously co-occurring quantum and relativistic relationships comprising any single or multiple USCF's)...

Now, in order to understand the far reaching theoretical implications of recognizing this UCP (singular) higher-ordered 'A-Causal Computation' let us turn our attention to the two (overarching) conceptual models of the "probabilistic interpretation of Quantum Mechanics" (i.e., represented by the 'probability wave function' and its "collapse" following any given subatomic probe measurement) and Einstein's (famous) General Relativity Einstein field equations (EFE): *Rμ<sup>ν</sup>* <sup>−</sup> <sup>1</sup> <sup>2</sup> *gμ<sup>ν</sup> <sup>R</sup>* <sup>+</sup> *gμ<sup>ν</sup> <sup>Λ</sup>* <sup>=</sup> <sup>8</sup>*π<sup>G</sup> <sup>c</sup>* <sup>4</sup> *Tμ<sup>ν</sup>* (describing the dynamic interaction that exists between massive object's curvature of the fabric of 'space-time' which in return determines their travelling pathway, and vice versa...) It is suggested that in both of these cases, their current theoretical formulation represents the above mentioned "materialistic-reductionistic" assumption – i.e., whereby it is the direct or indirect physical interaction/s between a given subatomic 'probe' and 'probability wave function' target element which determines the "collapse" of that wave function and hence the value of the measured 'target particle; or it is the direct physical interaction of a given (massive) object with the 'space-time' which deter‐ mines its curvature – and this curvature of 'space-time' (in return) interacts with this given (massive) object thereby determining its pathway movement...

Indeed, the 'materialistic-reductionistic' structure of any such (hypothetical) quantum 'probetarget' or relativistic 'observer-phenomenon' relationships was analyzed earlier, and proven to comprise a SROCS computational structure, e.g., being negated by the CUFT 'Duality Principle' – pointing at the necessity to reformulate both quantum and relativistic computa‐ tional systems based on the singularity of the Universal Computational Principle...

But, what becomes apparent here, e.g., based on the recognition of the UCP "A-Causal Computation" is that the basic (overarching) theoretical model of both Quantum Mechanics and (General) Relativity Theory must be revised based on this 'UCP A-Causal Computation'! This is because once we accept the CUFT assertion that there exists only one singular (com‐ putationally invariant) UCP "reality" and that this UCP singular 'reality' solely computes the "simultaneous co-occurrence" of all exhaustive (quantum and relativistic) relationships comprising a (minimal time-point) USCF frame/s (e.g., termed: 'UCP A-Causal Computation'), then we cannot (any longer) retain either QM's current model regarding the 'materisalisticreductionistic collapse of the probability wave function' or General Relativity's EFE. This is because both QM's assumed (SROCS) collapse of the probability wave function, as well as RT's assumed SROCS (massive) object –space-time curvature computational structure – are based on this "materialistic-reductionistic SROCS' assumption; In the case of QM this 'materialisticreductionistic' SROCS computational structure is represented in the assumption wherein the determination of the values of any subatomic 'target' (probability wave function) element is contingent upon its direct (or indirect) physical interaction with another subatomic 'probe' element – i.e., which "causes" the "collapse" of the probability wave function (see earlier description of the Quantum 'probe-target' SROCS structure and its violation of the Duality Principle); In the case of (General) Relativity Theory, this 'materialistic-reductionistic SROCS assumption' is represented by through Relativity's EFE which determines the 'curvature of space-time' based on its direct physical interaction with 'massive objects' and vice versa determines the movement of these 'massive objects' based on their interaction with the curvature of 'space-time' (also see the earlier Duality Principle's analysis of relativistic 'observer-phenomenon' SROCS computational structure);

However, based on the above recognition of the singular reality of the UCP which solely computes the "simultaneous co-occurrence" of all (exhaustive) quantum and relativistic interactions, i.e., comprising any single or multiple USCF's frames (e.g., termed: the UCP 'A-Causal Computation), we must revise this basic 'materialistic-reductionistic SROCS assump‐ tion' based on this higher-ordered singular UCP A Causal Computation! In other words, since the sole production- sustenance- and development- of any of the four (phenomenal) 'physical' features (e.g., say in the quantum domain) is based on the UCP 'A-Causal Computation', i.e., of all exhaustive 'probe-probability wave function target' interactions comprising a single (or multiple) USCF frame/s, then we must also conclude that the apparent "collapse" of the probability wave function – cannot be "caused" by the direct interaction between any given probe element and given probability wave function! This is simply due to the fact that according to the above 'UCP A-Causal Computation' all exhaustive values of all quantum subatomic 'probe-probability wave function target' interactions are *computed simultaneously by the UCP – comprising all spatial pixels comprising any single or multiple USCF frame/s...* And since the only "transference" of any of the four (phenomenal) 'physical' features from one USCF frame to another – can only be carried out through the singular operation of the UCP's production- sustenance- and development- of any spatial pixel in te universe (across USCF's frames), then we cannot attribute the "collapse of the wave function" to any physical interac‐ tion taking place between any 'probe' and 'probability wave function target' entities (e.g., at any particular USCF frame/s)... Hence, the UCP's A-Causal Computation which produces simultaneously all exhaustive quantum 'probe-target' interactions – at any single or multiple USCF frame/s negates the validity of the "materialistic-reductionistic SROCS" assumption of the "collapse of the probability wave function" (target element) as a result of its direct interaction with another subatomic 'probe' element.

(overarching) conceptual models of the "probabilistic interpretation of Quantum Mechanics" (i.e., represented by the 'probability wave function' and its "collapse" following any given subatomic probe measurement) and Einstein's (famous) General Relativity Einstein field

between massive object's curvature of the fabric of 'space-time' which in return determines their travelling pathway, and vice versa...) It is suggested that in both of these cases, their current theoretical formulation represents the above mentioned "materialistic-reductionistic" assumption – i.e., whereby it is the direct or indirect physical interaction/s between a given subatomic 'probe' and 'probability wave function' target element which determines the "collapse" of that wave function and hence the value of the measured 'target particle; or it is the direct physical interaction of a given (massive) object with the 'space-time' which deter‐ mines its curvature – and this curvature of 'space-time' (in return) interacts with this given

Indeed, the 'materialistic-reductionistic' structure of any such (hypothetical) quantum 'probetarget' or relativistic 'observer-phenomenon' relationships was analyzed earlier, and proven to comprise a SROCS computational structure, e.g., being negated by the CUFT 'Duality Principle' – pointing at the necessity to reformulate both quantum and relativistic computa‐

But, what becomes apparent here, e.g., based on the recognition of the UCP "A-Causal Computation" is that the basic (overarching) theoretical model of both Quantum Mechanics and (General) Relativity Theory must be revised based on this 'UCP A-Causal Computation'! This is because once we accept the CUFT assertion that there exists only one singular (com‐ putationally invariant) UCP "reality" and that this UCP singular 'reality' solely computes the "simultaneous co-occurrence" of all exhaustive (quantum and relativistic) relationships comprising a (minimal time-point) USCF frame/s (e.g., termed: 'UCP A-Causal Computation'), then we cannot (any longer) retain either QM's current model regarding the 'materisalisticreductionistic collapse of the probability wave function' or General Relativity's EFE. This is because both QM's assumed (SROCS) collapse of the probability wave function, as well as RT's assumed SROCS (massive) object –space-time curvature computational structure – are based on this "materialistic-reductionistic SROCS' assumption; In the case of QM this 'materialisticreductionistic' SROCS computational structure is represented in the assumption wherein the determination of the values of any subatomic 'target' (probability wave function) element is contingent upon its direct (or indirect) physical interaction with another subatomic 'probe' element – i.e., which "causes" the "collapse" of the probability wave function (see earlier description of the Quantum 'probe-target' SROCS structure and its violation of the Duality Principle); In the case of (General) Relativity Theory, this 'materialistic-reductionistic SROCS assumption' is represented by through Relativity's EFE which determines the 'curvature of space-time' based on its direct physical interaction with 'massive objects' and vice versa determines the movement of these 'massive objects' based on their interaction with the curvature of 'space-time' (also see the earlier Duality Principle's analysis of relativistic

tional systems based on the singularity of the Universal Computational Principle...

*<sup>c</sup>* <sup>4</sup> *Tμ<sup>ν</sup>* (describing the dynamic interaction that exists

<sup>2</sup> *gμ<sup>ν</sup> <sup>R</sup>* <sup>+</sup> *gμ<sup>ν</sup> <sup>Λ</sup>* <sup>=</sup> <sup>8</sup>*π<sup>G</sup>*

(massive) object thereby determining its pathway movement...

'observer-phenomenon' SROCS computational structure);

equations (EFE): *Rμ<sup>ν</sup>* <sup>−</sup> <sup>1</sup>

186 Selected Topics in Applications of Quantum Mechanics

Likewise, based on the recognition of the UCP's singular 'A-Causal Computation' which is solely responsible for the production- sustenance- and development- of all exhaustive 'observer – (space-time, energy-mass) phenomenon' interactions comprising any (single or multiple) USCF frame/s, we must revise the current EFE representing a "materialisticreductionistic SROCS assumption"; Once again, this is due to the fact that contrary to this 'materialistic-reductionistic SROCS' assumption represented by Relativity Theory EFE according to the UCP's singular 'A-Causal Computation' the UCP computes the "simulta‐ neous co-occurrence" of all exhaustive relativistic 'observer – (space-time or energy-mass) phenomenon' relationships comprising any (single or multiple) USCF frame/s! Therefore, contrary to (General) Relativity Theory's currently assumed 'materialistic-reductionistic SROCS' assumption, wherein the 'curvature of space-time' is determined through its direct physical interaction with 'massive objects' (and vice versa, the movement of these 'mas‐ sive objects' is determined strictly based on the 'curvature of space-time') – the UCP's A-Causal Computation asserts that it is solely the singularity of the UCP which computes the "simultaneous co-occurrences" of all exhaustive relativistic 'observer – (space-time or energy-mass) phenomenon' interactions (e.g., comprising any single or multiple USCF/s)...

Hence, the fundamental necessary revision of both (contemporary) probabilistic interpretation of QM and of (General) Relativity Theory involves a shift from the current 'materialistic-

reductionistic SROCS' assumption underlying both QM and RT – towards the UCP's singular 'A-Causal Computation'! Essentially, this revision implies that instead of the currently assumed (quantum or relativistic) 'materialistic-reductionistic SROCS' assumption wherein the "collapse of the target probability wave function' is "caused" by its direct physical interaction with the subatomic 'probe' element, and the 'curvature of space-time' is "caused" by its direct physical interaction with 'massive object/s' (and vice versa the movement of these 'massive object/s' is "caused" by the 'curvature of 'space-time'; the UCP 'A-Causal Computa‐ tion' negates any such 'materialistic-reductionistic' "causal" relationships, instead pointing at the fact that it is only the singularity of the UCP which computes the "simultaneous cooccurrence" of all (exhaustive) quantum 'probe-target' and relativistic 'observer-phenomenon' interactions comprising any (single or multiple) USCF frame/s... Perhaps another (lucid) manner of demonstrating the UCP's negation of contemporary Quantum and Relativistic 'materialistic-reductionistic SROCS' computational structure can be given through an analysis of the minimal-temporal, i.e., USCF's frames dynamics representing Quantum Mechanics' currently assumed "collapse of the target probability wave function", as well of the USCF's frames' dynamics representing Relativity's EFE (e.g., describing the interactive effect of 'massive objects' on the 'curvature of space-time' and vice versa as explained above); Accord‐ ing to the contemporary 'probabilistic interpretation of QM' the target's probability wave function "collapses" as a result of its direct physical interaction with another subatomic probe element: this means that at a particular minimal-time USCF frame there occurs a direct physical interaction between the 'target's probability wave function' and the 'probe element' – and that based on this direct 'probe-target probability wave function' physical interaction this 'target probability wave function' "collapses"; But since each (single) USCF frame comprises the "minimal time-point" (possible) at which the (singular) UCP produces all exhaustive (quan‐ tum and relativistic) spatial-pixels in the universe, then necessarily the initial direct physical interaction between the 'probe' element and the 'target's probability wave function' – takes place at a given USCF frame, whereas the (assumed) "resulting collapse" of this 'target's probability wave function' must occur at a subsequent USCF frame! But, since according to the above mentioned 'Computational Invariance Principle' the sole and singular (computa‐ tionally invariant) principle which exists both "during" and "in-between" any two subsequent USCF frames is the UCP (whereas the four phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' exist only "during" any given USCF frame), then the sole 'production' sustenance- and 'development' of any quantum (or relativistic) (phenomenal) 'physical' feature may only be carried out by the singularity of the UCP – which indeed computes the "simultaneous co-occurrence" of all exhaustive quantum (or relativistic) 'probe-target' interactions comprising any single or multiple USCF's... Hence, since it is not possible for any physical interaction (say) between the 'subatomic probe' and (assumed) 'target's probability wave function' at a given USCF frame (i) to have any effect on their (phenomenal) 'physical' features at a subsequent USCF frame (i+1) – but rather it is the sole (and singular) computation of the UCP of the "simultaneous co-occurrences" of any exhaustive 'probe-target' subatomic interaction/s (at any single or multiple USCF/s frames) which produces- sustains- and develops- any exhaustive subatomic probe-target relationships! Likewise, we can show that based on such 'minimal time-point' USCF's frames analysis, that the currently assumed 'materialistic-reductionistic SROCS' General Relativity Theory model's interactive 'curvature of space-time' based on its direct physical interaction with 'massive objects' (and vice versa) is negated – instead, pointing at the sole production- sustenance- and development- of any of the four phenomenal 'physical' features (of 'space', 'time', 'energy' and 'mass') including the phenomena of the curvature of 'space-time' or of the (apparent) movement of massive objects solely based on the singular UCP 'A-Causal Computation'; In order to demonstrate the impossibility of the currently assumed Relativistic 'materialistic-reductionistic SROCS' assumption, let us (once again) imagine the 'minimal time-point dynamics' of the currently assumed (General) Relativity 'materialistic-reductionistic SROCS' direct physical interaction between certain massive object/s and the curvature of space time (and vice versa, as outlined above): According to this 'materialistic-reductionistic SROCS' Relativistic assumption, it is the direct physical interaction that exists between (one or more) 'massive object/s' and 'space-time' which "causes" this 'space-time' (fabric) to curve, and vice versa, this 'curvature of space-time' "causes" any given 'massive object/s' to travel in a particular (curved) space-time pathway... But, when analyzed from the perspective of a 'minimal time-point' USCF frame/s, then we see that in the first (hypothetical) USCF frame, there is a direct physical interaction between the given 'massive object' and the (fabric of) 'space-time', whereas this 'curving of space-time' may affect the space-time movement pathway of this 'massive object' – only in a subsequent USCF frame/s... However, such 'materialistic-reductionistic' (SROCS) relativistic assumption – not only violates the Duality Principle (as shown earlier and previously), but in the context of this 'minimal time-point' USCF's frames analysis seems to negate the (proven) singularity of the UCP's 'A-Causal Computation': This is because, according to the above 'materialisticreductionisitc' relativistic assumption it is the direct physical interaction between the given 'massive object' and the (fabric of) 'space-time' – in the first USCF frame/s which "causes" this (fabric of 'space-time') to curve in subsequent USCF frame/s, and this 'curvature of space-time' in turn "causes" this given "massive object" to travel in a "curved space-time pathway" in still later USCF frame/s... However, based on the UCP's 'A-Causal Computation', which computes the "simultaneous co-occurrences" of all exhaustive relativistic 'observer-phenomenon' relationships, we have to negate this relativistic 'materialistic-reductionistic SROCS' assump‐ tion – since there cannot exist any "cause and effect" relationship between any direct (or indirect) physical interactions (e.g., such as 'massive object/s' which 'curve the fabric of spacetime') at an initial USCF frame/s, and its effect on the 'space-time movement pathway of that massive object' at a subsequent USCF/s... Instead, the singularity of UCP's 'A-Causal Com‐ putation' forces us to recognize its computation of the 'simultaneous co-occurrences' of all (four) phenomenal 'physical' features – i.e., including the simultaneous computation of any 'massive object' and any 'curvature of space-time'!

reductionistic SROCS' assumption underlying both QM and RT – towards the UCP's singular 'A-Causal Computation'! Essentially, this revision implies that instead of the currently assumed (quantum or relativistic) 'materialistic-reductionistic SROCS' assumption wherein the "collapse of the target probability wave function' is "caused" by its direct physical interaction with the subatomic 'probe' element, and the 'curvature of space-time' is "caused" by its direct physical interaction with 'massive object/s' (and vice versa the movement of these 'massive object/s' is "caused" by the 'curvature of 'space-time'; the UCP 'A-Causal Computa‐ tion' negates any such 'materialistic-reductionistic' "causal" relationships, instead pointing at the fact that it is only the singularity of the UCP which computes the "simultaneous cooccurrence" of all (exhaustive) quantum 'probe-target' and relativistic 'observer-phenomenon' interactions comprising any (single or multiple) USCF frame/s... Perhaps another (lucid) manner of demonstrating the UCP's negation of contemporary Quantum and Relativistic 'materialistic-reductionistic SROCS' computational structure can be given through an analysis of the minimal-temporal, i.e., USCF's frames dynamics representing Quantum Mechanics' currently assumed "collapse of the target probability wave function", as well of the USCF's frames' dynamics representing Relativity's EFE (e.g., describing the interactive effect of 'massive objects' on the 'curvature of space-time' and vice versa as explained above); Accord‐ ing to the contemporary 'probabilistic interpretation of QM' the target's probability wave function "collapses" as a result of its direct physical interaction with another subatomic probe element: this means that at a particular minimal-time USCF frame there occurs a direct physical interaction between the 'target's probability wave function' and the 'probe element' – and that based on this direct 'probe-target probability wave function' physical interaction this 'target probability wave function' "collapses"; But since each (single) USCF frame comprises the "minimal time-point" (possible) at which the (singular) UCP produces all exhaustive (quan‐ tum and relativistic) spatial-pixels in the universe, then necessarily the initial direct physical interaction between the 'probe' element and the 'target's probability wave function' – takes place at a given USCF frame, whereas the (assumed) "resulting collapse" of this 'target's probability wave function' must occur at a subsequent USCF frame! But, since according to the above mentioned 'Computational Invariance Principle' the sole and singular (computa‐ tionally invariant) principle which exists both "during" and "in-between" any two subsequent USCF frames is the UCP (whereas the four phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' exist only "during" any given USCF frame), then the sole 'production' sustenance- and 'development' of any quantum (or relativistic) (phenomenal) 'physical' feature may only be carried out by the singularity of the UCP – which indeed computes the "simultaneous co-occurrence" of all exhaustive quantum (or relativistic) 'probe-target' interactions comprising any single or multiple USCF's... Hence, since it is not possible for any physical interaction (say) between the 'subatomic probe' and (assumed) 'target's probability wave function' at a given USCF frame (i) to have any effect on their (phenomenal) 'physical' features at a subsequent USCF frame (i+1) – but rather it is the sole (and singular) computation of the UCP of the "simultaneous co-occurrences" of any exhaustive 'probe-target' subatomic interaction/s (at any single or multiple USCF/s frames) which produces- sustains- and develops- any exhaustive subatomic probe-target relationships! Likewise, we can show that based on such 'minimal time-point' USCF's frames analysis, that the currently assumed

188 Selected Topics in Applications of Quantum Mechanics

Hence, we reach the inevitable conclusion whereby both Quantum and Relativistic models have to be revised in terms of their basic 'materialistic-reductionistic SROCS' assumption, i.e., recognizing the fact that either Quantum Mechanics' assumed 'collapse of target's probability wave function' as 'caused' by its direct physical interaction with another subatomic 'probe' element; or Relativity's assumed 'curvature of space-time' as "caused" by its direct physical interaction with 'massive objects' – is negated by the CUFT's recognition of the singularity of the Universal Computational Principle's (UCP) (higher-ordered) 'A-Causal Computation', which computes the 'simultaneous co-occurrences' of all quantum and relativistic (exhaustive) 'probe-target' and 'observer-phenomenon' interactions. The key revision brought about by the UCP (higher-ordered) 'A-Causal Computation' is that it negates, i.e., in principle, the existence of any 'materialistic-reductionistic SROCS "causal" relationships in QM or Relativity Theory; this is because once we accept the UCP's (higher-ordered) 'A-Causal Computation' as the sole and singular source for producing- sustaining- and evolving- any of the four phenomenal 'physical' features (of 'space', 'time', 'energy' and 'mass'), e.g., across all exhaustive spatialpixels (in the universe) comprising any single or multiple USCF frame/s (at the minimal USCF time-point), then we must reject any 'materialistic-reductionistic SROCS' physical relation‐ ship/s between any hypothetical quantum 'probe' and 'target' or between any relativistic 'observer' and 'phenomenon' entities as "causing" any hypothetical change or effect in that subatomic target (e.g., such as the assumed "collapse of the target's probability wave function" as "caused" by its interaction with the subatomic probe element) or as "causing" any effect in the given relativistic 'phenomenon' (e.g., such as in Relativity's assumed "curvature of spacetime" as "caused" by its interaction with 'massive objects'). Thus, the necessary revision in both QM and RT brought about by the UCP's 'A-Causal Computation' is to base all 'phenom‐ enal' (quantum or relativistic) physical features of 'space', 'time', 'energy' and 'mass' on the singularity of the UCP which solely produces- sustains- or evolves- all of these physical features at all spatial-pixels in the universe comprising any (single or multiple) exhaustive USCF's frame/s...

#### **7. The CUFT's embedding & transcendence of QM and RT models**

It should, nevertheless, be made clear that this necessary revision of both Quantum Mechanical and Relativistic Models – does not negate any of the validated empirical phenomena or known quantum or relativistic laws and relationships, but rather broadens our theoretical under‐ standing of these quantum and relativistic phenomena, as embedded in- and (indeed) transcended by- the CUFT theoretical framework; This is due to the fact that whereas Relativity Theory may represent the characterization of single spatial-temporal (relativistic) objects and phenomena, and Quantum Mechanics represents multi spatial-temporal 'probability wave function' (subatomic) entities (e.g., which also embeds 'single' multi spatial-temporal 'particle' elements) – the CUFT expands the theoretical framework to include all single- multiple- and indeed *exhaustive*- spatial-pixels comprising any minimal time-point USCF frame/s... By doing so, and based on the CUFT's identification of this minimal time-point (extremely rapid-series: 'c2 /h') series of USCF's produced solely by the singular 'Universal Computational Principle', the CUFT is capable of fully integrating between quantum and relativistic components and phenomena – which is made most apparent in the CUFT's Universal Computational Formula (e.g. that fully integrates between quantum and relativistic relationships, as well as between the four basic 'physical' features of 'space', 'time', 'energy' and 'mass'):

$$\frac{\text{"{e}"}}{\text{h}} = \frac{\text{s}}{\text{t}} \times \frac{\text{e}}{\text{m}}$$

This embedding- and transcendence- of both quantum and relativistic phenomena and relationships within the broader (higher-ordered) CUFT is made most apparent in the two (above mentioned) Relativistic and Quantum formats, which include the known relativistic 'energy and mass equivalence' (E = Mc2 ) and the quantum 'complimentary pairs';

$$\text{I. Relativistic Format: e} \times \frac{\text{s}}{\text{t}} \text{=m} \times \frac{\text{c}^2}{\text{h}}$$

which computes the 'simultaneous co-occurrences' of all quantum and relativistic (exhaustive) 'probe-target' and 'observer-phenomenon' interactions. The key revision brought about by the UCP (higher-ordered) 'A-Causal Computation' is that it negates, i.e., in principle, the existence of any 'materialistic-reductionistic SROCS "causal" relationships in QM or Relativity Theory; this is because once we accept the UCP's (higher-ordered) 'A-Causal Computation' as the sole and singular source for producing- sustaining- and evolving- any of the four phenomenal 'physical' features (of 'space', 'time', 'energy' and 'mass'), e.g., across all exhaustive spatialpixels (in the universe) comprising any single or multiple USCF frame/s (at the minimal USCF time-point), then we must reject any 'materialistic-reductionistic SROCS' physical relation‐ ship/s between any hypothetical quantum 'probe' and 'target' or between any relativistic 'observer' and 'phenomenon' entities as "causing" any hypothetical change or effect in that subatomic target (e.g., such as the assumed "collapse of the target's probability wave function" as "caused" by its interaction with the subatomic probe element) or as "causing" any effect in the given relativistic 'phenomenon' (e.g., such as in Relativity's assumed "curvature of spacetime" as "caused" by its interaction with 'massive objects'). Thus, the necessary revision in both QM and RT brought about by the UCP's 'A-Causal Computation' is to base all 'phenom‐ enal' (quantum or relativistic) physical features of 'space', 'time', 'energy' and 'mass' on the singularity of the UCP which solely produces- sustains- or evolves- all of these physical features at all spatial-pixels in the universe comprising any (single or multiple) exhaustive

**7. The CUFT's embedding & transcendence of QM and RT models**

It should, nevertheless, be made clear that this necessary revision of both Quantum Mechanical and Relativistic Models – does not negate any of the validated empirical phenomena or known quantum or relativistic laws and relationships, but rather broadens our theoretical under‐ standing of these quantum and relativistic phenomena, as embedded in- and (indeed) transcended by- the CUFT theoretical framework; This is due to the fact that whereas Relativity Theory may represent the characterization of single spatial-temporal (relativistic) objects and phenomena, and Quantum Mechanics represents multi spatial-temporal 'probability wave function' (subatomic) entities (e.g., which also embeds 'single' multi spatial-temporal 'particle' elements) – the CUFT expands the theoretical framework to include all single- multiple- and indeed *exhaustive*- spatial-pixels comprising any minimal time-point USCF frame/s... By doing so, and based on the CUFT's identification of this minimal time-point (extremely rapid-series:

/h') series of USCF's produced solely by the singular 'Universal Computational Principle', the CUFT is capable of fully integrating between quantum and relativistic components and phenomena – which is made most apparent in the CUFT's Universal Computational Formula (e.g. that fully integrates between quantum and relativistic relationships, as well as between

<sup>2</sup> `c se = × h tm

the four basic 'physical' features of 'space', 'time', 'energy' and 'mass'):

USCF's frame/s...

190 Selected Topics in Applications of Quantum Mechanics

'c2

II. Quantum Format: t <sup>×</sup> <sup>m</sup> <sup>×</sup> <sup>c</sup><sup>2</sup> <sup>h</sup> =s <sup>×</sup> <sup>e</sup>

Note, however, that in both quantum and relativistic formats the Universal Computational Formula transcends these (known) quantum and relativistic relationships based on the incorporation of these known relationships within the broader (quantum and relativistic) formats computational structure: Specifically, it becomes apparent that these known quantum and relativistic relationships may represent "special cases" – within the broader Relativistic or Quantum Formats, which in fact represent the complete integration of both "quantum" and "relativistic" computational components within the singular higher-ordered (fully integrated) CUFT's Universal Computational Formula; *Indeed, a more comprehensive mathematical (and empirical) validation of these two Quantum and Relativistic Formats – as different from the (above‐ mentioned) known quantum and relativistic relationships constituted one of the three (abovementioned) "differential-critical predictions" differentiating the CUFT from both QM and RT* [2], *and should be further investigated and validated – i.e., both mathematically and empirically*.

On the principle theoretical level, it can be pointed out that whereas the current 'Quantum Mechanical' and 'Relativistic' models represent particular phenomena and relationships – corresponding to 'single' spatial-temporal relativistic objects (and phenomena) or subatomic 'multi spatial-temporal' "wave" (and embedded 'single spatial-temporal' "particle") elements and phenomena, the CUFT fully integrates these apparently distinct, particular phenomena within the higher-ordered series of exhaustive Universal Simultaneous Computational Frames (USCF's) which is produced by the Universal Computational Principle based on its singular 'A-Causal Computation' – which produces- sustains- and evolves- all four (phenomenal) 'physical' features of 'space', 'time', 'energy' and 'mass'; Indeed, as we've seen, this singular (higher-ordered) UCP 'A-Causal Computation' negates any "materialistic-reductionistic" (SROCS) quantum or relativistic physical relationships (e.g., such as RT's SROCS assumption regarding the 'curvature of space-time' as "caused" by 'mass', or as QM's SROCS assumed 'collapse of the target's probability wave function' as "caused" by its direct physical interaction with another 'subatomic probe element'); Instead, the singularity of this higher-ordered UCP brings about the complete integration- and indeed transcendence- of the four (quantum and relativistic) phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass'; A such, the CUFT goes beyond RT's integration of 'space-time' and 'energy-mass', and its curvature of space-time by mass (and vice versa) – by fully integrating 'space', 'time', 'energy' and 'mass' as four secondary computational (phenomenal) 'physical' features produced- sustained- and evolved- by the singular (computationally invariant) UCP... Likewise, the CUFT goes beyond QM's complimentary pairs of 'space and energy', 'time and mass' as constrained by Planck's constant ('h' simultaneous accuracy measurement) – as representing the exhaustive (compli‐ mentary) computational levels of the two UCP's Computational Dimensions of (Framework and Consistency) (as explained in [1]. Ultimately, the CUFT completely integrates the appa‐ rently "distinct" aspects of Quantum and Relativistic models as comprising integral compu‐ tational aspects of the same singular higher-ordered UCP 'A-Causal Computation' – i.e., such as the complete integration of Relativity's 'c2 ' (associated with the speed of light constraint imposed on the transmission of any signal) with Quantum's Planck's constant ('h', associated with subatomic complimentary pairs' simultaneous measurement accuracy constraint) to signify the CUFT's identified rate of UCP rapid production of the series of USCF's frames. Finally, as noted above, the CUFT's unique recognition of the singularity of the UCP's (e.g., computationally invariant) higher-ordered 'A-Causal Computation' which solely producessustains- and evolves- all four (e.g., computationally variant) phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' negates the current basic "materialistic-reductionistic" (SROCS) assumption underlying QM and RT – and forces Physics to recognize this singular UCP 'A-Causal Computation' as the sole reality giving rise to the phenomenology of all 'physical' features, including all quantum and relativistic phenomena and relationships.

### **8. The CUFT revision of 'dark energy/matter' & 'second law of thermodynamics'**

One of the initial theoretical implications of the acceptance of the CUFT as a satisfactory TOE and the acceptance of its singular (higher-ordered) UCP 'A-Causal Computation' as revising contemporary Physics' Quantum and Relativistic "materialistic-reductionistic" SROCS assumption – is its capacity to explain the unresolved "enigma" of 'Dark Matter' and 'Dark Energy' and its potential revision of the 'Second Law of Thermodynamics' (and its associated 'Arrow of Time' enigma); Essentially, the 'Dark Matter, Dark Energy' enigma constitutes the inability of contemporary Physics to account for the acceleration in the rate of expansion of the physical universe – solely based on the observed (and calculated) total mass and energy associated with all planetary object comprising this physical universe... According to these calculations roughly 70-90% of all the mass and energy in the universe is "missing", i.e., cannot be observed! Hence, the working assumption (of Contemporary Physics) is that this (70-90%) of the "missing" mass and energy in the universe is "dark", i.e., it cannot be observed empir‐ ically (for some unexplained reason)...

Interestingly, this 'Dark Matter, Dark Energy' enigma is closely connected with the above mentioned UCP 'A-Causal Computation' constraining Relativity's SROCS (interactive) determination of 'massive objects' "causing" the 'curvature of space-time' and vice versa: 'curved space-time' "causing" these 'massive objects' to travel along curved space-time pathways... As delineated above, both the CUFT's 'Duality Principle' theoretical postulate and the discovery of the UCP's singular 'A-Causal Computation' prove (unequivocally) the impossibility of any such Relativity's SROCS "materialistic-reductionistic" assumption: e.g., due to such Relativistic SROCS inevitably leading to both 'logical inconsistency' and ensuing 'computational indeterminacy' which are contradicted by Relativistic Systems empirical capacity to determine both the curvature of space-time and the movement pathways of massive objects; as well as due to the UCP's 'A-Causal Computation' "minimal time-point" USCF's analysis which indicates that based on the 'Computational Invariance Principle' proof that only the 'computationally invariant' UCP exists constantly both "during" the USCF frames and also "in-between" USCF frames, whereas the four phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' only exist "during" the USCF frames as produced by the singular UCP but cease to exist "in-between" these USCF frames – we must conclude that it is not possible for any of these four phenomenal 'physical' features to "cause" any change across USCF's frames... In other words, the discovery of the UCP singular 'A-Causal Computation' (alongside the Duality Principle) negates the basic "materialistic-reductionistic" SROCS (Relativistic) current assumption, wherein it may be possible for any direct (phenomenal) 'physical' interaction, e.g., of any of these four phenomenal 'physical' features (of 'space', 'time', 'energy' and 'mass') to "cause" any change (or effect) upon another physical attribute (e.g., such as the abovementioned 'curvature of space-time by massive objects' or its vice versa: 'curved space-time' "causing" 'massive object to move along these curved space-time path‐ ways...) Instead, the UCP asserts that the only singular 'A-Causal Computation' solely responsible for the production- sustenance- and development- of any of these four 'phenom‐ enal physical' features (e.g., at any hypothetical 'spatial pixels' comprising any single or multiple USCF's) is singularly conducted by the UCP's computation of the "simultaneous cooccurrence" of all spatial pixels comprising any such USCF.

mentary) computational levels of the two UCP's Computational Dimensions of (Framework and Consistency) (as explained in [1]. Ultimately, the CUFT completely integrates the appa‐ rently "distinct" aspects of Quantum and Relativistic models as comprising integral compu‐ tational aspects of the same singular higher-ordered UCP 'A-Causal Computation' – i.e., such

imposed on the transmission of any signal) with Quantum's Planck's constant ('h', associated with subatomic complimentary pairs' simultaneous measurement accuracy constraint) to signify the CUFT's identified rate of UCP rapid production of the series of USCF's frames. Finally, as noted above, the CUFT's unique recognition of the singularity of the UCP's (e.g., computationally invariant) higher-ordered 'A-Causal Computation' which solely producessustains- and evolves- all four (e.g., computationally variant) phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' negates the current basic "materialistic-reductionistic" (SROCS) assumption underlying QM and RT – and forces Physics to recognize this singular UCP 'A-Causal Computation' as the sole reality giving rise to the phenomenology of all 'physical' features, including all quantum and relativistic phenomena and relationships.

**8. The CUFT revision of 'dark energy/matter' & 'second law of**

One of the initial theoretical implications of the acceptance of the CUFT as a satisfactory TOE and the acceptance of its singular (higher-ordered) UCP 'A-Causal Computation' as revising contemporary Physics' Quantum and Relativistic "materialistic-reductionistic" SROCS assumption – is its capacity to explain the unresolved "enigma" of 'Dark Matter' and 'Dark Energy' and its potential revision of the 'Second Law of Thermodynamics' (and its associated 'Arrow of Time' enigma); Essentially, the 'Dark Matter, Dark Energy' enigma constitutes the inability of contemporary Physics to account for the acceleration in the rate of expansion of the physical universe – solely based on the observed (and calculated) total mass and energy associated with all planetary object comprising this physical universe... According to these calculations roughly 70-90% of all the mass and energy in the universe is "missing", i.e., cannot be observed! Hence, the working assumption (of Contemporary Physics) is that this (70-90%) of the "missing" mass and energy in the universe is "dark", i.e., it cannot be observed empir‐

Interestingly, this 'Dark Matter, Dark Energy' enigma is closely connected with the above mentioned UCP 'A-Causal Computation' constraining Relativity's SROCS (interactive) determination of 'massive objects' "causing" the 'curvature of space-time' and vice versa: 'curved space-time' "causing" these 'massive objects' to travel along curved space-time pathways... As delineated above, both the CUFT's 'Duality Principle' theoretical postulate and the discovery of the UCP's singular 'A-Causal Computation' prove (unequivocally) the impossibility of any such Relativity's SROCS "materialistic-reductionistic" assumption: e.g., due to such Relativistic SROCS inevitably leading to both 'logical inconsistency' and ensuing 'computational indeterminacy' which are contradicted by Relativistic Systems empirical capacity to determine both the curvature of space-time and the movement pathways of massive

' (associated with the speed of light constraint

as the complete integration of Relativity's 'c2

192 Selected Topics in Applications of Quantum Mechanics

**thermodynamics'**

ically (for some unexplained reason)...

Hence, the UCP's singular (proven) computation of the 'simultaneous co-occurrence' of all (four) phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' comprising all spatial pixels in the universe at any minimal time-point (single or multiple) USCF's necessarily negates the possibility of any 'materialistic-reductionistic' relationship existing between any of these four (secondary computational) phenomenal 'physical' features – i.e., including both the curvature of space time by massive objects (or vice versa) as well as the "expansion of the physical universe" – as "caused" by the phenomenal features of the amount of "mass" or "energy" comprising any single or multiple 'USCF' frame/s... In other words, based on the CUFT's proven singularity of the UCP – in producing- sustaining- and developing- all (four) phenomenal 'physical' features of 'space', 'time', energy' and 'mass' (across all exhaustive spatial pixels comprising the totality of the physical universe at any minimal time-point single or multiple USCF frame/s), the UCP's 'A-Causal Computation' is seen as solely responsible for all quantum, relativistic and CUFT known (or predicted) phenomena: This includes also the observed accelerated expansion of the physical universe – i.e., which indeed cannot be accounted for through any 'materialistic-reductionistic ' interactions between any of these secondary computational phenomenal 'physical' features. Indeed, viewed from this singular (higher-ordered) perspective of the UCP'S sole production- sustenance- and evolution- of all spatial pixels in the universe (comprising any minimal time-point single or multiple USCF frame/s) all quantum and relativistic phenomena, e.g., including the accelerated expansion of the physical universe must be accounted for solely through the UCP Causal Computation; Hence, according to the CUFT the relativistic phenomenon of the accelerated expansion of the physical universe cannot be accounted for by the currently assumed "Dark Energy and Dar Matter" – which represent a "materialistic-reductionistic" assumption (as explained in detail above), but instead must be explained as arising from the sole and singular productionsustenance- and evolution- of the physical universe by the UCP...

Another interesting potential theoretical ramification of the adoption of the CUFT as a satisfactory 'TOE' – including its discovery of the UCP singular 'A-Causal Computation', may be its potential revision of the 'Second Law of Thermodynamic' (and associated 'Arrow of Time' phenomenon). The 'Arrow of Time' enigma refers to the observation that the laws of Physics are "biased" in such a manner that events (and phenomena) always occur in a unidirectional temporal direction: thus, for instance, a glass may break into a hundred pieces – but those hundred pieces will not (of themselves) revert back to form a single unitary glass... Indeed, closely associated with this 'Arrow of Time' unidirectional temporal characteristic of physical phenomena is the (famous) 'Second Law of Thermodynamics' which states that in any given physical system the degree of entropy always increases with time... However, based on the CUFT's discovery of the singular UCP 'A-Causal Computation' – which was shown (above) to negate the basic "materialistic-reductionistic" assumption (underlying both Quantum and Relativistic models of physical reality), and one of the CUFT's (previous: [2] 'differential-critical' predictions regarding the possibility of "reversing the sequence of spatialelectromagnetic pixels" based on the application of certain electromagnetic effects, we may need to revise this Second Law of Thermodynamics (and associated 'Arrow of Time' enigma); This is because as explained earlier, none of the quantum or relativistic physical phenomena, relationships (or even laws) can continue to be based on any "materialistic-reductionistic" assumption/s. Hence, as shown above, neither the curvature of space-time by massive objects or (vice versa) the determination of the movement of massive objects based on the curvature of space-time, nor the observed accelerated expansion of the physical universe – can be explained by the current 'materialistic-reductionistic' relativistic (or quantum) assumption, but must be based on the singularity of the UCP 'A-Causal Computation'; Indeed, as we've shown (above), a fine temporal analysis of the dynamics of this singular UCP's productionsustenance- and evolution- of every (exhaustive) spatial pixel in the universe (comprising any single or multiple USCF frame/s) indicates that there cannot exist any 'materialistic-reduc‐ tionistic' effect of any of the four phenomenal 'physical' features (of 'space', 'time', 'energy' or 'mass') between any two (or more) spatial pixels, i.e., either within the same USCF frames or across different USCF frames. This is due to the UCP's singular asserted computation of the "simultaneous co-occurrence" of all spatial pixels in a given USCF frame (which prohibits any "causal" materialistic-reductionistic" effects existing between any two or more spatial pixels in the same USCF frame), as well as the UCP 's 'A-Causal Computation' associated 'Compu‐ tational Invariance Principle' which indicates that the sole and singular reality existing invariantly "during" the USCF's frame/s (e.g., as producing, sustaining and evolving any of the four phenomenal 'physical' features of all of its spatial pixels) and "in-between" these USCF's frames is the UCP. Hence, the only source for producing- sustaining- and evolvingany spatial pixel in the physical universe (e.g., at any given USCF frame/s) is the singular UCP, but not any of its (computationally variant) phenomenally produced 'physical' features...

Therefore, also the 'Second Law of Thermodynamics' which asserts the increase in entropy of any physical system with the progression of time – must be revised based on this new higherordered recognition of the singularity of the UCP 'A-Causal Computation': Hence, instead of the currently assumed 'materialsitic-reductionistic' basis for this Second Law of Thermody‐ namics, i.e., wherein it is the physical relationships that exist between a given physical system's material components which "causes" the degree of entropy in that system to necessarily increase with time, the UCP's 'A-Causal Computation' points unequivocally at the singularity of the UCP as producing- sustaining- and evolving- all spatial pixels in the universe and all associated physical phenomena and laws... Moreover, since all four phenomenal 'physical' features of 'space', 'time', 'energy' and 'mass' – are shown to comprise only 'computationally variant' features singularly produced by this UCP then we can foresee a condition in which the spatial-temporal sequence of a given USCF's frames can be reversed (i.e., at least when it is limited to a particular physical phenomenon); This was indeed predicted as one of the CUFT's 'differential-critical' predictions [2] – i.e., regarding the possibility of reversing the 'spatial-electromgnetic' sequence of a given phenomenon such as the growth and decay of a given amoeba. Essentially, this 'critical-differential' prediction of the CUFT states that it should be possible at least in principle) to reverse any given physical phenomenon by recording its precise USCF's spatial-electromagnetic values (e.g., of each of its constituting spatial pixels across a give number of USCF's frames), and then applying a specific electromagnetic stimulation (to each of this phenomenon or physical object's spatial pixels) in such a manner as to produce the "reversed spatial electromagnetic sequence" across the same number of given USCF's frames! Therefore, this CUFT's 'differential-critical' prediction predicted that it should be possible (at least in principle) to "cause" an 'amoeba' to "go back in time" – reversing its spatial-electromagnetic spatial pixels' sequence (by applying the particular electromagnetic stimulation to each of its spatial-pixels across a given number of USCF's frames... More generally then, the CUFT asserts the possibility of reversing the sequence of temporal events comprising any physical phenomenon! Therefore, it should be possible to increase the degree of entropy in any given physical system – contrary to the (currently accepted) 'Second Law of Thermodynamics'!

above), but instead must be explained as arising from the sole and singular production-

Another interesting potential theoretical ramification of the adoption of the CUFT as a satisfactory 'TOE' – including its discovery of the UCP singular 'A-Causal Computation', may be its potential revision of the 'Second Law of Thermodynamic' (and associated 'Arrow of Time' phenomenon). The 'Arrow of Time' enigma refers to the observation that the laws of Physics are "biased" in such a manner that events (and phenomena) always occur in a unidirectional temporal direction: thus, for instance, a glass may break into a hundred pieces – but those hundred pieces will not (of themselves) revert back to form a single unitary glass... Indeed, closely associated with this 'Arrow of Time' unidirectional temporal characteristic of physical phenomena is the (famous) 'Second Law of Thermodynamics' which states that in any given physical system the degree of entropy always increases with time... However, based on the CUFT's discovery of the singular UCP 'A-Causal Computation' – which was shown (above) to negate the basic "materialistic-reductionistic" assumption (underlying both Quantum and Relativistic models of physical reality), and one of the CUFT's (previous: [2] 'differential-critical' predictions regarding the possibility of "reversing the sequence of spatialelectromagnetic pixels" based on the application of certain electromagnetic effects, we may need to revise this Second Law of Thermodynamics (and associated 'Arrow of Time' enigma); This is because as explained earlier, none of the quantum or relativistic physical phenomena, relationships (or even laws) can continue to be based on any "materialistic-reductionistic" assumption/s. Hence, as shown above, neither the curvature of space-time by massive objects or (vice versa) the determination of the movement of massive objects based on the curvature of space-time, nor the observed accelerated expansion of the physical universe – can be explained by the current 'materialistic-reductionistic' relativistic (or quantum) assumption, but must be based on the singularity of the UCP 'A-Causal Computation'; Indeed, as we've shown (above), a fine temporal analysis of the dynamics of this singular UCP's productionsustenance- and evolution- of every (exhaustive) spatial pixel in the universe (comprising any single or multiple USCF frame/s) indicates that there cannot exist any 'materialistic-reduc‐ tionistic' effect of any of the four phenomenal 'physical' features (of 'space', 'time', 'energy' or 'mass') between any two (or more) spatial pixels, i.e., either within the same USCF frames or across different USCF frames. This is due to the UCP's singular asserted computation of the "simultaneous co-occurrence" of all spatial pixels in a given USCF frame (which prohibits any "causal" materialistic-reductionistic" effects existing between any two or more spatial pixels in the same USCF frame), as well as the UCP 's 'A-Causal Computation' associated 'Compu‐ tational Invariance Principle' which indicates that the sole and singular reality existing invariantly "during" the USCF's frame/s (e.g., as producing, sustaining and evolving any of the four phenomenal 'physical' features of all of its spatial pixels) and "in-between" these USCF's frames is the UCP. Hence, the only source for producing- sustaining- and evolvingany spatial pixel in the physical universe (e.g., at any given USCF frame/s) is the singular UCP, but not any of its (computationally variant) phenomenally produced 'physical' features...

Therefore, also the 'Second Law of Thermodynamics' which asserts the increase in entropy of any physical system with the progression of time – must be revised based on this new higher-

sustenance- and evolution- of the physical universe by the UCP...

194 Selected Topics in Applications of Quantum Mechanics

Although apparently "radical" this 'differential-critical' prediction of the CUFT does not aim to "topple down" the foundations of theoretical Physics, but rather expand our understanding of the physical reality by incorporating both Quantum Mechanics and Relativity Theory within a broader (higher-ordered) theoretical framework based on the discovery (and initial empirical verification) of the CUFT and its associated singularity of the UCP 'A-Causal Computation'; This is simply because in light of contemporary Physics basic contradiction between its two primary theoretical pillars (e.g., Quantum Mechanics and Relativity Theory) which has been shown to be resolved by the CUFT, the (initial) empirical validation of the CUFT's 'differentialcritical' prediction associated with the 'Proton-Radius Puzzle', and its discovery of the singularity of the UCP's 'A-Causal Computation' – the fundamental concepts of 'space', 'time', 'energy' and 'mass' as representing merely secondary ('computationally variant') 'phenom‐ enal' features produced by the sole reality of the ('computationally invariant') UCP have to be revised: Specifically, since "time" (alongside all three other 'phenomenal' physical features) is conceptualized as being singularly produced by the UCP – e.g., representing the degree of change of any given object or phenomena across a series of USCF's frames, then it should be possible (at least in principle) to reverse the sequence of spatial-change across frames (through the application of specific electromagnetic stimulation to the relevant spatial pixels comprising this physical phenomenon), thereby reversing the temporal events comprising this physical phenomenon... Hence, it may be said that the Second Law of Thermodynamics accurately represents the "natural progression" or temporal phenomena – but must be revised to include the possibility of reversing these 'natural phenomena' (thereby increasing their measured degree of entropy) across a series of USCF's frames. In a broader theoretical sense, the discovery of the CUFT's 'A-Causal Computation' necessitates us to revise our basic 'materi‐ alistic-reductionistic' assumptions underlying contemporary Physics, in such a manner that Quantum Mechanics and Relativity Theory will be anchored and based on the singular higherordered operation of the UCP's A-Causal Computation...

Therefore, we see that there is an urgent need to revise both quantum and relativistic models (laws and phenomena) based on the CUFT's discovery of the singularity of the UCP's pro‐ duction- sustenance- and evolution- of the physical universe; This important task involves several future steps, including: an empirical and mathematical verification of all of the CUFT's "differential-critical" predictions (e.g., beyond the initial empirical validation of one of its 'differential-critical' predictions associated with the 'Proton-Radius Puzzle' findings, men‐ tioned earlier), a revision of the laws of Physics based on the CUFT's 'Universal Computational Formula' (which in fact fully embeds and integrates the key quantum and relativistic compo‐ nents) and further explication and exploration of the new theoretical vistas offered by the CUFT higher-ordered and broader theoretical framework (including the potential connection between this singular Universal Computational/Consciousness Principle and individual human Consciousness).

#### **Author details**

#### Jehonathan Bentwich

Address all correspondence to: drbentwich@gmail.com

'BLIS' LTD, Israel

#### **References**


the application of specific electromagnetic stimulation to the relevant spatial pixels comprising this physical phenomenon), thereby reversing the temporal events comprising this physical phenomenon... Hence, it may be said that the Second Law of Thermodynamics accurately represents the "natural progression" or temporal phenomena – but must be revised to include the possibility of reversing these 'natural phenomena' (thereby increasing their measured degree of entropy) across a series of USCF's frames. In a broader theoretical sense, the discovery of the CUFT's 'A-Causal Computation' necessitates us to revise our basic 'materi‐ alistic-reductionistic' assumptions underlying contemporary Physics, in such a manner that Quantum Mechanics and Relativity Theory will be anchored and based on the singular higher-

Therefore, we see that there is an urgent need to revise both quantum and relativistic models (laws and phenomena) based on the CUFT's discovery of the singularity of the UCP's pro‐ duction- sustenance- and evolution- of the physical universe; This important task involves several future steps, including: an empirical and mathematical verification of all of the CUFT's "differential-critical" predictions (e.g., beyond the initial empirical validation of one of its 'differential-critical' predictions associated with the 'Proton-Radius Puzzle' findings, men‐ tioned earlier), a revision of the laws of Physics based on the CUFT's 'Universal Computational Formula' (which in fact fully embeds and integrates the key quantum and relativistic compo‐ nents) and further explication and exploration of the new theoretical vistas offered by the CUFT higher-ordered and broader theoretical framework (including the potential connection between this singular Universal Computational/Consciousness Principle and individual

[1] Bentwich, J. (2012a) "Harmonizing Quantum Mechanics and Relativity Theory". *The‐ oretical Concepts of Quantum Mechanics*, Intech (ISBN 979-953-307-377-3), Chapter 22,

[2] Bentwich, J. (2012b) "Theoretical Validation of the Computational Unified Field Theory". Theoretical Concepts of Quantum Mechanics, (ISBN 979-953-307-3773),

ordered operation of the UCP's A-Causal Computation...

196 Selected Topics in Applications of Quantum Mechanics

Address all correspondence to: drbentwich@gmail.com

human Consciousness).

**Author details**

'BLIS' LTD, Israel

**References**

pp. 515-550.

Chapter 23, pp. 551-598.

Jehonathan Bentwich


#### **A Lie-QED-Algebra and their Fermionic Fock Space in the Superconducting Phenomena** <sup>1</sup>**A Lie-QED-Algebra and their Fermionic Fock**  <sup>2</sup>**Space in the Superconducting Phenomena**

#### Francisco Bulnes 3 Francisco Bulnes

5

6

[18] Stephen W. Hawking Godel and the end of Physics. (Public lecture on March 8 at

[19] On the right track. Interview with Professor Edward Witten, *Frontline*, Vol. 18, No. 3,

Texas A&M University) 2003.

198 Selected Topics in Applications of Quantum Mechanics

February 2001.

Additional information is available at the end of the chapter 4 Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59078

#### 7 **1. Introduction**

8 It's created a canonical Lie algebra in electrodynamics with all the "nice" algebraic and 9 geometrical properties of an universal enveloping algebra with the goal of can to obtain 10 generalizations in electrodynamics theory of the TQFT (*Topological Quantum Field Theory*) 11 and the Universe based in *lines and twistor bundles* to the obtaining of *orbital spaces* [1] that 12 will be useful in the study of superconducting phenomena. The obtained object haves the 13 advantages to be an algebraic or geometrical space at the same time. This same space of 14 certain *L -* modules can explain and model different electromagnetic phenomena as 15 superconductor and quantum processes where is necessary an organized transformation of 16 the electromagnetic nature of the space-time and obtain nanotechnology of the space-time 17 and their elements when this is affected by the superconducting fields created by the 18 different *electrodynamic Majorana states* in the matter and space [2]. Then using the second 19 quantizing formalism to the description of the behavior of the particles of the affected space 20 for superconducting fields created by the BSC-theory (*Bardeen-Cooper-Schrieffer-theory*) in 21 condensation of the matter where is conformed a fermionic space called set of *pairs of Cooper*, 22 which are comported as bosons will be the domain of the QED (*Quantum Electro-Dynamics*) 23 transformations of the space to different actions as magnetic levitation, electromagnetic 24 impulse, etc. Then considering the Hilbert space of their multiple fermionic modes we 25 obtain the *fermionic Fock space* which describes the *quantum organized transformation* of the 26 particles to a *Fock space* of an arbitrary number of identical fermions explaining the 27 superconductivity as a Bose-Einstein condensation in this process. The fluidity obtained in 28 the Bose-Einstein condensation obeys to a Bogoliubov transformation which in this case, is 29 our organized transformation required to produce the micro-electromagnetic effects from 30 the actions of the operators of Lie-QED-algebra whose fermionic Fock space is the given for 31 the affecting of the energy space where exist the fermionic modes as a Plasmon resonances.

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2014 Author(s). Licensee InTech. This is an open access chapter distributed under the terms of the

2 BookTitle

1 From this study are mentioned some applications to photonics using the *boson link-wave* to 2 hyper-telecommunications and superconducting materials.

#### 3 **2. A Lie-quantum electrodynamical-algebra and their corollaries**

#### 4 **2.1. A Lie Algebra in Electro-dynamics**

5 We consider the electromagnetic field or Maxwell field defined as the differential 2 form of the forms space <sup>2</sup> <sup>4</sup> <sup>6</sup> ( ) R ;

$$F = F\_{ab} dx^a \wedge dx^b \,, \tag{1}$$

which can be described in the endomorphism space of *M*, by the matrix (where , *<sup>a</sup>* and , *<sup>b</sup>* 8 9 are equal to 1,2,3 ):

$$F\_{ab} = \begin{pmatrix} 0 & B^3 & -B^1 & E^1 \\ -B^3 & 0 & B^1 & E^2 \\ B^2 & -B^1 & 0 & E^3 \\ -E^1 & -E^2 & -E^3 & 0 \end{pmatrix}' \tag{2}$$

11 where *E* (respectively *B* ) the corresponding forms of electric field (respectively magnetic 12 field).

13 We want to obtain a useful form to define the actions of the group *L* , on the space of electromagnetic fields *F*, which is resulted of generalize to the space <sup>2</sup> 14 ( ) *M* , as an *anti-*15 *symmetric tensor algebra* through from induce to the product in the product , shape that will 16 be useful to the localizing and description of the irreducible unitary representations of the 17 groups *SO O* (4), (1,3), and *representations of spinor fields* in the space-time furthermore of 18 their characterizing as principal *G* bundle of *M* . In the context of the gauge theories (that 19 is to say, in the context of *bundles with connection* as the principal *G* bundles) we first observe that *F*, is an *exact form* and thus there exists a 1 form *<sup>b</sup>* 20 *A* (*electromagnetic potential*) that defines a connection in a *U*(1) bundle on *M* , and such that1 21

```
( ). ab a b b a ba b a a b F A AF A A
```
Likewise, the electromagnetic field is the 2-form given by (6) with the property of the transformation '' '' ' () . *ab a b b a ac bd c d d c ac bd cd F A A aa A A aaF* 

In R<sup>3</sup> , said 2-form match with the 3 3- matrix to *ab <sup>B</sup>* . Remember that *B A* .

<sup>1</sup> The anti-symmetric nature of this form results obvius:

A Lie-QED-Algebra and their Fermionic Fock Space in the Superconducting Phenomena 3 A Lie-QED-Algebra and their Fermionic Fock Space in the Superconducting Phenomena http://dx.doi.org/10.5772/59078 201

$$F\_{ab} = \partial\_a A\_b - \partial\_b A\_{a'} \tag{3}$$

Consider the *<sup>K</sup>* invariant *<sup>G</sup>* structure ( ), *GS M* of the differentiable manifold *<sup>M</sup>* <sup>4</sup> <sup>2</sup>R , with Lorentzian metric (and thus pseudo-Riemannian) *g*, on R<sup>4</sup> 3 , with 4 *Diag g*( ) (1,1,1, 1), in the system of canonical coordinates

$$\mathfrak{sp}\{U, \mathbf{x}, \mathbf{y}, \mathbf{z}, t\},\tag{4}$$

6 and let the spaces E H, , two free R modules (modules belonging to a commutative ring 7 with unit R) such that

$$\mathbb{E} = \{ E \in \mathfrak{P}(\mathbb{R}^4) \Big| E^b = \iota \frac{\partial}{\partial t} F \},\tag{5}$$

9 and

2 BookTitle

1 From this study are mentioned some applications to photonics using the *boson link-wave* to

5 We consider the electromagnetic field or Maxwell field defined as the differential 2 form

*ab* 7 *F F dx dx* (1)

which can be described in the endomorphism space of *M*, by the matrix (where ,

0

*ab*

that defines a connection in a *U*(1) bundle on *M* , and such that1 21

1 The anti-symmetric nature of this form results obvius:

( ). *ab a b b a ba b a a b F A AF A A*

'' '' ' ()

*ab a b b a ac bd c d d c*

*F A A aa A A*

.

*ac bd cd*

*aaF*

*<sup>b</sup>* 8

*B BE <sup>F</sup> BB E EEE*

10 (2)

11 where *E* (respectively *B* ) the corresponding forms of electric field (respectively magnetic

13 We want to obtain a useful form to define the actions of the group *L* , on the space of electromagnetic fields *F*, which is resulted of generalize to the space <sup>2</sup> 14 ( ) *M* , as an *anti-*15 *symmetric tensor algebra* through from induce to the product in the product , shape that will 16 be useful to the localizing and description of the irreducible unitary representations of the 17 groups *SO O* (4), (1,3), and *representations of spinor fields* in the space-time furthermore of 18 their characterizing as principal *G* bundle of *M* . In the context of the gauge theories (that 19 is to say, in the context of *bundles with connection* as the principal *G* bundles) we first observe that *F*, is an *exact form* and thus there exists a 1 form *<sup>b</sup>* 20 *A* (*electromagnetic potential*)

Likewise, the electromagnetic field is the 2-form given by (6) with the property of the transformation

In R<sup>3</sup> , said 2-form match with the 3 3- matrix to *ab <sup>B</sup>* . Remember that *B A* .

, *a b*

3 11 3 12 21 3 123

*B BE*

<sup>0</sup> , <sup>0</sup>

0

*<sup>a</sup>* and ,

3 **2. A Lie-quantum electrodynamical-algebra and their corollaries** 

2 hyper-telecommunications and superconducting materials.

4 **2.1. A Lie Algebra in Electro-dynamics** 

200 Selected Topics in Applications of Quantum Mechanics

of the forms space <sup>2</sup> <sup>4</sup> <sup>6</sup> ( ) R ;

9 are equal to 1,2,3 ):

12 field).

$$\mathbb{H} \vdash \mathbb{H} = \{ B \in \mathfrak{M}^\*(\mathbb{R}^4) \Big| B^b = -\iota \frac{\partial}{\partial t} F \},\tag{6}$$

where , *<sup>b</sup>* is Euclidean in <sup>3</sup> R , and , is Loretzian in R<sup>4</sup> 11 . Such R modules are *<sup>L</sup>* – 12 modules where *L* ( ) *M O*(1,3), is the orthogonal group of range 4. The two modules in (5) 13 and (6) intrinsically define all electric and magnetic fields *E,* and *B* , in terms of *F* . Thus 14 also their tensor, exterior, and scalar products between elements must be expressed in terms 15 of *<sup>F</sup>* . To it we consider the tensor product of (5) and (6) as free R modules elements, to know2 16 ,

$$E^{\flat} \otimes B^{\flat} = \frac{1}{c} F \otimes \overline{F} \,, \tag{7}$$

where *c* , is light speed and *F* <sup>3</sup> 18 , is the dual electromagnetic tensor of *F* . Then, what must 19 be *E B* ?

<sup>2</sup>This is valid since tensor product of free R modules is a free R module[5]. Here , . *b b b b b b E E dx dt B B dx dt*

<sup>3</sup>The Levi-Civita tensor can be used to construct the dual electromagnetic tensor in which the electric and magnetic components exchange roles (conserving the symmetry, characteristic that can be seen in the matrices of the electromagnetic tensor *F* , and their dual *F* ):

<sup>1</sup>**Proposition 2. 1 (F. Bulnes) [3].** Said R modules are invariant under Euclidean 2 movements of the group *O*(1,3), and thus are *L* modules.

3 *Proof*. [3]. �

4 BookTitle

4 Now we consider electro-strength field algebra given by the *U*(1) *gauge field* coupled to a 5 charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" QED.

Let ( ) E , be the tensor algebra generated by the elements 1 22 1 6 *F FF F* . Let *J*, be the two-seated ideal generated by the elements 1 2 2 1 12 7 *F F F F FF* [ , ]. Let e, be the Lie 8 algebra whose composition rule is [,] . Its wanted to construct an associative algebra with 9 unity element corresponding to e, such that

$$\{F\_1, F\_2\}\_{\bigotimes} = F\_1 \otimes F\_2 - F\_2 \otimes F\_1 \tag{8}$$

11 We want describe energy flux in liquid and elastic media in a completely generalized 12 diffusion of electromagnetic energy from the source (*particles of the space-time "infected" for*  13 *this electromagnetic energy*), which must be very seemed as a *multi-radiative tensor insights*  14 *space* or a *electromagnetic insights tensor space*. This will permits us to express and model the 15 flux of electromagnetic energy through pure tensor product of Maxwell fields F, which will 16 be useful in the symplectic structure subjacent in the quantum version of this algebra and 17 their actions of group. After and inside of the demonstration of one result where is related 18 the structure of this quantum version algebra with the superconducting phenomena , the 19 quantum macroscopic effects obtained result of this inheritance of structure, having that for 20 the energy conservation and the use of Lagrangians [4, 5]:

21 "*The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a*  22 *charge distribution plus the energy flux leaved in that region*"

Of fact these are elements *<sup>b</sup> E <sup>b</sup>* 23 *B* , that are constructed from a power space given in by 24 E B , and that conforms the *electromagnetic multi-radiative space which will be the region space*  25 *with the "transmittance" of the fermions effects obtained one time that let be quantized the space* 26 E H ,

27 to obtain our QED-Lie algebra necessary whose operators will act in the wrapped space (of 28 the electromagnetic type) to get the superconducting effects accord to the Bogoliubob 29 transformation [6] required to produce the quantum electromagnetic effects (electro-anti-30 gravitational effects) from the actions of the operators of Lie-QED-algebra whose fermionic

$$
\tilde{F}\_{\mu\nu} = \frac{1}{2} \varepsilon\_{\mu\nu\omega\ell} F^{\mu\nu}{}\_{\nu}
$$

where , is the rank-4 Levi-Civita tensor density in Minkowski space. 1 Fock space is the given for the affecting of the energy space where exist the fermionic modes 2 as a *Plasmon resonances*.

3 First we demonstrate the nature of Lie algebra of this space of tensors to electro-physics.

4 **Proposition (F. Bulnes) 3. 1.** The electrodynamical space E H , is a closed algebra under the composition law , 5 of the *U*(1) connections.

*Proof*. Let 11 11 <sup>1</sup> , *ab ba FAA* and 22 22 <sup>2</sup> , *ab ba* 6 *FAA* two elements of E H , *A* an *U*(1) connection. Then the composition 1 2 7 [ , ], *F F* takes the form in function of the *U*(1) 8 connections as

11 11 22 22 1 2 22 22 11 11 11 22 11 22 11 22 11 22 22 11 22 11 11 22 22 11 11 2 [,]( ) ( ) ( )( ) ( ) ( *ab ba ab ba ab ba ab ba ab ab ba ab ab ba ba ba ab ab ba ab ba ab ab ba ab ab FF A A A A AA AA A AA AA A A A A AA A AAA A A A* 2 22 11 22 22 22 11 11 22 11 11 11 22 11 22 11 22 11 22 22 11 22 11 11 22 22 11 1 1 )( ) { ( ) ( )} *ba ba ab ba ab ab ba ba ab ba ab ab ab ba ba ab ba ba ab ab ba ab ba ab ab ba b a A A AA A AA A AA A AA AA A A AA AA A AAA A A* 22 22 11 , *ab ba ab AA A*

Since , *ab ba F F* in R<sup>4</sup> 10 , then

9

11

4 BookTitle

3 *Proof*. [3]. �

26 E H ,

where

<sup>1</sup> , <sup>2</sup> *F*

 *F*

> 

<sup>1</sup>**Proposition 2. 1 (F. Bulnes) [3].** Said R modules are invariant under Euclidean

4 Now we consider electro-strength field algebra given by the *U*(1) *gauge field* coupled to a 5 charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" QED.

Let ( ) E , be the tensor algebra generated by the elements 1 22 1 6 *F FF F* . Let *J*, be the two-seated ideal generated by the elements 1 2 2 1 12 7 *F F F F FF* [ , ]. Let e, be the Lie 8 algebra whose composition rule is [,] . Its wanted to construct an associative algebra with

12 1 2 2 1 [,] *FF F F F F* 10 (8)

11 We want describe energy flux in liquid and elastic media in a completely generalized 12 diffusion of electromagnetic energy from the source (*particles of the space-time "infected" for*  13 *this electromagnetic energy*), which must be very seemed as a *multi-radiative tensor insights*  14 *space* or a *electromagnetic insights tensor space*. This will permits us to express and model the 15 flux of electromagnetic energy through pure tensor product of Maxwell fields F, which will 16 be useful in the symplectic structure subjacent in the quantum version of this algebra and 17 their actions of group. After and inside of the demonstration of one result where is related 18 the structure of this quantum version algebra with the superconducting phenomena , the 19 quantum macroscopic effects obtained result of this inheritance of structure, having that for

21 "*The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a* 

Of fact these are elements *<sup>b</sup> E <sup>b</sup>* 23 *B* , that are constructed from a power space given in by 24 E B , and that conforms the *electromagnetic multi-radiative space which will be the region space*  25 *with the "transmittance" of the fermions effects obtained one time that let be quantized the space*

27 to obtain our QED-Lie algebra necessary whose operators will act in the wrapped space (of 28 the electromagnetic type) to get the superconducting effects accord to the Bogoliubob 29 transformation [6] required to produce the quantum electromagnetic effects (electro-anti-30 gravitational effects) from the actions of the operators of Lie-QED-algebra whose fermionic

, is the rank-4 Levi-Civita tensor density in Minkowski space.

2 movements of the group *O*(1,3), and thus are *L* modules.

9 unity element corresponding to e, such that

202 Selected Topics in Applications of Quantum Mechanics

20 the energy conservation and the use of Lagrangians [4, 5]:

22 *charge distribution plus the energy flux leaved in that region*"

$$\begin{aligned} -\nabla^1\_b A^1\_a \otimes \nabla^2\_a A^2\_b + \nabla^2\_b A^2\_a \otimes \nabla^1\_a A^1\_b &= \nabla^1\_a A^1\_b \otimes \nabla^2\_b \\ \nabla^2\_a A^2\_b - \nabla^2\_b A^2\_a \otimes \nabla^1\_b A^1\_a &\in \mathbb{E} \otimes \mathbb{H}\_\prime \end{aligned}$$

Thus <sup>2</sup> 1 2 1 2 12 [ , ] , , ( ). *FF FF M* E H

13 Due to that we are using a *torsion-free connection* (e.g. the *Levi Civita connection*), then the partial derivative , *<sup>a</sup>* used to define *F*, can be replaced with the covariant derivative *<sup>a</sup>* 14 . 15 The Lie derivative of a tensor is another tensor of the same type, i.e. even though the 16 individual terms in the expression depend on the choice of coordinate system, the expression as a whole result in a tensor in R<sup>4</sup> 17 .

18 **Proposition (F. Bulnes) 3. 2.** The closed algebra ( , [,] E H ), is a Lie algebra.

1 *Proof*. Consider

6 BookTitle

2

$$\begin{aligned} \{F, F\} &= \nabla\_{\sf a} A\_b \otimes \nabla\_b A\_a - \nabla\_b A\_a \otimes \nabla\_a A\_b - \{\nabla\_a A\_b\} \\ &\otimes \nabla\_b A\_a - \nabla\_b A\_a \otimes \nabla\_a A\_b \end{aligned}$$

3 Then the other properties of Lie algebra are trivially satisfied. Thus E H , haves structure 4 of Lie algebra under the operation[,] .

#### 5 **2.2. Lie-QED-algebra to superconducting phenomena**

If first, we consider the Maxwell tensors given by , *ab ba* 6 *FA A* and thus of the Lie-EM-7 algebra given in the before section, these comply the following variation principle given by 8 the Maxwell Lagrangian.

$$\mathcal{L}\_{\text{MAXWHLL}} = -\frac{1}{4} F\_{ab} F^{ab} \, , \tag{9}$$

10 Then the due action to this Maxwell Lagrangian is

$$\begin{split} \mathfrak{J} &= -\frac{1}{4} \int d\mathbf{x}^{4} F\_{ab} F^{ab} = -\frac{1}{4} \int d\mathbf{x}^{4} \langle \nabla\_{a} A\_{b} - \nabla\_{b} A\_{a} \rangle (\nabla^{a} A^{b} - \nabla^{b} A^{a}) \\ &= \frac{1}{2} \int d^{4}k A\_{a}(k) [-k^{2} g^{ab} + k^{a} k^{b}] A\_{b}(-k), \end{split} \tag{10}$$

where is expected that the inner product ( ), ( ') *Aa a* 12 *kAk* , must be equivalent to an expression where the inverse of the differential operator defined by <sup>2</sup> [ ] *ab a b kg kk* , <sup>4</sup> 13 appears

$$\int [d\rho] \exp\left\{-\frac{1}{2}(\rho k\_{ab}\rho) + (f\rho)\right\} \approx \frac{1}{\sqrt{\det K}} \exp(J K^{-1} K)\_{\rho}$$

However in our case the operator , *K kab* comes given by

$$K = k\_{ab}(\boldsymbol{\chi} - \boldsymbol{y}) = [-k^2 \boldsymbol{g}^{ab} - k^a k^b] \boldsymbol{\delta}^4(\boldsymbol{\chi} - \boldsymbol{y})\_{\boldsymbol{\epsilon}}$$

Has the property of the projection operator, that is to say,

$$\int d^4 y k\_{ab}(\mathfrak{x} - \mathfrak{y}) k\_\varkappa^\nu(\mathfrak{y} - \mathfrak{z}) \propto k\_{\varkappa\sharp}(\mathfrak{x} - \mathfrak{y})\_\varkappa$$

and has not inverse. This means that the Gaussian integral diverges. The reason that the free-field part of the action integral given in (10) is singular is due to the gauge invariance which projects out the transverse gauge fields. In the path integral for the free-field part given by

 <sup>0</sup> 4 4 ( )exp [ ] , *<sup>a</sup> a a dA i dx JA <sup>L</sup>*

<sup>4</sup>The general formula for the Gaussian integral of the last integral of (10) takes the form:

1 in the product given in the covariant rules of Feynman diagrams. The field equation that 2 must be to solve is

6 BookTitle

2

1 *Proof*. Consider

4 of Lie algebra under the operation[,] .

204 Selected Topics in Applications of Quantum Mechanics

8 the Maxwell Lagrangian.

5 **2.2. Lie-QED-algebra to superconducting phenomena** 

10 Then the due action to this Maxwell Lagrangian is

1 1 <sup>1</sup> [ ]exp ( ) ( ) exp( ), <sup>2</sup> det *ab <sup>d</sup> k J JK K*

However in our case the operator , *K kab* comes given by

Has the property of the projection operator, that is to say,

transverse gauge fields. In the path integral for the free-field part given by

 

 

2 4 ( ) [ ] ( ), *ab a b K k x y kg kk x y ab*

<sup>4</sup> ( ) ( ) ( ), *ab d yk x y k y z k x y* 

 <sup>0</sup> 4 4 ( )exp [ ] , *<sup>a</sup> a a dA i dx JA <sup>L</sup>*

[,] [

3 Then the other properties of Lie algebra are trivially satisfied. Thus E H , haves structure

If first, we consider the Maxwell tensors given by , *ab ba* 6 *FA A* and thus of the Lie-EM-7 algebra given in the before section, these comply the following variation principle given by

MAXWELL

4 4

*dx dx*

<sup>1</sup> ( )[ ] ( ), <sup>2</sup>

*d kA k k g k k A k*

4The general formula for the Gaussian integral of the last integral of (10) takes the form:

*K*

*ab a b a b*

4 2

*ab* 9 *L F F* (9)

*ba ba ab FF A A A A A AA A* 

] 0,

*ab ba ba ab ab*

<sup>1</sup> , <sup>4</sup> *ab*

*ab ab ba*

1 1 ( )( ) 4 4

*F F A AA A*

*ab ab ba*

11 (10)

where is expected that the inner product ( ), ( ') *Aa a* 12 *kAk* , must be equivalent to an expression where the inverse of the differential operator defined by <sup>2</sup> [ ] *ab a b kg kk* , <sup>4</sup> 13 appears

and has not inverse. This means that the Gaussian integral diverges. The reason that the free-field part of the action integral given in (10) is singular is due to the gauge invariance which projects out the

$$[-k^2g^{ab} + k^ak^b]k\_b\alpha(k) = [-k^2k^a + k^2k^a]\alpha = 0, \dots$$

4 which is particular case of the Bulnes´s equation in the *curvature context* [4] to *G U* (1). Here , *a a k* , *a a* 5 *k* and *k* .

We consider the model that consists of a complex scalar field ( ), *x* 6 5minimally coupled to a 7 gauge field given by 1 forms (*U*(1) gauge field) "coupled to a charged spin 0 scalar 8 field" and that satisfy:

$$\mathcal{L} = \frac{1}{2} (D\_{\mu}\phi)^{\*} \, D^{\mu}\phi - \mathcal{U}(\phi^{\*}\,\phi) - \frac{1}{4} F\_{\phi} F^{ab} \, , \tag{11}$$

where , *ab F* has been defined in the section 2. 1. We define to ( ), 10 *D ieA aa a* as the 11 covariant derivative of the field , also *e* , is the electric charge and *U*( , \*) , is the 12 potential for the complex scalar field. This model is invariant under gauge transformations 13 parametrized by ( ), *x* that is to say, are had the following transformations to the fields:

$$
\varphi'(\mathbf{x}) = e^{i\iota\mathcal{L}(\mathbf{x})}\varphi(\mathbf{x}), \qquad A\_{\iota}^{\;\;\;}(\mathbf{x}) = A\_{\iota}(\mathbf{x}) + \partial\_{\iota}\mathcal{L}(\mathbf{x}), \tag{12}
$$

15 If the potential is such that their minimum occurs at non-zero value of , this model 16 exhibits the Higgs mechanism. This can be seen studying the fluctuations about the lowest 17 energy configuration, one sees that gauge field behaves as a massive field with their mass 18 proportional to the *e* , times the minimum value of . As shown by Nielsen and Olesen 19 [7], this model, in 2 1 , dimensions, admits time-independent finite energy configurations 20 corresponding to vortices carrying magnetic flux. The magnetic flux carried by these

$$\mathcal{L} = -\frac{1}{4} \int d\boldsymbol{\chi}^{d \cdot 1} \boldsymbol{\chi} dt \, \mathcal{L} = \int d^{d \cdot 1} \boldsymbol{\chi} dt [\boldsymbol{\eta}^{ab} \nabla\_a \boldsymbol{\rho}^\* \, \nabla\_b \boldsymbol{\rho} - V(\left| \boldsymbol{\rho} \right|^2) \mathbf{J}]$$

This has a *U*(1), or, equivalently *SU*(2), symmetry, whose action on the space of fields rotates , *<sup>i</sup> e* for some real phase angle.

we have summed over all field configurations including "orbits" that are related by gauge transformations. This over counting is the root of the divergent integral. Thus we have to remove this "volume" of the orbit in this quantization. In the case where the quantizing is realized by the scalar field theory to our superconducting phenomena, the "orbits" are considered as part of the interactions spinorbit, and the Lie-QED structure to the orbits will be conserved.

<sup>5</sup>In a complex scalar field theory, the scalar field takes values in the complex numbers, rather than the real numbers. The action considered normally takes the form

8 BookTitle

vortices is quantized (in units of <sup>2</sup> *e* <sup>6</sup> 1 ) and appears as a topological charge associated with 2 the topological current [8]:

$$J\_{top}^{a} = \in^{abc} F\_{bc'} \tag{13}$$

4 These vortices are similar to the vortices appearing in type-II superconductors. These in the 5 superconducting theory are acquaintance as *fluxoids*. There exist some thermo-dynamical conditions established to the existence the superconductor of type II7 6 .

**Theorem. (F. Bulnes) [1, 9, 10] 2. 2. 1.** We consider <sup>1</sup> <sup>2</sup> ( )/4 , <sup>2</sup> *i k ik F HH H* <sup>8</sup> 7 and <sup>2</sup> /8 , *S ext F Hn* 8 with the Hamiltonian foreseen in the *Appendix A* given by (A. 12), of the

6If S , is away from of the borders and rounds to the hollow (see the Figure 1) and suppose that is have applied a magnetic field to this superconductor S, then

$$\oint\_{\Gamma} \frac{m^\star}{\hbar n\_s e^\star} J + \frac{e^\star}{\hbar c} A \Big) dl = n \, 2\pi \,, \rho$$

But *J* 0, (inside the superconductor (or ring in the experimental Figure 1) there not are currents) then *<sup>c</sup> Adl <sup>n</sup>*

$$\begin{aligned} \bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \\ ^\square \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \, ^\bullet \end{aligned}$$

For the Stokes theorem is had that , S *Adl BdS*

To it is necessary remember that the superconducting current *<sup>s</sup> J* , haves an unique value in each point, which is equivalent to that the density of superconductor electrons is injective in each point. This bring as consequence that in a close circuit , of length 2 , we have

$$
\phi(2\pi) - \phi(0) = n2\pi\_{\prime\prime}
$$

For the circulation around a close circuit , and considering that

, *<sup>a</sup> A* 

we have that on the close circuit , <sup>2</sup> , *dl n <sup>a</sup>* that in our case is

$$
\Phi = \mathfrak{m} \Phi\_0.
$$

7Its considered the superconductor of the type I, in the intermediate state (1 ) , *n H HH c c* (ellipsoidal superconductor) and we calculate the transition to type II, with ( ) ( )[9,10]. *T T* 

8Where the term <sup>1</sup> <sup>2</sup> , <sup>2</sup> *H ik* is the term of the tensor *F*, corresponding to the free or total energy of the magnetic field of the superconductor, which involves the thermodynamic effects foreseen in (9), for "compression", to which is subject the surface of the object *O* (**see theAppendix A**).

1 *Lemma. A. 1.,* and their proof, where is satisfied the inequality on magnetic energy necessary 2 to all magnetic process of superconducting

4 **Figure 1.** Ring region where is applied a magnetic field in a superconductor S. This magnetic field 5 could realize iso-rotations and levitation force and impulse. The magnetic flow is intense in this region 6 and the lines of intensity of magnetic field behave like it is pointed out. Inside of hollow the current is 7 zero. The region of *fluxoids* will be generated from the inner of the ring where not exists current but yes 8 magnetic flows doing it in discrete form.

$$8\pi \Phi\_d n H\_v \ge \int\_{\mathcal{O}} (H(1 - h\nabla^2 \vec{H}))^2 dV \ge 8\pi \int\_{\mathcal{O}} H^2 dV. \tag{14}$$

10 Then a sensorchip to magnetic flux (pressure on the surface of *O* ) to super-currents is 11 defined by the inequality

$$\int\_{V} j\_s(x, y)\delta(z)dV \le \int\_{\mathcal{S}} \mathbb{B}dS \le n\Phi\_{0'} \tag{15}$$

where 0 , is a fluxoid ( 7 2 *c e*/ \* 2.07 10 *gauss cm* 13 ), being *e e* \* 2, where *e*, is the 14 charge of electron.

15 *Proof*. [9]. �

16

3

8 BookTitle

vortices is quantized (in units of <sup>2</sup>

206 Selected Topics in Applications of Quantum Mechanics

2 the topological current [8]:

<sup>2</sup> /8 , *S ext F Hn*

\* \* 

*<sup>e</sup> <sup>J</sup> <sup>n</sup> <sup>e</sup> m <sup>s</sup>*

 , *<sup>e</sup>* \* *<sup>c</sup> Adl <sup>n</sup>*

 , S *Adl BdS*

(2 ) (0) 2 , 

*n*

 

 

 *dl n* 

 

, *<sup>a</sup> A* 

<sup>2</sup> ,

*<sup>a</sup>*

that in our case is <sup>0</sup> *n* .

8Where the term

<sup>2</sup> , \*

 

*A dl n*

For the Stokes theorem is had that

 

we have that on the close circuit ,

<sup>1</sup> <sup>2</sup> , <sup>2</sup> *H ik* 

*c*

*e* 

conditions established to the existence the superconductor of type II7 6 .

have applied a magnetic field to this superconductor S, then

as consequence that in a close circuit , of length 2 ,

For the circulation around a close circuit , and considering that

(ellipsoidal superconductor) and we calculate the transition to type II, with

"compression", to which is subject the surface of the object *O* (**see theAppendix A**).

<sup>6</sup> 1 ) and appears as a topological charge associated with

4 These vortices are similar to the vortices appearing in type-II superconductors. These in the 5 superconducting theory are acquaintance as *fluxoids*. There exist some thermo-dynamical

**Theorem. (F. Bulnes) [1, 9, 10] 2. 2. 1.** We consider <sup>1</sup> <sup>2</sup> ( )/4 , <sup>2</sup> *i k ik F HH H*

<sup>8</sup> 7 and

8 with the Hamiltonian foreseen in the *Appendix A* given by (A. 12), of the

6If S , is away from of the borders and rounds to the hollow (see the Figure 1) and suppose that is

But *J* 0, (inside the superconductor (or ring in the experimental Figure 1) there not are currents) then

To it is necessary remember that the superconducting current *<sup>s</sup> J* , haves an unique value in each point, which is equivalent to that the density of superconductor electrons is injective in each point. This bring

> we have

7Its considered the superconductor of the type I, in the intermediate state (1 ) , *n H HH c c*

magnetic field of the superconductor, which involves the thermodynamic effects foreseen in (9), for

is the term of the tensor *F*, corresponding to the free or total energy of the

( ) ( )[9,10]. *T T* 

, *a abc top bc* 3 *J F* (13)

17 **Figure 2.** One has a view of profile of the magnetic flow of a plate under magnetic field (this simulation 18 was published in the Proceedings of Fluid Flow, Heat Transfer and Thermal Systems of ASME in the 19 paper IMECE2010-37107, British Columbia, Canada with all rights reserved ® [9]).

Developing these topological electromagnetic elements using the tensor *abc* 20 , we have to 21 two Maxwell tensors:

$$J\_{a\emptyset} = \nabla\_b^1 A\_c^1 \otimes \nabla\_b^2 A\_c^2 - \nabla\_c^2 A\_b^2 \otimes \nabla\_c^1 A\_b^1 = (F\_1 \otimes F\_2 - F\_2 \otimes F\_1),\tag{16}$$

23 precisely is our tensor algebra given in the *proposition 3. 1*., with their conserved Lie 24 structure.

10 BookTitle

1 The essential difference between both versions consists in the coupling to a charged *spin*0, 2 scalar field, that in this case is a scalar magnetic field corresponding to a magnetic flow associated to the supercurrent *<sup>s</sup>* 3 *J* .

Considering the supercurrent *<sup>s</sup>* 4 *J* in presence of magnetic field of vector potential, this takes 5 the form

$$J\_s = \frac{e^{\ast \cdot} \hbar}{2m^{\ast} \, ^\ast \text{i}} (\psi^{\ast \cdot} \nabla \, \psi - \psi \nabla \, \psi^{\ast \cdot}) - \frac{\langle e^{\ast \cdot} \rangle^2}{m^{\ast} \, ^\ast \text{c}} \left| \nu \right|^2 A\_\prime \tag{17}$$

7 where , is a function very general of complex type that are changing spatially and that in 8 an any point this function depends of the order parameter (as coherent length, penetration length, etc parameters that are useful to characterize a superconductor [11]) and <sup>2</sup> , *<sup>s</sup>* 9 *n* is 10 the density of the superconducting electrons.

11 Considering the action (10) and the proper to the Lagrangian (11), the electromagnetic total 12 action to all electromagnetic phenomena (included the given in ordinary electromagnetism) 13 using the scalar field theory takes the form:

$$\begin{aligned} S\_{\frac{\nu}{4}} &= \int \boldsymbol{\rho}^{\*} \left[ -\nabla\_{\boldsymbol{a}} \nabla^{\boldsymbol{a}} - \boldsymbol{m}^{2} \right] \boldsymbol{\rho} + \frac{1}{2} A\_{\boldsymbol{a}} [\boldsymbol{g}^{\boldsymbol{a}} - (\boldsymbol{1} - \frac{1}{\xi}) \nabla^{\boldsymbol{a}} \nabla^{\boldsymbol{a}}] A\_{\boldsymbol{b}} \\ &+ i e A\_{\boldsymbol{a}} ( (\nabla^{\boldsymbol{a}} \boldsymbol{\rho}^{\star}) \boldsymbol{\rho} - \boldsymbol{\rho}^{\star} \nabla^{\boldsymbol{a}} \boldsymbol{\rho} ) + e^{2} A\_{\boldsymbol{a}} A^{\boldsymbol{a}} \boldsymbol{\rho}^{\star} \cdot \boldsymbol{\rho}, \end{aligned} \tag{18}$$

15 in where and under certain physical conditions of symmetries we can establish the 16 following relations between the complex functions , and \*, and their covariant derivatives , \*, and the proper to the scalar fields given by , ( \*) *<sup>a</sup>* , \*, and *<sup>a</sup>* 17 18 obtaining the pure action to superconductors given by the integral:

$$S\_s = \int i e A\_i (\psi^\* \nabla \psi - \psi \nabla \psi^\*) + e^2 A\_i A^\* \nabla \psi^\* \nabla \psi,\tag{19}$$

20 which appears in the superconductor under energy regime given by their corresponding 21 Hamiltonian defined in (A. 12) of the *appendix A*. The representation spaces that appear are the 22 Fock states spaces and are corresponded in the superconductivity with the photon states 23 spaces where there exist the interaction *photon-phonon-photon* [12] under frame study of the 24 microscopic theory of the superconductivity. In a second affirmation, is necessary consider 25 that electrons in superconductivity are moving in a very special enthrone, that is the crystalline 26 net formed by ions that constitute the solid which we want that be superconductor under the 27 application of field actions as given in (19) on the solid of object *O* .

28 We consider a particles system, all identical, that is to say, undistinguishable, that is to say, the interchange of two of them not change the measurable properties of the system. Let 1 29 *e* ,

and 2 *e* , two electrons and we suppose that the electron 1 1 *e* , is in a state that comes represented by the function , *<sup>n</sup>* (*wave function*), for other side, the electron 2 2 *e* , comes represented by the wave function , *<sup>m</sup>* 3 and we suppose for last that the direct interaction 4 between these two electrons is quasi-vanishing. Then we can describe the system of these two electrons by the wave function 1 2 ( ) ( ). *n m* 5 *e e* Remember that the two particles are 6 identical, thus the interchange of the two let us equal the system. Then also is valid as before that 2 1 ( ) ( ). *n m* 7 *e e* Then the total wave function with the interchanges realized by the two 8 electrons takes the form:

12 21 <sup>1</sup> [ ( ) ( ) ( ) ( )], <sup>2</sup> *nm nm* <sup>9</sup> *ee ee* (20)

10 and their conjugated

10 BookTitle

5 the form

7 where

derivatives

 , 

associated to the supercurrent *<sup>s</sup>* 3 *J* .

208 Selected Topics in Applications of Quantum Mechanics

10 the density of the superconducting electrons.

13 using the scalar field theory takes the form:

16 following relations between the complex functions

18 obtaining the pure action to superconductors given by the integral:

27 application of field actions as given in (19) on the solid of object *O* .

1 The essential difference between both versions consists in the coupling to a charged *spin*0, 2 scalar field, that in this case is a scalar magnetic field corresponding to a magnetic flow

Considering the supercurrent *<sup>s</sup>* 4 *J* in presence of magnetic field of vector potential, this takes

6 (17)

8 an any point this function depends of the order parameter (as coherent length, penetration length, etc parameters that are useful to characterize a superconductor [11]) and <sup>2</sup> , *<sup>s</sup>* 9

11 Considering the action (10) and the proper to the Lagrangian (11), the electromagnetic total 12 action to all electromagnetic phenomena (included the given in ordinary electromagnetism)

2

 

 14 (18)

15 in where and under certain physical conditions of symmetries we can establish the

\*, and the proper to the scalar fields given by

*s a <sup>a</sup> S ieA*

*eAA* 19 (19)

20 which appears in the superconductor under energy regime given by their corresponding 21 Hamiltonian defined in (A. 12) of the *appendix A*. The representation spaces that appear are the 22 Fock states spaces and are corresponded in the superconductivity with the photon states 23 spaces where there exist the interaction *photon-phonon-photon* [12] under frame study of the 24 microscopic theory of the superconductivity. In a second affirmation, is necessary consider 25 that electrons in superconductivity are moving in a very special enthrone, that is the crystalline 26 net formed by ions that constitute the solid which we want that be superconductor under the

28 We consider a particles system, all identical, that is to say, undistinguishable, that is to say, the interchange of two of them not change the measurable properties of the system. Let 1 29 *e* ,

\*, and *<sup>a</sup>* 17

 

 

1 1 \* [ ] [ (1 ) ] <sup>2</sup>

*S m Ag A*

*ieA eAA*

(( \*) \* ) \* ,

 

<sup>2</sup> ( \* \*) \* , *<sup>a</sup>*

*a ab a b aa b a aa a a*

<sup>2</sup> \* ( \*) <sup>2</sup> ( \* \*) , 2 \* \* *<sup>s</sup> e e J A m i m c* 

, is a function very general of complex type that are changing spatially and that in

 

2

, and

 

 

, ( \*) *<sup>a</sup>* ,  *n* is

\*, and their covariant

28

$$\Psi^{\ast} = \frac{1}{\sqrt{2}} [\boldsymbol{\nu}\_n(\boldsymbol{e}\_1)\boldsymbol{\nu}\_m(\boldsymbol{e}\_2) - \boldsymbol{\nu}\_n(\boldsymbol{e}\_2)\boldsymbol{\nu}\_m(\boldsymbol{e}\_1)]\_{\boldsymbol{\nu}} \tag{21}$$

12 But if the two states are the same state, then only 0, since \* 0 . Then the system is 13 anti-symmetric in the interchange of two particles, that is to say, the wave function is anti-14 symmetric under the interchange of electron coordinates. But by the *Pauli Exclusion Principle* 15 the situation described in the total wave function , is incorrect, being the correct by \* .

16 But the before help us to establish the anti-symmetric structure of the interaction between 17 pair of particles in the microscopy superconductivity theory and reflected this anti-18 symmetry property also in every *spin-orbit interaction* of every wave function to the two pair 19 electrons that satisfy the total wave function \* .

20 Considering to an electron field, a representation : E *V*, where *V*, , is a Hilbert space 21 and whose correspondence rule is

$$e \mapsto \xi(e),\tag{22}$$

and let *J* , <sup>9</sup> 23 the two-sided ideal in the tensor algebra defined in the section 2. 1, ( ), E, generated by the elements of the form 1 22 1 *e ee e* , where 1 2 24 *e e*, E .

25 **Proposition 2. 2. 1.** There is a natural one-to-one correspondence between the set of all 26 representations of E , on *V*, and the set of all representations of E E / *J,*on *V*. . If , is a 27 representation of E , on *V*, and , \*, is a representation of E E / *J* , on *V*, then

$$
\varphi^\circ\_\circ(e) = \xi^\circ \text{ (}e^\bullet\text{)}, \qquad \forall \, e \in \mathfrak{C}, \tag{23}
$$

<sup>9</sup>Remember that *J* , from a point of view of the superconductors is a topological current associated with the topological charge defined related with the magnetic flux carried by the fluxoinds.

12 BookTitle

8

9

1 *Proof.* Let , be a representation of E , on *V*. Then there exists a unique representation , 2 of ( ), E, on *V* , satisfying that () () *e e* , . *e* E Then mapping , , vanishes on the 3 ideal *J* , , became

$$
\overline{\tilde{\xi}}(e\_1 \otimes e\_2 - e\_2 \otimes e\_1 - [e\_1, e\_2]) = \tilde{\xi}(e\_1)\tilde{\xi}(e\_2) - \tilde{\xi}(e\_2)\tilde{\xi}(e\_1) - \tilde{\xi}([e\_1, e\_2]) = 0,\tag{24}
$$

5 Thus we can define a representation \*, of factor algebra E E / *J* , on *V* , by the condition \* . 6 10Then (23) is satisfied and determines \*, uniquely. Also is unique, indeed 7 suppose other representation, for example , E E / *J* of on *V* . If *e*E , we put ( ) ( \*) *e e* . Then the mapping *e e* ( ), is linear and is a representation of E on *V* , since

$$\begin{aligned} \xi([e\_1, e\_2]) &= \sigma([e\_1, e\_2]^\*) = \sigma(\pi(e\_1 \otimes e\_2 - e\_2 \otimes e\_1)) \\ &= \sigma(e\_1^\* e\_2^\* - e\_2^\* e\_1^\*) = \xi(e\_1)\xi(e\_2) - \xi(e\_2)\xi(e\_1) \Box \xi \end{aligned}$$

Considering in particular the representation given for \*, 10 11we have that Lie-QED-algebra 11 structure to the spin and orbits is conserved. Indeed the spin and orbit parts can be separated in each wave function as for example, from (24) we have that if 1 11 ( ) ( ) ( ), *e rs <sup>n</sup>* 12 and 2 22 ( ) ( ) ( ), *e rs <sup>m</sup>* 13 then (24) defines \* . Remember that the electron is a *fermion* and 14 the electrons are described for the anti-symmetric wave functions. This property will be 15 fundamental in the section relative to the construction of the fermionic Foch space 16 corresponding to the Lie-QED-algebra.

17 **Def. 2. 2. 1. [**10**]. A** E H - field is an element of a bi-sided ideal of the Maxwell fields [1, 6]. 18 Explicitly is the formal space

$$\mathfrak{C}\otimes\mathfrak{B}\mathfrak{f} = \{(F\_1, F\_2) \in \Omega^2(O) \times \Omega^2(O) \Big| F\_1 \otimes F\_2 - F\_2 \otimes F\_1 - \{F\_1, F\_2\}, \text{with } \otimes = \otimes\_{\mathfrak{e}\_n} \},\tag{25}$$

20 Before of this, we pass to the fundamental lemma to characterize the algebra E H , as the 21 fundamental algebra of all movements and electromagnetic phenomena (*for example,*  22 *magnetic levitation, electromagnetic matter condensation, Eddy currents, etc*) produced to

$$\begin{aligned} \Psi^{\bullet} &= \frac{1}{\sqrt{2}} \begin{vmatrix} \phi\_n(r\_1)\alpha(s\_1) & \phi\_n(r\_2)\alpha(s\_2) \\ \phi\_m(r\_1)\alpha(s\_1) & \phi\_m(r\_2)\alpha(s\_2) \end{vmatrix} = \frac{1}{\sqrt{2}} [\phi\_n(r\_1)\alpha(s\_1)\phi\_m(r\_2)\alpha(s\_2) - \\ [\phi\_m(r\_1)\alpha(s\_1)\phi\_n(r\_2)\alpha(s\_2)] \end{aligned}$$

<sup>10</sup> Its realized the following descend map , from : , E EE to \*: E E E / . *J*

<sup>11</sup>It is the Slater determinant (that helps to construct wave functions to start of the expressions 11 2 2 ()() ()( ) *n m rs rs* where , ( 1,2) *i i rsi* are the radius of the orbit and spin respectively) , for example:

1 quantum level by their electromagnetic fields satisfying the variation principle in their field 2 actions.

3 **Lemma (F. Bulnes) [10] 2. 2. 1.** All electromagnetic actions and their effects (microscopic and 4 macroscopic) on the superconductor object *O* , comes from the E H - fields.

#### 5 *Proof*. [10].

12

20

,

, , vanishes on the

(24)

\*, uniquely. Also is unique, indeed

 

, E E / *J* of on *V* . If *e*E , we put

 

\*, of factor algebra E E / *J* , on *V* , by the condition

( ), is linear and is a representation of E on *V* , since

12 BookTitle

1 *Proof.* Let

 \* . 

( ) ( \*) *e e* 

and 2 22

18 Explicitly is the formal space

11 2 2 ()() ()( ) *n m*

11 22

( ) ( ) ( ) ( )]

*rs rs*

 

16 corresponding to the Lie-QED-algebra.

10 Its realized the following descend map

*rs rs* where , ( 1,2) *i i*

11 22

 

*rs rs*

1 1 ()() ()( ) \* [ ()() ()( )

11 22

*rs rs*

 

2 2 ()() ()( )

 

*n n*

*m m*

 

 

 

example:

*m n*

8

9

3 ideal *J* , , became

2 of ( ), E, on *V* , satisfying that

210 Selected Topics in Applications of Quantum Mechanics

5 Thus we can define a representation

6 10Then (23) is satisfied and determines

. Then the mapping *e e*

7 suppose other representation, for example

, be a representation of E , on *V*. Then there exists a unique representation

 (*e e e e ee e e e e ee* [ , ]) ( ) ( ) ( ) ( ) ([ , ]) 0, 

12 12 1 22 1

 

> 

([ , ]) ([ , ]\*) ( ( ))

*ee ee e e e e*

Considering in particular the representation given for \*, 10 11we have that Lie-QED-algebra 11 structure to the spin and orbits is conserved. Indeed the spin and orbit parts can be separated in each wave function as for example, from (24) we have that if 1 11

( ) ( ) ( ), *e rs <sup>n</sup>* 12

 ( ) ( ) ( ), *e rs <sup>m</sup>* 13 then (24) defines \* . Remember that the electron is a *fermion* and 14 the electrons are described for the anti-symmetric wave functions. This property will be 15 fundamental in the section relative to the construction of the fermionic Foch space

17 **Def. 2. 2. 1. [**10**]. A** E H - field is an element of a bi-sided ideal of the Maxwell fields [1, 6].

20 Before of this, we pass to the fundamental lemma to characterize the algebra E H , as the 21 fundamental algebra of all movements and electromagnetic phenomena (*for example,*  22 *magnetic levitation, electromagnetic matter condensation, Eddy currents, etc*) produced to

> , from

> >

1 2 1 2 2 1 12 *F F O O F F F F F F* , , with , <sup>R</sup> 2 2 19 E H {( ) ( ) ( ) [ ] }, (25)

11It is the Slater determinant (that helps to construct wave functions to start of the expressions

*n m*

11 22

 

*rs rs*

: , E EE to

*rsi* are the radius of the orbit and spin respectively) , for

\*: E E E / . *J*

() () *e e* , . *e* E Then mapping

 

12 21 1 2 2 1

 

( ) ( ) ( ) ( ) ( ).

*ee ee e e e e*

 �

1 2 2 1 12 1 2 2 1 12 4

 

 

6 One important fact inside the demonstration of the lemma 2. 2. 1, was consider the bi-sided 7 ideal given by the space (25) whose actions are extended to all space from the 8 superconductor *O* , until the infinite (ambient of *O* ) through the gauge transformations 9 used by the Lie-QED-algebra . Then by the lemma A. 1 (F. Bulnes), given in the *appendix A*, 10 of this work [9], the quantum effects underlying in superconducting phenomena satisfies 11 that

$$\begin{aligned} H(A\_\prime \mathfrak{T}\_M) = \mathfrak{T}\_M - H(A\_\prime B) &= \int\_{\mathcal{O}} \left[ F\_i F\_k - \frac{1}{2} H^2 \delta\_{ik} \right] dV \, \, \Big|\, \mathbf{4}\pi \\ &= \int\_{\mathcal{O}} L\_M(\mathbf{x}(\mathbf{s})) d(\mathbf{x}(\mathbf{s})) - \int\_{\mathcal{O}} H^2 \, \, \Big|\, \mathbf{8}\pi dV \, \, \boldsymbol{\mu} \end{aligned}$$

where <sup>2</sup> 13 *H* / 8 , is the free magnetic energy and the integral of the Lagrangian, of the 14 expulsion by the action , and that is useful to establish the macroscopic wave functions that 15 give place to a microscopic quantum current Js. By the mathematical electrodynamics [3] we 16 can define using the structure of E H , (that is to say, a module of the exterior algebra 17 which is deduced by the universal map applied to each term of the element 1 2 1 12 , *<sup>2</sup>* 18 *F F F F F F* [ ] ) that:

\* , \* , *O dS dS O* 19 ( , ) ( ) *J J - J \** (26)

where *J* , , and \* ( ), *conj* , such that <sup>2</sup> , \* , which is (19) in the 21 supercurrents modality. As the last integral (18) measures effects due to the macroscopic 22 actions to level quantum, this proved the affirmation of the lemma 2. 2. 1, in microscopic 23 theory of superconductivity and also the macroscopic effects due to the Eddy currents must 24 satisfy the magnetic force equation [9] to magnetic levitation.

#### 25 **2.3. Photon spin algebra from** E H

26 The same Lie structure is conserved to the electromagnetic spin algebra. The Lie structure of 27 the macroscopic level given and demonstrated to the space E H , in the before subsection 14 BookTitle

2.1, and using after through the path integral quantization on Lie bracket 1 2 1 [ , ], *e e* given in 2 the *subsection 2. 2* can establish a version of this quantized Lie algebra to quantum spin number (parts ( )*<sup>i</sup>* 3 *s* ) associated to the photons that interact in the superconducting 4 phenomena on the quantum macroscopic effects generated by the superconducting currents 5 (for quantized electromagnetic fields). Then the QED- algebra obtained conserves the Lie 6 structure to spin electromagnetic operators.

7 The photon can be assigned a triplet spin with spin quantum number 1, *s* accord to the 8 classification of particles and their spin. This is similar to, say, the nuclear spin of the *N* isotope, but with the important difference that the state with 0, *Ms* 9 is zero, only the states with 1, *Ms* 10 are non-zero. We consider the electro-spin operator as the vector with their Pauli matrices associated12 11 :

$$S\_k = -i\hbar \varepsilon\_{l|k} = \frac{\hbar}{2} \sigma\_{k'} (k = 1, 2, 3), \tag{27}$$

13 Then we can define the analogous of E H to the quantum spin context as the algebra:

$$\mathbf{e} \otimes \mathbf{b} = \left( S\_k \in \left| S\_\prime, S\_\prime \right. \right) = -i\hbar \varepsilon\_{\parallel k} \Big|\_{\parallel} \tag{28}$$

which is closed under the bracket [,] , 15 operation . Indeed, we consider two elements , , *i j* 16 *S S* e h given by the relations

$$\mathcal{S}\_{i} = -i\hbar(\boldsymbol{\varepsilon}\_{\cdot}\otimes\boldsymbol{\varepsilon}\_{i} - \boldsymbol{\varepsilon}\_{i}\otimes\boldsymbol{\varepsilon}\_{\cdot}), \qquad \mathcal{S}\_{\cdot} = -i\hbar(\boldsymbol{\varepsilon}\_{\cdot}\otimes\boldsymbol{\varepsilon}\_{i} - \boldsymbol{\varepsilon}\_{\cdot}\otimes\boldsymbol{\varepsilon}\_{i}), \tag{29}$$

18 satisfying the cyclically rule *ijki* . Then their operation under [,], is

$$\begin{split} [\mathbb{S}\_{l}, \mathbb{S}\_{j}] &= -\hbar^{2} (\boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{k}} - \boldsymbol{\varepsilon}\_{\boldsymbol{k}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}}) (\boldsymbol{\varepsilon}\_{\boldsymbol{k}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} - \boldsymbol{\varepsilon}\_{\boldsymbol{\imath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{k}}) + \\ &\hbar^{2} (\boldsymbol{\varepsilon}\_{\boldsymbol{k}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} - \boldsymbol{\varepsilon}\_{\boldsymbol{\imath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{k}}) (\boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{k}} - \boldsymbol{\varepsilon}\_{\boldsymbol{k}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}}) = i\hbar \Big[ -i\hbar (\boldsymbol{\varepsilon}\_{\boldsymbol{\imath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} - \boldsymbol{\varepsilon}\_{\boldsymbol{\jmath}} \otimes \boldsymbol{\varepsilon}\_{\boldsymbol{\imath}}) \Big] = i\hbar S\_{\boldsymbol{k}} \in \mathbf{e} \otimes \mathbf{h}. \end{split}$$

20 For simple inspection it follows that

19

$$-i\hbar (\mathfrak{c}\_{\rangle} \otimes \mathfrak{c}\_{\rangle} - \mathfrak{c}\_{\rangle} \otimes \mathfrak{c}\_{\rangle}) \bullet \mathfrak{c}^{(\mu)} = \mu \mathfrak{c}^{(\mu)},\tag{30}$$

12In the special case of spin 1/2 particles, , *<sup>x</sup>* , *<sup>y</sup>* and , *<sup>z</sup>* are the three Pauli matrices given by:

$$
\sigma\_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\
\sigma\_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \\
\sigma\_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.
$$

with 1 or 1, and therefore labels the photon spin, , , , *<sup>z</sup>* 1 *S* **k k** with 1 or 2 1. Then due the vector potential is a transverse field the photon has no forward ( 0 ) 3 spin component.

#### 4 **3. Fermionic Fock spaces of the superconductivity**

14 BookTitle

number (parts ( )*<sup>i</sup>* 3

Pauli matrices associated12 11 :

, , *i j* 16 *S S* e h given by the relations

2

19

*S S*

 

20 For simple inspection it follows that

212 Selected Topics in Applications of Quantum Mechanics

6 structure to spin electromagnetic operators.

2.1, and using after through the path integral quantization on Lie bracket 1 2 1 [ , ], *e e* given in 2 the *subsection 2. 2* can establish a version of this quantized Lie algebra to quantum spin

4 phenomena on the quantum macroscopic effects generated by the superconducting currents 5 (for quantized electromagnetic fields). Then the QED- algebra obtained conserves the Lie

7 The photon can be assigned a triplet spin with spin quantum number 1, *s* accord to the 8 classification of particles and their spin. This is similar to, say, the nuclear spin of the *N* isotope, but with the important difference that the state with 0, *Ms* 9 is zero, only the states with 1, *Ms* 10 are non-zero. We consider the electro-spin operator as the vector with their

,( 1,2,3), <sup>2</sup> *k ijk k Si k*

 

13 Then we can define the analogous of E H to the quantum spin context as the algebra:

14 e h *Sk* [,] , *SS i i j*

2

12In the special case of spin 1/2 particles, , *<sup>x</sup>*

01 0 1 0 , , . 10 0 0 1 *xy z i*

*i*

 

> 

21

 

( ), ( ), *i j kk j j k ii k* 17 *S i*

[,] ( )( )

*ij j k k j k i i k*

 

 

18 satisfying the cyclically rule *ijki* . Then their operation under [,], is

<sup>12</sup> (27)

which is closed under the bracket [,] , 15 operation . Indeed, we consider two elements

*S i*

 

( )( ) ( )

 , *<sup>y</sup>* 

*k i i kj k k j i jj i k*

e h

() () ( ), *i jj i i*

 and , *<sup>z</sup>* 

 

*s* ) associated to the photons that interact in the superconducting

 

*i i i S*

 

 

 

*ijk* (28)

(29)

.

(30)

are the three Pauli matrices given by:

 

5 By the BSC-theory we have a Cooper pair is a magnitude whose spin is zero. But spin zero 6 are bosons, then is easy fall in the temptation of treat a Cooper pairs as bosons. Furthermore, 7 we have indicated that few many more Cooper pairs, better energy will be the process of 8 superconductivity. However, the Pauli principle remains in force and the state formed, for 9 example by ( , ), *k k* cannot be occupied for more than a pair of electrons at the same time 10 (see figure 3).

12 **Figure 3.** BSC fundamental state with three Cooper pairs.

11

20

13 Also the Hamiltonian in the BSC-theory is constructed by operators (that is to say, their 14 formalism with that is calculated the energy of the *fundamental superconductor state are anti-*15 *commutative*) follows anti-commutative rules as was discussed in the introduction of this paper. But involve a boson that is created an interaction between fermions13 16 . The wave 17 function such and as is proposed by the BSC-theory to , electrons (*foreseen in* (21) *to the case*  18 *of only pairs of electrons*) is the product of wave functions of pair conveniently anti-19 symmetrized, that is to say:

$$
\psi(1,2,\ldots,\Xi) \approx \psi(1,2)\psi(2,3)\cdots\psi(1-\Xi,\Xi),\tag{31}
$$

21 If we not write explicitly the part of spin and only we do with the orbital part we have:

$$\left|\psi(1,2,\ldots,\Xi)\right\rangle \approx \sum\_{k\_1} \sum\_{k\_2} \cdots \sum\_{k\_{\overline{\pi}\mathbb{Z}}} \zeta\_{k\_1} \cdots \zeta\_{k\_{\overline{\pi}\mathbb{Z}}} e^{i(k\_1\varepsilon\_1 - k\_2\varepsilon\_2 \ast \ldots \ast k\_{\overline{\pi}\mathbb{Z}}\tau\_{k\_1} - k\_{\overline{\pi}\mathbb{Z}}\tau\_{\overline{\pi}})},\tag{32}$$

<sup>13</sup>To difference of the London superconductivity where a charged gas of bosons produces naturally a Meissner effect.

16 BookTitle

8

1 where each term of this wave function describes a configuration where the , electrons is 2 grouped in / 2, pairs that are:

$$(k\_1, -k\_1) \cdots (k\_{\Xi/2}, -k\_{\Xi/2}),\tag{33}$$

4 The spin part is immediate, each electron of each pair haves opposite spines. The wave 5 function is a complicated function that covers all the related pairs between them. This takes 6 the form from excited states are obtained as linear combinations of the ground state excited

by some creation operators † 1 0 0 *<sup>k</sup> n i k a* <sup>7</sup> ,to the wave functions as:

$$\Psi = \prod\_{k} \Psi\_{k'} \tag{34}$$

9 Using the second quantizing formalism to the Fock space in *appendix B*,, we have that the 10 potential energy to said pairs is:

$$V = \sum\_{k,k^{\*}} V\_{k,k^{\*}} b\_{k}^{\dagger} b\_{k^{\*}} \tag{35}$$

12 The term of kinetic energy of the corresponding Hamiltonian considering the energy in the 13 Fermi level is:

$$E = \sum\_{k,k'} 2\varepsilon\_k \mathbf{b}\_k^\dagger \mathbf{b}\_k + \Delta \mathbf{b}\_k^\dagger \mathbf{b}\_{'} \tag{36}$$

15 where to the states of exited electrons (super-electrons) appear the trenches (to break the 16 pairs and get to superconductivity peak state (see the figure 4A) and 4B)).

17 The fermionic Fock space is (*B*. 1) where for second approximation we have

$$\mathcal{F} = \mathcal{H}\_0 \oplus \mathcal{H}\_1 \oplus \mathcal{H}\_{2'} \tag{37}$$

whose energies of the electron are , *<sup>k</sup>* and , *k q* 19 with momentums to two electrons in 20 superconducting states *hk*, and *hk q* ( ) respectively. This interaction is negative since is 21 attractive, Then the potential is:

$$V(k,k',q) = \frac{A^2 \hbar o\_q}{(\varepsilon\_k - \varepsilon\_{k-q})^2 - (\hbar o\_q)^2},\tag{38}$$

1 where *A*, is a coupling electron-fonon. Then

16 BookTitle

8

2 grouped in / 2, pairs that are:

214 Selected Topics in Applications of Quantum Mechanics

by some creation operators †

10 potential energy to said pairs is:

13 Fermi level is:

1

<sup>14</sup>*E bb bb*

whose energies of the electron are , *<sup>k</sup>*

21 attractive, Then the potential is:

*n i k a* <sup>7</sup> ,to the wave functions as:

0 0 *<sup>k</sup>*

1 where each term of this wave function describes a configuration where the , electrons is

1 1 /2 /2 ( , ) ( , ), *kk k k* 3 (33)

4 The spin part is immediate, each electron of each pair haves opposite spines. The wave 5 function is a complicated function that covers all the related pairs between them. This takes 6 the form from excited states are obtained as linear combinations of the ground state excited

> 

9 Using the second quantizing formalism to the Fock space in *appendix B*,, we have that the

*k k* <sup>11</sup>*V V bb* (35)

12 The term of kinetic energy of the corresponding Hamiltonian considering the energy in the

, '

15 where to the states of exited electrons (super-electrons) appear the trenches (to break the

<sup>012</sup> 18 *F H H H* , (37)

 and , *k q* 19 with momentums to two electrons in 20 superconducting states *hk*, and *hk q* ( ) respectively. This interaction is negative since is

2

*A*

*q k kq q*

 

2 2 ( , ', ) , , ( )( )

22 (38)

*k k*

16 pairs and get to superconductivity peak state (see the figure 4A) and 4B)).

17 The fermionic Fock space is (*B*. 1) where for second approximation we have

*Vkk q*

, *k k*

† , ' , '

, *kk k k*

† †

2 , *kkk k*

(34)

(36)

11

$$\left|\boldsymbol{\varepsilon}\_{k\pm q} - \boldsymbol{\varepsilon}\_{k}\right| < \hbar \boldsymbol{\alpha}\_{q},\tag{39}$$

3 Then after of realize some calculations in the fermionic Fock space is had that

$$
\mathcal{E}\_{\mathcal{F}} = \left(\boldsymbol{\Lambda}^2 + \boldsymbol{\varepsilon}\_k\right)^{1/2},\tag{40}
$$

5 where , is the minimum energy of excitation, that is to say, the value of energy trenches 6 that appears in the superconductor state (*see the* Figure 11A, *in the last section*). *These trenches* have a big variation with respect to the absolute temperature of the material. The energy , *<sup>k</sup>* 7 8 is the quasi-particle energy, that is to say, the energy of the holes of the fermions distribution 9 when happen the excited states. The trenches energy as the excited states shape the 10 orthogonal space of infinite dimension separated by the Fermi momentum.

12 **Figure 4.** A) Nano-wire device to break Cooper pairs. The Cooper pairs must be break to obtain the 13 maximum superconductor state. The super-electrons are transformed in Fermi liquid which established 14 the required transformation of the immediate region of the space-time where must be executive to 15 transformation due superconductivity [2, 13, 14, 15]. B). Spectral density of electron-phonon.

16 Indeed, consider a system of fermions with an one-body Hamiltonian of the form (accord to 17 (32) and (34)):

$$
\hat{H} = \sum\_{k} \varepsilon\_{k} \hat{a}\_{k}^{\dagger} \hat{a}\_{k} + E\_{0}.\tag{41}
$$

When all particle energies , *<sup>k</sup>* 19 are positive, the ground state of the system is the vacuum state vac , with all 0. *<sup>k</sup>* 20 *n* In terms of the creation and annihilation operators said state 21 vac , can be identified as the unique state killed by the the annihilation operators, that is to say, ˆ vac 0, . *<sup>k</sup>* 22 *a k* The excited states of the Hamiltonian (50) are particle states which obtain by applying creation operators to the vacuum, vac <sup>1</sup> † † <sup>1</sup> , , , ˆ ˆ *k k k kaa* 23 the energy of such a state is 1 0 0 . *k k E E E* 24

Now suppose for a moment that all the particle energies , *<sup>k</sup>* 1 are negative instead of positive. 2 In this case, adding particles decreases the energy, so the ground state of the system is not 3 the vacuum but rather the full-to-capacity state

$$\left| \text{full} \right> = \left| \text{all } n\_k = 1 \right> \prod\_{\text{all } k} \hat{a}\_k^\dagger \left| \text{vac} \right>\,,\tag{42}$$

5 with energy

18 BookTitle

$$E\_{\text{full}} = E\_0 + \sum\_{\text{full }k} \mathcal{E}\_{k \text{ } \prime} \tag{43}$$

7 Never mind whether the sum here is convergent; if it is not, we may add an infinite constant to the 0 8 *E* , cancel the divergence. What's important to us here are the energy difference 9 between this ground and the excited states.

10 The excited states of the system are not completely full but have a few *holes*. If we consider 1 0, *k k n n* for some modes 1 ( , , ), *k k* while all the other 1. *<sup>l</sup>* 11 *n* The energy of 12 such a state is

$$E = E\_0 + \sum\_{l \neq k\_1, \dots, k\_{\overline{\pi}}} \mathcal{E}\_l = E\_{\text{full}} - \sum\_{l=1}^{\overline{\pi}} \mathcal{E}\_{k\_l} > E\_{\text{full}}.\tag{44}$$

In other words, an un-filled hole in mode *k*, carries a positive energy *<sup>k</sup>* 14 .

15 In terms of the operator algebra, the full , state is the unique killed by all the creation operators, full † ˆ , *<sup>k</sup>* 16 *a k* . The holes can be obtained by acting on the full , state with the 17 annihilation operators that remove one particle at a time. Thus,

$$\left| \{ \text{1hole at } k \} \right\rangle = \left| \hat{n}\_k \text{ other } n = 1 \right\rangle = \hat{a}\_k \left| \{ \text{full} \} \right\rangle,\tag{45}$$

19 and likewise

$$\left| N \text{ holes at } k\_1, k\_2, \dots, k\_{\Xi} \right\rangle = \hat{a}\_{k\_{\Xi}} \cdots \hat{a}\_{k\_t} \left| \text{full} \right\rangle,\tag{46}$$

21 Altogether, when the ground state is full , the creation and annihilation operators Exchange their roles. Indeed, the ˆ , *<sup>k</sup>* 22 *a* make extra holes in the full or almost-full states, while the † ˆ , *<sup>k</sup>* 23 *a* operators annihilate those holes (by filling them up). Also the algebraic definition of the full , and vac , states are related by the exchange: vac vs full † ˆ ˆ 0, , , . *k k* 24 *a ka k*

1 To make this exchange manifest, let us define a new family of fermionic creation and 2 annihilation operators, to know,

$$
\hat{\boldsymbol{b}}\_{k} = \hat{\boldsymbol{a}}\_{k}^{\dagger}, \qquad \hat{\boldsymbol{b}}\_{k}^{\dagger} = \hat{\boldsymbol{a}}\_{k}, \tag{47}
$$

4 Unlike the bosonic commutation relations, the fermionic anti-commutation are symmetric between † *a a* ˆ ˆ, , so the ˆ ˆ† *b b*, , satisfy exactly the same anti-commutation relations as the ˆ , *<sup>k</sup>* <sup>5</sup>*<sup>a</sup>* and † ˆ , *<sup>k</sup>* 6 *a*

$$\{\hat{b}\_k, \hat{b}\_l\} = 0, \{\hat{b}\_k^\dagger, \hat{b}\_l^\dagger\} = 0, \{\hat{b}\_k, \hat{b}\_l^\dagger\} = \delta\_{k,l'} \tag{48}$$

Physically, the ˆ† , *<sup>k</sup> <sup>b</sup>* operators *create holes* while the <sup>ˆ</sup> , *<sup>k</sup>* <sup>8</sup>*b* operators *annihilate holes* and the holes 9 obey exactly the same Fermi statistics (as given in the Figure 4B) as the original particles. In 10 condensed-matter terminology, the holes are *quasi-particles*, but the only distinction between 11 the quasi-particles and true particles is that the later may exist outside the condensed 12 matter. When viewed from the inside of condensed matter, this distinction becomes 13 irrelevant.

14 Anyhow, from the hole point of view, the full , state is the *hole vacuum* which is the unique state with no holes at all, algebraically defined by full <sup>ˆ</sup> 0, *<sup>k</sup>* <sup>15</sup>*b k* . The excitations are hole states obtained by acting with *hole-creation operators* ˆ† *<sup>k</sup>* 16 *b* , on the hole-vacuum, 1 holes at full <sup>1</sup> † † <sup>2</sup> , , *k k kkbb* 17 *k* . Then the Hamiltonian operator (32) of the system becomes14 18

$$
\hat{H} = E\_0 + \sum\_k \varepsilon\_k (1 - \hat{b}\_k^\dagger \hat{b}\_k) = E\_{\text{tail}} + \sum\_k (-\varepsilon\_k) \hat{b}\_k^\dagger \hat{b}\_k. \tag{49}
$$

in accordance with individual holes having positive energies 0 *<sup>k</sup>* 20 .

21 The *k*, modes are eigenspaces of some conserved quantum numbers such as momentum or spin (or rather ˆ *<sup>z</sup>* 22 *S* ). When one makes a hole by removing a particle from mode ( , ), **p** *s* the net momentum of the system changes by **p**, while the net , *<sup>z</sup>* 23 *S* changes by *s*, so one can say that the hole in that mode has momentum **p**, and *<sup>z</sup>* 24 *S s* . Consequently the hole 25 operators are usually defined as

$$
\hat{b}\_{\mathbf{p},s} = \hat{a}\_{\cdot \cdot \mathbf{p}, \cdot s'}^{\dagger} \qquad \hat{b}\_{\mathbf{p},s}^{\dagger} = \hat{a}\_{\cdot \cdot \mathbf{p}, \cdot s'} \tag{50}
$$

<sup>14</sup> †† † ˆˆ ˆˆ ˆ ˆ (1 ). *kk kk kk aa bb bb*

18 BookTitle

5 with energy

1

12 such a state is

0, *k k n n*

operators, full †

19 and likewise

the †

Now suppose for a moment that all the particle energies , *<sup>k</sup>* 1

3 the vacuum but rather the full-to-capacity state

216 Selected Topics in Applications of Quantum Mechanics

<sup>6</sup>*<sup>E</sup> <sup>E</sup>*

9 between this ground and the excited states.

2 In this case, adding particles decreases the energy, so the ground state of the system is not

*<sup>a</sup> <sup>k</sup>* <sup>4</sup>*n* (42)

7 Never mind whether the sum here is convergent; if it is not, we may add an infinite constant to the 0 8 *E* , cancel the divergence. What's important to us here are the energy difference

10 The excited states of the system are not completely full but have a few *holes*. If we consider

for some modes 1 ( , , ), *k k* while all the other 1. *<sup>l</sup>* 11 *n* The energy of

, , 1

*lk k i E EE* 

 

15 In terms of the operator algebra, the full , state is the unique killed by all the creation

ˆ , *<sup>k</sup>* 16 *a k* . The holes can be obtained by acting on the full , state with the

1 , hole at other 1 full ˆ ˆ *k k* 18 *k n n a* (45)

<sup>1</sup> holes at full <sup>1</sup> <sup>2</sup> , , , ˆ ˆ *N kkaa k k* <sup>20</sup>*<sup>k</sup>* (46)

21 Altogether, when the ground state is full , the creation and annihilation operators Exchange their roles. Indeed, the ˆ , *<sup>k</sup>* 22 *a* make extra holes in the full or almost-full states, while

ˆ , *<sup>k</sup>* 23 *a* operators annihilate those holes (by filling them up). Also the algebraic definition of

the full , and vac , states are related by the exchange: vac vs full † ˆ ˆ 0, , , . *k k* 24 *a ka k*

1

 13 *E* (44)

0

In other words, an un-filled hole in mode *k*, carries a positive energy *<sup>k</sup>* 14

17 annihilation operators that remove one particle at a time. Thus,

full

all full all vac † 1 , ˆ*<sup>k</sup>*

> full <sup>0</sup> , *<sup>k</sup> k*

*k*

full full

 

, *<sup>i</sup> l k*

are negative instead of positive.

(43)

. 1 which leads to

20 BookTitle

$$
\hat{\mathbf{P}}\_{\text{tot}} = \mathbf{P}\_{\text{tail}} + \sum\_{\mathbf{p},s} \mathbf{p} \times \hat{\boldsymbol{\phi}}\_{\text{p},s}^{\dagger} \hat{\boldsymbol{\phi}}\_{\text{p},s'} \tag{51}
$$

3 and likewise

$$
\hat{S}\_{\text{rot}}^z = S\_{\text{tail}}^z + \sum\_{\mathbf{p},s} \mathbf{s} \times \hat{b}\_{\mathbf{p},s}^\dagger \hat{b}\_{\mathbf{p},s'} \tag{52}
$$

Finally, consider a system where the energies , *<sup>k</sup>* 5 take both signs: positive for some modes 6 *k*, but negative for other modes. For example, a free fermion gas with a positive chemical 7 potential , and free-energy operator

$$
\hat{H} = \sum\_{\mathbf{p},s} (\frac{\mathbf{p}^2}{2m} - \mu) \hat{a}\_{\mathbf{p},s}^{\dagger} \hat{a}\_{\mathbf{p},s} \tag{53}
$$

has positive , for , *<sup>f</sup>* **p** *p* where , *<sup>f</sup>* 9 *p* is the Fermi momentum defined by the threshold 0 <sup>2</sup>*<sup>m</sup> 2 <sup>f</sup> <sup>p</sup>* 10 (see figure 4). For this system the ground state is the *Fermi sea* where

$$m\_{\mathbf{p},s} = \Theta(\left|\mathbf{p}\right| < p\_{/f}) = \begin{cases} 1 & \text{for } \left|\mathbf{p}\right| < p\_{/f} \\ 0 & \text{for } \left|\mathbf{p}\right| > p\_{/f} \end{cases} \tag{54}$$

12 In terms of the creation and annihilation operators, the Fermi sea is the state (directly from † 1 0 0 *<sup>k</sup> n i k a* <sup>13</sup> ):

$$\left| \text{Fermi sea} \right\rangle = \prod\_{\mathbf{p},s}^{|\mathbf{p}| = p\_f \text{ only}} \hat{a}\_{\mathbf{p},s}^{\dagger} \left| \text{vac} \right\rangle\_{\text{\prime}} \tag{55}$$

15 which satisfies

$$
\hat{a}\_{p,s} \left| \text{Fermi sea} \right> = 0,\tag{56}
$$

for , *<sup>f</sup>* 17 **p** *p* and

$$\hat{a}\_{\mathbf{p},s}^{\dagger} \left| \text{Fermi sea} \right\rangle = 0,\tag{57}$$

for .*<sup>f</sup>* 1 **p** *p*

20 BookTitle

1 which leads to

3 and likewise

7 potential

0

has positive

<sup>2</sup>*<sup>m</sup>* 

> † 1

15 which satisfies

for , *<sup>f</sup>* 17 **p** *p* and

*n i k a* <sup>13</sup> ):

0 0 *<sup>k</sup>*

*2*

Tot full

Tot full

Finally, consider a system where the energies , *<sup>k</sup>* 5

218 Selected Topics in Applications of Quantum Mechanics

, and free-energy operator

† , ,

† , ,

† , ,

for for

**p**

**p**

*f*

*p*

*p*

*f*

1 ( ) , <sup>0</sup>

only

, Fermi sea vac †

**p** 14 (55)

*f p*

 *<sup>s</sup> s*

**p**

Fermi sea , ˆ 0, *<sup>s</sup> a* **<sup>p</sup>** 16 (56)

, Fermi sea † *a*ˆ 0, **<sup>p</sup>** *<sup>s</sup>* 18 (57)

,

**p**

*a*

ˆ ,

 **<sup>p</sup>** *<sup>s</sup>*

*s s*

take both signs: positive for some modes

, <sup>ˆ</sup> ˆ ˆ , *s s s b b* **p p p** 2 **P P p** (51)

> , ˆ ˆ ˆ , *z z*

*s S S sb b* **p p p** 4 (52)

6 *k*, but negative for other modes. For example, a free fermion gas with a positive chemical

2

<sup>ˆ</sup> ( ), ˆ ˆ <sup>2</sup> *s s*

**p p**

,

**p <sup>p</sup>** 8 (53)

*<sup>f</sup> <sup>p</sup>* 10 (see figure 4). For this system the ground state is the *Fermi sea* where

*n p*

**p**

,

*s H a a <sup>m</sup>*

, for , *<sup>f</sup>* **p** *p* where , *<sup>f</sup>* 9 *p* is the Fermi momentum defined by the threshold

*f*

11 (54)

12 In terms of the creation and annihilation operators, the Fermi sea is the state (directly from

2 We may treat this state as a quasi-particle vacuum if we re-define all the operators killing the Fermi sea , as annihilation operators. Thus we define new (59) for .*<sup>f</sup>* 3 **p** *p* only. But keep the original operators , ˆ *<sup>s</sup> a***<sup>p</sup>** , and , † *a*ˆ **<sup>p</sup>** *<sup>s</sup>* 4 acting with momentum outside the Fermi surface. 5 Despite the partial exchange the complete set or creation and annihilation operators satisfies 6 the fermionic anticommutation relations having:

$$\begin{aligned} \text{all} \{\hat{a}, \hat{a}\} &= \{\hat{b}, \hat{b}\} = \{\hat{a}, \hat{b}\} = 0, \\ \text{all} \{\hat{a}^+, \hat{a}^+\} &= \{\hat{b}^+, \hat{b}^+\} = \{\hat{a}^+, \hat{b}^+\} = 0, \\ \text{all} \{\hat{a}, \hat{b}^+\} &= \{\hat{b}^+, \hat{a}^+\} = 0, \end{aligned} \tag{58}$$

8 and also

11

$$\{\hat{a}\_{\mathbf{p},s'},\hat{a}\_{\mathbf{p},s}^{\dagger}\} = \delta\_{\mathbf{p},\mathbf{p}'}\delta\_{s,s'}, \qquad \{\hat{b}\_{\mathbf{p},s'},\hat{b}\_{\mathbf{p},s}^{\dagger}\} = \delta\_{\mathbf{p},\mathbf{p}'}\delta\_{s,s'} \tag{59}$$

if we restrict the , <sup>ˆ</sup> *<sup>s</sup> <sup>b</sup>***<sup>p</sup>** , and , ˆ† *<sup>b</sup>* **<sup>p</sup>** *<sup>s</sup>* , to , *<sup>f</sup>* **<sup>p</sup>** *<sup>p</sup>* only and the , <sup>ˆ</sup> ,*<sup>s</sup> <sup>a</sup>***p** and , † *a*ˆ **<sup>p</sup>** *<sup>s</sup>* , to . *<sup>f</sup>* 10 **p** *p*

12 **Figure 5.** A). Fermi surface to the gold (Au). All Au-quasi-particles must shape this surface with the 13 number of pairs corresponding to the metal to the superconducting state. This surface in the BCS-theory 14 shapes the quantum nucleus of the interaction electron-fonon-electron corresponding to the fermionic 15 Fock superconducting space [16, 17].. B). Fermionic Fock superconducting space conformed with for 16 Cooper pairs: (red particle *k* , blue particle *k* ). The net is obtained by the adding of quantum 17 Hilbert spaces respectively.

18 The Fermi sea Fermi sea , is the quasi-particle vacuum state of these fermionic operators. The two types of creation operators , † *a*ˆ **<sup>p</sup>** *<sup>s</sup>* , and , ˆ† *<sup>b</sup>* **<sup>p</sup>** *<sup>s</sup>* 19 , create two distinct types of quasi-20 particles (*respectively the extra fermions above the Fermi surface and the holes below the surface*).

22 BookTitle

1 Both types of quasi-*particles* have positive energies. Then in terms of our new fermionic 2 operators, the Hamiltonian takes the form:

$$\hat{H} = E\_{\text{fermiSurface}} + \sum\_{\mathbf{p},s} (\frac{\mathbf{p}^2}{2m} - \mu > 0) \times \hat{a}\_{\mathbf{p},s}^\dagger \hat{a}\_{\mathbf{p},s} + \sum\_{\mathbf{p},s} (\mu - \frac{\mathbf{p}^2}{2m} > 0) \times \hat{b}\_{\mathbf{p},s}^\dagger \hat{b}\_{\mathbf{p},s'} \tag{60}$$

with the domains to every sum , *<sup>f</sup>* **p** *p* only (in the first sum) and , *<sup>f</sup>* 4 **p** *p* only (in the 5 second sum).

6 Clearly (60) describes elements of the fermionic Fock space given in (37).

#### 7 **4. Fermionic C\*-Lie-QED-algebra**

8 **Theorem (F. Bulnes) 5. 1.** The *electro-anti-gravitational effect*s produced from 9 superconductivity have that to be governed by the actions of the *superconducting Lie-*QED*-*10 *algebra* E H .

11 To demonstrate the before result is necessary to define the electro-anti-gravity in the 12 formalism of the Lie-QED-algebra and their C\*-algebras associated to her. The electro-anti-13 gravity is obtained through of experiments where a fast rotating superconductor reduces the 14 gravitational effect. Of fact the rotation is fundamental and necessary to the complementing 15 of the anti-gravity effects searched through the magnetic levitation (see the Figure 6. where 16 were realized many experiments with rotating geometrical pin, using the high intense 17 magnetic field).

19 **Figure 6.** Magnetic levitation by fast rotating magnet.

18

20 The demostration of this theorem has been realized in part, by the lemma 2. 2. 1. However 21 we require other additional lemmas that have that to see with other aspects, as the iso-22 rotations (condition established and illustrated in the experiments realized in the Figure 6) 23 and the condensation effects of the matter required in the transmission process through a 1 "bosonic cloud" of the plasmons in the quantum transmission of the electro-anti-gravity 2 effect. The concluyent aspect in this last digression is that the rotation movement to very fast 3 velocity and the superconducting phenomena must be togheter and due to the Lie structure 4 of our QED-algebra, this rotations will be inside the E H - fields as images of orthogonal transformations of the special orthogonal group *SO*(2), <sup>15</sup> 5 and their topological essency let to 6 see the inherent geometrical properties to an application of our superconductor electro-7 fields as was demostrated in the following lemma [9] and mentioned in [18]:

**Lemma (F. Bulnes) 5. 1.** Let *G CT* / ( ), C <sup>8</sup>S be with *C T*( ), a space of orbits (*hypersurfaces*), 9 generated by the electro-fields on *O*, by the realization of movements given for *SU*(2),

through of the action of their Maxwell fields *F*, given by <sup>1</sup> <sup>2</sup> ( )/4 , <sup>2</sup> *i k ik* <sup>10</sup>*F HH H* in the superconductor. Then the orbits engendered by the actions , *<sup>M</sup>* 11 on *M*, are magnetic torus engendered by rotations *SO x s x M* (2) ( ), , generates by fluxoids 0 12 , in the vortex 13 zone 8.

14 *Proof.* [9].

22 BookTitle

5 second sum).

10 *algebra* E H .

17 magnetic field).

18

2 operators, the Hamiltonian takes the form:

220 Selected Topics in Applications of Quantum Mechanics

7 **4. Fermionic C\*-Lie-QED-algebra** 

19 **Figure 6.** Magnetic levitation by fast rotating magnet.

Fermi Surface

1 Both types of quasi-*particles* have positive energies. Then in terms of our new fermionic

, , <sup>ˆ</sup> ˆ ˆ ( 0) ˆ ˆ ( 0) , 2 2 *s s s s s s H E a a b b*

with the domains to every sum , *<sup>f</sup>* **p** *p* only (in the first sum) and , *<sup>f</sup>* 4 **p** *p* only (in the

8 **Theorem (F. Bulnes) 5. 1.** The *electro-anti-gravitational effect*s produced from 9 superconductivity have that to be governed by the actions of the *superconducting Lie-*QED*-*

11 To demonstrate the before result is necessary to define the electro-anti-gravity in the 12 formalism of the Lie-QED-algebra and their C\*-algebras associated to her. The electro-anti-13 gravity is obtained through of experiments where a fast rotating superconductor reduces the 14 gravitational effect. Of fact the rotation is fundamental and necessary to the complementing 15 of the anti-gravity effects searched through the magnetic levitation (see the Figure 6. where 16 were realized many experiments with rotating geometrical pin, using the high intense

20 The demostration of this theorem has been realized in part, by the lemma 2. 2. 1. However 21 we require other additional lemmas that have that to see with other aspects, as the iso-22 rotations (condition established and illustrated in the experiments realized in the Figure 6) 23 and the condensation effects of the matter required in the transmission process through a

6 Clearly (60) describes elements of the fermionic Fock space given in (37).

**p p p p** 3 (60)

 

2 2

*m m*

† † , , , ,

 **p p p p**

> 15 Then an analogous to QED of the fields *F*, will have that consider in the states generated in 16 a Fock space *F* , the corresponding transformation of a subgroup of *O n*( ), that is to say, the 17 automorphism of the group must act on fermionic states of the space, where the electro-antigravity comes established to change of spin-orbit from 1, *Ms* to 1, *Ms* 18 (or viceversa), 19 in a bose-Einstein distribution in the matter condensation phenomena to produce an electro-20 anti-garavity wrapping of the object *O* .

> 21 One important fact is that there exist orthogonal invariance of the CAR-algebra on a Fock 22 space, that is to say, the Fermionic Fock space is invariant under rotations, that is tosay, 23 *O n*() , *F* , where explicitely the orthogonal group is:

$$\mathbf{O}(n,K) = \{ \mathbf{Q} \in \text{GL}(n,K) | \mathbf{Q}^T \mathbf{Q} = \mathbf{Q} \mathbf{Q}^T = I \},\tag{61}$$

25 If we consider the subgroup *SO*(2), of *O K* (2, ), we have that the group *U*(1), of the 2-forms *ab* 26 *F* , satisfy:

$$LI(1) = Spin(2) = SO(2),\tag{62}$$

<sup>15</sup> <sup>2</sup> cos sin (2) , [0,2 ] . sin cos *SO A GL A* 

24 BookTitle

1 having the considered in the section 2. 3. Then all particle represented for their energy (by 2 their wave function ) can change their behavior using a gauge group as *U*(1), , or *SU*(2). 3 This last enclose the all electromagnetic phenomena around of the superconductivity that 4 we want cover. Remember that we required to obtain anti-gravity from the E H - fields of 5 our superconducting Lie-QED-algebra.

Then considering two elements of the group *SO*(2), , for example 1 2 6 *e e*, E H, the 7 representation fulfils (by proposition 2. 2. 1) is

$$
\zeta'(e\_1)\zeta'(e\_2) - \zeta'(e\_2)\zeta'(e\_1) = \zeta'(e\_1 \otimes e\_2 - e\_2 \otimes e\_1),
\tag{63}
$$

9 and the field is transformed as

$$
\Psi \mapsto \Psi',\tag{64}
$$

where explicitly the image ' () *J* 11 . From this always is possible construct a second 12 representation defined by:

$$
\zeta^\*(I\_{a\beta}) = \zeta'((I\_{a\beta}^{\ \ \ \ \ \ \ \ \ \beta})^{\ \ \ \ \ \ \beta}),\tag{65}
$$

14 which belongs to the charge-conjugated particle. The anti-particle is obtained of accord to 15 the contragradient , representation, which is:

$$
\overline{\zeta}'(f\_{a\theta}) = \zeta'(f\_{a\theta}^{-1}),
\tag{66}
$$

17 There are not charge-conjugated in gravity, since if the gauge group is Lorentz group *SO*(3,1), then elements <sup>1</sup> , *<sup>T</sup> J J* 18 , which means that the second representation \* , is 19 equivalent to .

20 But we need affect the immediate space-time at least locally through of these E H - fields, 21 such that we will have the anti-particles given in (75). Also we need a mapping that involves 22 and include in their image the spin connection that is involved in this anti-gravity process 23 from superconductivity.

24 We define the field , as a vector field whose application is as given in (64)

$$
\Psi' = \Gamma \Psi, \qquad \overline{\Psi}' = \overline{\Psi} \Gamma^{-1}, \tag{67}
$$

1 under a general diffeomorphism , that is to say, the mapping belonging to the space 2 *Diff TM TM* ( , \*) , where *TM*, is the dual to *T M*\* . But we required local transformations at 3 least in the immediate enthrone of object *O* , such that be anti-gravitational and this local 4 enthrone acts with the space-time to create levitation in *O* .

24 BookTitle

2 their wave function

222 Selected Topics in Applications of Quantum Mechanics

7 representation fulfils (by proposition 2. 2. 1) is

where explicitly the image ' () *J*

*SO*(3,1), then elements <sup>1</sup> , *<sup>T</sup> J J*

.

11

1 2 2 1 1 22 1 8 

 

, representation, which is:

24 We define the field , as a vector field whose application is as given in (64)

5 our superconducting Lie-QED-algebra.

9 and the field is transformed as

12 representation defined by:

15 the contragradient

19 equivalent to

23 from superconductivity.

1 having the considered in the section 2. 3. Then all particle represented for their energy (by

3 This last enclose the all electromagnetic phenomena around of the superconductivity that 4 we want cover. Remember that we required to obtain anti-gravity from the E H - fields of

Then considering two elements of the group *SO*(2), , for example 1 2 6 *e e*, E H, the

10 ', (64)

14 which belongs to the charge-conjugated particle. The anti-particle is obtained of accord to

17 There are not charge-conjugated in gravity, since if the gauge group is Lorentz group

20 But we need affect the immediate space-time at least locally through of these E H - fields, 21 such that we will have the anti-particles given in (75). Also we need a mapping that involves 22 and include in their image the spin connection that is involved in this anti-gravity process

<sup>1</sup> ',' , 25 (67)

18 , which means that the second representation

<sup>1</sup> \* ( ) (( ) ), *<sup>T</sup> J J*

<sup>1</sup> ( ) ( ), *J J* 

 16 (66)

 

 13 (65)

 

) can change their behavior using a gauge group as *U*(1), , or *SU*(2).

( )( ) ( )( ) ( *e e e e eee e* ), (63)

. From this always is possible construct a second

\* , is 5

6 **Figure 7.** Rotations and anti-gravitational wrapping energy. An example of this idea is the cloud energy 7 in the formatting iso-rotations in a galaxy, in condensed matter to sidereal objects with autonomous 8 energy.

9 Then the principal equivalence requires that the fields on our manifold locally transform be as in special relativity, that is to say, if , is an element of the Lorentz group ,16 10 the fields are 11 transformed like Lorentz-vectors. Of fact this property is extended to all electro-physical modules E , and H, like *L* modules17 12 .

13 However, the generalization to a general diffeomorphism is not unique. We could have 14 chosen the field , as a vector field whose applications *Diff TM TM* ( , \*) are

$$
\overline{\mathbf{h}}' = (\mathbf{I}\_{\mathbb{Z}})^{-1} \overline{\mathbf{h}}' \qquad \overline{\mathbf{h}}' = \overline{\mathbf{h}} \mathbf{I}\_{\mathbb{Z}}' \tag{68}
$$

But as , is an element of *L* , that is to say <sup>1</sup> *<sup>T</sup>* 16 , both representations (67) and (68) 17 agree. For general diffeomorphism that will not be the case, although introducing a new 18 field that have a modified scaling behavior, this can be possible to affected to the space-time 19 by E H - fields. Then is considered Isom( , ), *TM TM* such that to fields 20 , ' ' ', one finds the behavior

$$
\boldsymbol{\pi}^\* = (\boldsymbol{\Gamma}^T)^{-1} \boldsymbol{\pi}^{-1},\tag{69}
$$

22 It will be useful to clarify the emerging picture of space-time properties by having a close look at a contravariant vector field 23 , as depicted in the wrapping energy around *O* , (see

16 4 4 *L* { ( ) ( , ) *GL g p q* R R *gpq pq* ( , ), , },

<sup>17</sup>**Proposition 2.1 (F. Bulnes) [3].** <sup>E</sup> , and H, like R modules are invariant under Euclidean movements of the group *O*(1,3), and thus are *L* modules.

26 BookTitle

1 the figure 7). This field in blue is a cut in the tangent bundle, that is the set of tangent spaces , , *<sup>p</sup>* 2 *TM p M* which describes our space-time. The field is mapped to their covariant field , which is a cut in the cotangent bundle \*, *<sup>q</sup>* 3 *T M* , by the metric tensor [19]

$$
\Psi\_{\nu} = \mathbf{g}\_{\nu\nu} \Psi^{\nu} \,, \tag{70}
$$

Newly introducing the fields 5 (from here anti-graviting) this is transformed under the local Lorentz transformations like a Lorentz-vector in special relativity18 6 . Then we can have 7 (after of involve the relations of Isom( \*, \*) *TM TM* ):

$$
\boldsymbol{\tau} = \left(\boldsymbol{\Lambda}^{\mathrm{T}}\right)^{-1} \hat{\boldsymbol{\tau}} \boldsymbol{\Lambda}^{-1} = \left(\boldsymbol{\Lambda} \boldsymbol{\Lambda}^{\mathrm{T}}\right)^{-1} \boldsymbol{\lambda} \tag{71}
$$

where 1, in the space Isom(,) *TM TM* . Then (1,3) 1 , *SO gg g* , and *g* 9 , thus 10 the properties of the vector fields are transformed directly to those of fermionic fields by 11 using the fermionic representatives and the transformation of, in this case is the mapping † <sup>0</sup> , 19 12 instead of the metric, is used to relate a particle to the particle transforming under 13 the contravariant or contragradient representation.

14 Then using the notation , to covariant derivative we have:

$$
\nabla\_{\underline{\kappa}} = \pi\_{\underline{\kappa}}^{\kappa} \nabla\_{\underline{\kappa}\prime} \tag{72}
$$

16 which is a new connection. Then the *Maxwell-anti-gravity Lagrangian* (that is to say, for antigravitational pendants *<sup>a</sup>* 17 *A* , of gauge fields) is introduced via the field tensors:

$$\underline{F}\_{\underline{\omega\nu}}^{a} = \nabla\_{\underline{\kappa}} A\_{\underline{\omega}}^{\kappa} - \nabla\_{\underline{\omega}} A\_{\underline{\omega}}^{a} + \varepsilon f^{abc} A\_{\underline{\kappa}}^{b} A\_{\underline{\omega}\prime}^{c} \tag{73}$$

$$
\underline{\tau^{\underline{K}}\_{\underline{\nu}}} \underline{\tau^{\underline{\nu}}\_{\underline{\underline{\pi}}}} = \delta^{\underline{\underline{K}}\_{\underline{\underline{\pi}}}}\_{\underline{\underline{\pi}}} \qquad \underline{\tau^{\underline{\underline{\pi}}}\_{\underline{\underline{\pi}}}} \underline{\tau^{\underline{K}}\_{\underline{\underline{\nu}}}} = \delta^{\underline{\underline{\pi}}\_{\underline{\underline{\nu}}}}\_{\underline{\underline{\pi}}}.
$$

Then for completeness, let us also define the combined mappings through the relations:

$$
\begin{split}
\boldsymbol{\tau}\_{\boldsymbol{\nu}\underline{\boldsymbol{\nu}}} &= \boldsymbol{\tau}\_{\boldsymbol{\nu}}^{\underline{\boldsymbol{\kappa}}} \mathcal{g}\_{\underline{\boldsymbol{\kappa}}\underline{\boldsymbol{\nu}}\prime} \qquad \boldsymbol{\tau}^{\boldsymbol{\nu}\underline{\boldsymbol{\kappa}}} = \boldsymbol{\mathcal{g}}^{\boldsymbol{\nu}\boldsymbol{\kappa}} \boldsymbol{\tau}\_{\boldsymbol{\kappa}}^{\underline{\boldsymbol{\nu}}}. \\
\boldsymbol{\tau}^{\boldsymbol{\alpha}} &\text{ is the canonical Dirac matrix} \begin{pmatrix} \boldsymbol{0} & \boldsymbol{I} \\ \boldsymbol{I} & \boldsymbol{0} \end{pmatrix}.
\end{split}
$$

<sup>18</sup> The underlined indices on these quantities do not refer to the coordinates of the manifols, but to the local basis in the tangential. All of these fields still are functions of the space-time coordinates *x* . As diffeomorphism , maps the basis of one space into the other. We can expand it as *dx* , or *dx* , respectively, such that (have inverses):

Staying an Lagrangian of the type <sup>2</sup> [ \* ( )] *ab a b V* (see the section 2. 2. 1). Here *abc* 1 *f* , 2 are the structure constants of the group and *e* , is the charge electron coupling with the 3 Planck scale. Then the corresponding electro-anti-gravitational-Lie-QED-algebra is that with 4 supercurrents

$$J\_{\underline{a}\underline{\omega}}^{\;\;\nu} = \underline{F}\_{\underline{\omega}\underline{\omega}}^{a} \otimes \underline{F}\_{\underline{\omega}\underline{\omega}}^{b} - \underline{F}\_{\underline{\omega}\underline{\omega}}^{b} \otimes \underline{F}\_{\underline{\omega}\underline{\omega}}^{a} - \left\{ \underline{F}\_{\underline{\omega}\underline{\omega}\prime}^{a}, \underline{F}\_{\underline{\omega}\underline{\omega}}^{b} \right\}\_{\prime} \tag{74}$$

6 Then the Lagrangian of fermionic fields can now be composed from the new ingredients as 7 [19]:

$$
\mathcal{L}\_{\texttt{ZZECHO}\cdot\texttt{ZWT-GAN}\cdot\texttt{XYZDO}\cdot\texttt{ZMT-CNOT}\prime\texttt{2D}} = \mathcal{L}\_F + \underline{\mathcal{L}}\_F \tag{75}
$$

where using the fields , , , and , the Lagrangian *L L*, , *<sup>F</sup> <sup>F</sup>* 9 take the form (using of 10 Feynman symbols):

$$\mathcal{L}\_{\overline{\mathsf{F}}} = (\overline{\mathsf{D}^{\mathsf{V}}}\mathsf{P})\Psi + \overline{\Psi}(\mathsf{D}^{\mathsf{V}}\mathsf{P}), \qquad \underline{\mathcal{L}\_{\overline{\mathsf{F}}}} = (\overline{\mathsf{D}^{\mathsf{V}}\overline{\mathsf{V}}})\overline{\Psi} + \overline{\Psi}(\mathsf{D}^{\mathsf{V}}\overline{\mathsf{V}}), \tag{76}$$

where ( ,) *g g L L* , and ( ,) *g g* 12 *L L* . This prove the *theorem 5. 1.*

13 Testing the Lagrangian we can see that there not is direct interaction between gravitational 14 and anti-gravitational particles. However, both of the particles-species will interact with the 15 gravitational field, which mediates an interaction between them. But this coupling is 16 suppressed with the Planck scale. Thus the production of anti-gravitational matter (which is 17 not observable today) can be is explained as ones condensation matter obtained in the 18 scattering process when the anti-gravitational wrapping is created. This usually could see as 19 a cloud or other haze type.

#### 20 What happen with the energy states Fock space?

26 BookTitle

Newly introducing the fields

224 Selected Topics in Applications of Quantum Mechanics

where 1, 

 † <sup>0</sup> 

 

 

 

 

<sup>19</sup> 0 

 

diffeomorphism

*dx* 

> 

   

, ,

*g g* , .

, is the canonical Dirac matrix

 

 

 

 

 

, respectively, such that (have inverses):

 

 

7 (after of involve the relations of Isom( \*, \*) *TM TM* ):

13 the contravariant or contragradient representation.

15

14 Then using the notation , to covariant derivative we have:

1 the figure 7). This field in blue is a cut in the tangent bundle, that is the set of tangent spaces , , *<sup>p</sup>* 2 *TM p M* which describes our space-time. The field is mapped to their covariant field

> *g* ,

11 1 ( ) ( ), ˆ *T T*

, 

> 

, and *g*

(72)

, thus

. As

 

 , or

 

 4 (70)

5 (from here anti-graviting) this is transformed under the local Lorentz transformations like a Lorentz-vector in special relativity18 6 . Then we can have

 8 (71)

10 the properties of the vector fields are transformed directly to those of fermionic fields by 11 using the fermionic representatives and the transformation of, in this case is the mapping

, 19 12 instead of the metric, is used to relate a particle to the particle transforming under

 

16 which is a new connection. Then the *Maxwell-anti-gravity Lagrangian* (that is to say, for anti-

18 (73)

 

Then for completeness, let us also define the combined mappings through the relations:

<sup>0</sup> . <sup>0</sup> *I*

*I* 

gravitational pendants *<sup>a</sup>* 17 *A* , of gauge fields) is introduced via the field tensors:

 , *<sup>a</sup> a abc b c F A A ef A A*

18 The underlined indices on these quantities do not refer to the coordinates of the manifols, but to the local basis in the tangential. All of these fields still are functions of the space-time coordinates *x*

, maps the basis of one space into the other. We can expand it as *dx*

in the space Isom(,) *TM TM* . Then (1,3) 1 , *SO gg g*

9

, which is a cut in the cotangent bundle \*, *<sup>q</sup>* 3 *T M* , by the metric tensor [19]

21 States of the Fermionic particles entering go interact through of the corresponding C\*-CAR-22 algebra [20, 21]. Likewise, for example if we consider the anti-symmetric Fock space <sup>a</sup> *F H*( ), and let *<sup>a</sup>* 23 *p* , the othogonal projection on to anti-symmetric vectors then C\*-CARalgebra is represented on <sup>a</sup> 24 *F H*( ), by settings

$$b^\*(\boldsymbol{\varrho}) p\_a(\boldsymbol{\wp}\_1 \otimes \boldsymbol{\wp}\_2 \otimes \dots \otimes \boldsymbol{\wp}\_n) = p\_a(\boldsymbol{\wp} \otimes \boldsymbol{\wp}\_1 \otimes \boldsymbol{\wp}\_2 \otimes \dots \otimes \boldsymbol{\wp}\_n),\tag{77}$$

26 This means that the action of orthogonal group *O*(2), stay restricted to the Hilbert space 27 corresponding to the C\*-CAR-algebra becoming the immediated finite region of the space-28 time in a fermionic Fock space that is mixture of particles and anti-particles (*at least until that*  28 BookTitle

10

1 *is converted all space*). We could call to this restrinction of orthogonal group as *O*(2, ), *H* where a new operator is obtained by the composition \*( ) *<sup>a</sup> Tb p* 2 , acting on a module Fock space that we can write as ( ), *A H*<sup>2</sup> <sup>a</sup> [22] 20 3 which represent the new energy space 4 whose elements are the second side of (77). Using the CAR-algebra of creation and annhilitation operators and 2 † *Db b* ( ) ( ), *H*<sup>a</sup> 5 . The canonical anti-commutation 6 relations are equivalent to the commutator relation:

$$\{D\_{\rho}, D\_{\rho}\} = D\_{\rho}D\_{\rho} - D\_{\rho}D\_{\rho} = -2i o(\varphi, \varphi),\tag{78}$$

8 with the anti-symmetrical form of the Weyl relations given by (, ) . If we extend the operators before to linear R-operators on Hilbert space 2 *H*<sup>a</sup> 9 , we obtain the relation ( , ) ( , ), *A A* which defines a fermionic orthogonal group

$$\mathfrak{S}^{\mathcal{O}\_2} = \{ A \in \mathcal{O}(\mathcal{Z}, \mathcal{H}\_{\mathbf{R}}) \, \Big|\, o(A\varphi, A\nu) = o(\varphi, \nu\nu), \forall \,\varphi, \nu \in \mathcal{H}\_{\mathbf{u}\mathbf{v}^2} \}\,\tag{79}$$

where appear the Bogoliubov transformation21 12 .

13 Finally, the orbital spaces created by the superconductivity in the quantum regime satisfy 14 the corresponding orbital integrals due F. Bulnes [17] to cuspidal surfaces in the generating 15 chirality inversion through a Dirac node(with Hamiltonian † † † *H i* () \* 16 [23]):

$$J\_{\iota}(\mathbf{E}) = \int\_{\underline{N}} a\_{\mathbf{f}}(a\_{\iota}\underline{\mathbf{n}}\underline{\mathbf{n}}\_{-\iota})^{\rho+\mu} < \sigma(m\_{\mathbf{f}}(a\_{\iota}\underline{\mathbf{n}}\underline{\mathbf{n}}\_{-\iota}))^{-1} m\_{\mathbf{f}}(\underline{\mathbf{n}}) \underline{\mathbf{g}}(k(a\_{\iota}\underline{\mathbf{n}}\underline{\mathbf{n}}\_{-\iota})) > d\underline{\mathbf{n}},\tag{80}$$

18 where *E*, is the total Fermi energy in all Fermi surface including the proper kinetic energies, the term ( ) *t t k a na* 19 ,is the momentum created in the chirality inversion through the node of automorphism *<sup>F</sup> n N* , where the space *NF* 20 is the normal subgroup defined to the action 21 created by fermions in the transit electron-phonon-electron, which is normed by the product 22 of logarithms of the actions of their automorphisms [24].

$$[\hat{\mathbf{b}}, \hat{\mathbf{b}}^{\dagger}] = [\mathbf{s}\hat{\mathbf{a}} + t\hat{\mathbf{a}}^{\dagger}, \mathbf{s}^{\*}\hat{\mathbf{a}}^{\dagger} + t^{\*}\hat{\mathbf{a}}] = \dots \dots = (\left|\mathbf{s}\right|^{2} + \left|t\right|^{2})[\hat{\mathbf{a}}, \hat{\mathbf{a}}^{\dagger}].$$

<sup>20</sup> ( ), *A H*<sup>2</sup> <sup>a</sup> is a algebra of operators from *H* , in the super-algebra <sup>2</sup> a .

<sup>21</sup>The Bogoliubov transformation is a canonical transformation of these operators. To find the conditions on the constants *s*, and *t*, such that the transformation remains canonical, the commutator is expanded:

2 **Figure 8.** A). Fermionic distribution of probability density *r*2|Ψpair(*r*)|2 of the neutron Cooper pairs as 3 a function of the neutron Fermi momentum *kFn* and the relative distance *r* between the pair partners in 4 symmetric nuclear matter [22]. This can produces Eddy currents with the property anti-gravitational 5 current given in (83). B). The part colored in blue determines the absence of magnetic flow, such like is 6 wanted that it happens for the super-currents existence on the surface of a levitating vehicle, making 7 that this everything behaves as a diamagnetic, except in the central ring colored in red and yellow 8 where exists an intense magnetic field (*this simulation was published in the Proceedings of Fluid Flow, Heat*  9 *Transfer and Thermal Systems of* ASME *in the paper* IMECE2010-37107*, British Columbia, Canada with all*  10 *rights reserved* ® [9]).

One example of this automorphisms *t t a na* 11 , in action are the quantum operators given by the product † † , *kq kq k k a a aa* 12 , which acts on pairs and not change the electrons in *k*, and 13 *k*, and transits to *k q* , and *k q* , letting equals spins. The energy *E*, is given by † k , ' 2 , *k k k k E ee* where † †† <sup>k</sup> () , () , , *k k <sup>k</sup> k k e f a a e g a a fg* 14 *H.*

Relating the meaning of these operators with the Debye energy to photons given by , *<sup>D</sup>* 15 , 16 we can to obtain a complete criteria to the energies given by (40) considering the 17 Coulumbian repulsion, obtaining a precise wide measure of trenches where to some real 18 superconductor we consider the term *N V* (0) 0.3 , is the magnetic momentum developed by the free electrons in the formatting of the Fermi liquid [25, 26, 27]. The integral ( ) *<sup>t</sup>* 19 *J E* , is 20 bounded [1, 25].

#### 21 **5. Applications**

1

28 BookTitle

10

 

<sup>20</sup> ( ), *A H*<sup>2</sup>

expanded:

 

( , ) ( , ), *A A*

† † † *H i*

1 *is converted all space*). We could call to this restrinction of orthogonal group as *O*(2, ), *H*

2 , acting on a module

<sup>a</sup> [22] 20 3 which represent the new energy space 4 whose elements are the second side of (77). Using the CAR-algebra of creation and

 *H*<sup>a</sup> 5 . The canonical anti-commutation

> 

<sup>2</sup> { (2, ) ( , ) ( , ), , }, *AO AA*

 

> 

operators before to linear R-operators on Hilbert space 2 *H*<sup>a</sup> 9 , we obtain the relation

(, ) 

(78)

. If we extend the

where a new operator is obtained by the composition \*( ) *<sup>a</sup> Tb p*

*A H*<sup>2</sup>

† *Db b* ( ) ( ),

 

which defines a fermionic orthogonal group

 

 *L H* R *<sup>H</sup>* <sup>a</sup> 11 (79)

13 Finally, the orbital spaces created by the superconductivity in the quantum regime satisfy 14 the corresponding orbital integrals due F. Bulnes [17] to cuspidal surfaces in the generating 15 chirality inversion through a Dirac node(with Hamiltonian

*t Ft t Ft t F t t*

18 where *E*, is the total Fermi energy in all Fermi surface including the proper kinetic energies, the term ( ) *t t k a na* 19 ,is the momentum created in the chirality inversion through the node of automorphism *<sup>F</sup> n N* , where the space *NF* 20 is the normal subgroup defined to the action 21 created by fermions in the transit electron-phonon-electron, which is normed by the product

 17 (80)

<sup>1</sup> ( ) ( ) ( ( )) ( ) ( ( )) ,

21The Bogoliubov transformation is a canonical transformation of these operators. To find the conditions on the constants *s*, and *t*, such that the transformation remains canonical, the commutator is

*J E a a na m a na m n g k a na dn*

Fock space that we can write as ( ),

226 Selected Topics in Applications of Quantum Mechanics

6 relations are equivalent to the commutator relation:

2

where appear the Bogoliubov transformation21 12 .

() \*

16 [23]):

 

*F*

22 of logarithms of the actions of their automorphisms [24].

† 2 2 † ˆ ˆ † † [ , ] [ , \* \* ] ( )[ , ]. *b b sa ta s a t a s t a a* ˆˆ ˆ ˆ ˆ ˆ

*N*

annhilitation operators and 2

[,] *D D DD DD i* 2 ( , ), 7 

8 with the anti-symmetrical form of the Weyl relations given by

 

> 

<sup>a</sup> is a algebra of operators from *H* , in the super-algebra <sup>2</sup> a .

> 22 **Proposition (F. Bulnes) 6. 1.** Using organized transformations as given in ( ) ( ) , *T T* M M *<sup>n</sup>* 23 we can to establish that the state of all particles in set, is their 24 corresponding *Fock image* [15, 28].

> 25 Inside of the Fock space begins a realization of the potential of the superconductivity, since 26 the Fock pure state involves all the states of particles of the space, object of the 27 transformation [15], that in this case is *superconducting state*. We want organize the particles 28 in two the phases that define our Fock space then the proposition is the shape to do it!

30 BookTitle

16

1 **Theorem V. 1 (F. Bulnes, R. Goborov).** The organized transformation given by [15]

$$
\sigma\_!\mathcal{T}(\mathbb{M}) \otimes \dots \otimes \sigma\_\*\mathcal{T}(\mathbb{M}) \otimes \dots,\tag{81}
$$

3 to *electro-anti-gravitational effect* produced from superconductivity must have a fermionic Fock space [10]22 4

$$\mathcal{H}\_1 \otimes \mathcal{H}\_2 = \{ \hat{e}\_k^\dagger \hat{e}\_k \} \Big| \{ \hat{e}\_k, \hat{e}\_l \} = 0,\\ \{ \hat{e}\_k^\dagger, \hat{e}\_l^\dagger \} = 0,\\ \{ \hat{e}\_k^\dagger, \hat{e}\_l \} = \delta\_{k,l} \} \,\tag{82}$$

6 with rule of transformation in an inherent context of the space-time with Hamiltonian 7 (*transforming each particle around of the source that produces this transformation*):

$$\hat{H} = E\_{\text{Fermi surface}} + \sum\_{\mathbf{p},s} (\frac{\hbar^2 k\_{\text{r}}}{m^\*} - \mu > 0) \times \hat{e}\_{\text{p,s}}^{\dagger} \hat{e}\_{\text{p,s}} + \sum\_{\mathbf{p},s} (\mu - \frac{\hbar^2 k\_{\text{r}}}{m^\*} > 0) \times \hat{e}\_{\text{p,s}}^{\dagger} \hat{e}\_{\text{p,s}'} \tag{83}$$

where *<sup>F</sup>* 9 *k* , is the Fermi sphere radios (their super-electron momentum) given by

$$k\_F = \left(\frac{2\,mE\_{\text{Fermi Star}}}{\hbar^2}\right)^{1/2},\tag{84}$$

11 Their demonstration of the theorem needs more studies and experimental results. This 12 theorem is by way of *conjecture*. But we think that fermionic Fock space of electromagnetic 13 nature can be who can express the phase change in all particles beginning from the structure 14 of metal and transmitting to the immediate ambient space of the metal object (see figure 15 9B)).

17 **Figure 9.** A) The quasi-particle region: holes. The fermionic Fock superconducting space for one 18 particle: observe the two phases of fermion spaces, upper surface corresponds to the holes zone. Of fact 19 this zone is like volcano, since in their interior are holes. The below surface is the free fermions whose 20 behavior is seemed to the Bose-Einstein distribution. B). Structure of the ship transmitting the change 21 phase of the particles that come from of the ship reactor [29, 30]. C) Electro-twistor generated by the 22 magnetic field-superconducting interaction [9]. D) Appearing of the creation operators that shape the 23 wrapping space over structure of the ship. This is defined by a fermionic Fock space, for example under 24 the ship as the impeller twistor [23].

<sup>22</sup>A electromagnetic case is given bythe algebra: ( )\* E H ( , \*) \* \* [ , ] .

**Figure 10.** Electro-anti-gravitational Fields , E*<sup>t</sup>* 4 to levitation (see also figure 11 B)) [18]. Disk (thermal 5 cloud forming the saucer) experiments to magnetic levitation showing the fermionic Fock space in 6 nano-seconds [31] from the orbit-spin interaction.

#### 7 **6. Conclusions**

1

2 3

30 BookTitle

Fock space [10]22 4

15 9B)).

16

24 the ship as the impeller twistor [23].

1 **Theorem V. 1 (F. Bulnes, R. Goborov).** The organized transformation given by [15]

1 2

Fermi Surface

228 Selected Topics in Applications of Quantum Mechanics

*k k k l k l k l kl* 5 *H H ee e e e e e e*

7 (*transforming each particle around of the source that produces this transformation*):

where *<sup>F</sup>* 9 *k* , is the Fermi sphere radios (their super-electron momentum) given by

( ) ( ) , *T T* M M *<sup>n</sup>* 2 (81)

3 to *electro-anti-gravitational effect* produced from superconductivity must have a fermionic

6 with rule of transformation in an inherent context of the space-time with Hamiltonian

, , <sup>ˆ</sup> ( 0) ˆ ˆ ( 0) , ˆ ˆ \* \* *F F*

11 Their demonstration of the theorem needs more studies and experimental results. This 12 theorem is by way of *conjecture*. But we think that fermionic Fock space of electromagnetic 13 nature can be who can express the phase change in all particles beginning from the structure 14 of metal and transmitting to the immediate ambient space of the metal object (see figure

17 **Figure 9.** A) The quasi-particle region: holes. The fermionic Fock superconducting space for one 18 particle: observe the two phases of fermion spaces, upper surface corresponds to the holes zone. Of fact 19 this zone is like volcano, since in their interior are holes. The below surface is the free fermions whose 20 behavior is seemed to the Bose-Einstein distribution. B). Structure of the ship transmitting the change 21 phase of the particles that come from of the ship reactor [29, 30]. C) Electro-twistor generated by the 22 magnetic field-superconducting interaction [9]. D) Appearing of the creation operators that shape the 23 wrapping space over structure of the ship. This is defined by a fermionic Fock space, for example under

<sup>22</sup>A electromagnetic case is given bythe algebra: ( )\* E H ( , \*) \* \* [ , ] .

**p p** 8 (83)

*s s k k H E e e e e*

 

† †† † , { { , } 0,{ , } 0,{ , } }, ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ

2 2

*m m*

Fermi Surface 2

<sup>2</sup> , *<sup>F</sup> mE <sup>k</sup>* 10 (84)

† † , , , ,

 

1/2

*s s s s*

**p p p p**

(82)

8 The *sections 4 and 5*, establishes the general conditions to construct the fermionic Fock 9 superconducting space which born from organized transformation of fermions and bosons 10 from the actions of the Lie-QED-algebra E H , [3, 18, 32] of this way in the next section we 11 establish this and also their transformation to obtain the two phases signed in the 12 Hamiltonian (83) (see figure 9 A)).

13 Important is do note, that the energy of a quasi-particle depends of the distribution of all the 14 other quasi-particles that haves in the system. Simplified, is can to say, that a free electron, 15 or "naked electron", that is to say, (*outside of interactions*); have as difference with a quasi-16 particle, or electron with interactions the different masses. The principal effect of the 17 interaction between electrons in normal state consists in change the effective mass of the 18 electron; for example, the specific heat of a Fermi liquid have formally the same expression 19 that the of a ideal Fermi gas changing so only the effective mass, *m*\*, for the mass of the free 20 electron *m*.

21 The fermionic Fock space is a useful topological space picture to describe the interaction 22 obtained for electrons and their link-wave as fonon (boson to describe the quasi-particles) 23 inside the Fermi fluid. In the next work we need to demonstrate this interaction and related 24 with the proposed Hamiltonian in (83). The following region (figure 9 A)), must be the free 25 fermions that realize the transformation in the particles of immediate space moving their 26 spins. Of fact, this change of phase happens inside the superconductor material where the 27 superconductor phenomena happen.

1 One careful analysis establish certain relations between orbit-spin, saying orbit to the two 2 surfaces that begin in certain step of superconducting process as Rasbha Effect (see figure 4 3 a)), from the Majorana field produced for this coupling in intermediate state of semi-4 conductor-superconductor. This can help to design an inter-phase with the reactor of a 5 vehicle of magnetic levitation [23].

#### 6 **Appendices**

32 BookTitle

#### 7 **A. Variation principles to EM in superconductors**

**Lemma. A. 1. (Bulnes, F)** [24, 33]. The energy of action given by , *<sup>M</sup>* 8 *E* of the *O* , all like a diamagnetic given by , *<sup>M</sup>* 9 satisfies to all vector magnetic potential *A*, ( 1 form of the 10 corresponding Maxwell equations to the levitation: *rotB j* 4 (1/ ), *c* and *B* **0**, 33), the 11 following Hamiltonian

$$H(A, \mathfrak{I}\_M) = \int\_{\mathcal{O}} \left\{ L\_M - H^2 / 8\pi \right\} dV\_{\prime} \tag{85}$$

13 *Proof.* [33, 34].

#### 14 **B. Fermionic Fock space**

15 Now suppose there is an infinite but discrete set of fermionic modes , corresponding to some 1 particle quantum states , with wave functions ( ) 16 . In the vector , we 17 include the spin and other non-spatial quantum numbers into ( , , , , ). *x y z spin etc* In this 18 case, the fermionic Hilbert space is

$$\mathcal{F} = \mathop{\otimes}\limits\_{a} \mathcal{H}\_{\text{mode}} \text{ (spanning } \left| n\_a = 0 \right\rangle \text{ and } \left| n\_a = 1 \right\rangle \text{)},\tag{86}$$

20 which has infinite dimension and we may interpret this as a Fock space or arbitrary number 21 of identical fermions. This is our space of study to organized transformations that we 22 require [15, 35, 36] ].

#### 23 **Author details**

24 Francisco Bulnes\*

25 \*Address all correspondence to: francisco.bulnes@tesch.edu.mx

26 Research Depatment in Mathematics and Engineering, TESCHA, Federal Highway Mexico-27 Cuautla Tlapala "La Candelaria" Chalco, State of Mexico, Mexico

#### 1 **References**

32 BookTitle

6 **Appendices** 

11 following Hamiltonian

14 **B. Fermionic Fock space** 

22 require [15, 35, 36] ].

23 **Author details** 

24 Francisco Bulnes\*

some 1 particle quantum states

18 case, the fermionic Hilbert space is

13 *Proof.* [33, 34].

5 vehicle of magnetic levitation [23].

230 Selected Topics in Applications of Quantum Mechanics

7 **A. Variation principles to EM in superconductors** 

10 corresponding Maxwell equations to the levitation: *rotB j* 4 (1/ ),

15 Now suppose there is an infinite but discrete set of fermionic modes

, with wave functions ( ) 16

17 include the spin and other non-spatial quantum numbers into

25 \*Address all correspondence to: francisco.bulnes@tesch.edu.mx

27 Cuautla Tlapala "La Candelaria" Chalco, State of Mexico, Mexico

19 *F H n n* (86)

20 which has infinite dimension and we may interpret this as a Fock space or arbitrary number 21 of identical fermions. This is our space of study to organized transformations that we

26 Research Depatment in Mathematics and Engineering, TESCHA, Federal Highway Mexico-

1 One careful analysis establish certain relations between orbit-spin, saying orbit to the two 2 surfaces that begin in certain step of superconducting process as Rasbha Effect (see figure 4 3 a)), from the Majorana field produced for this coupling in intermediate state of semi-4 conductor-superconductor. This can help to design an inter-phase with the reactor of a

**Lemma. A. 1. (Bulnes, F)** [24, 33]. The energy of action given by , *<sup>M</sup>* 8 *E* of the *O* , all like a diamagnetic given by , *<sup>M</sup>* 9 satisfies to all vector magnetic potential *A*, ( 1 form of the

<sup>2</sup> (, ) /8 , *<sup>M</sup> <sup>M</sup>*

mode spanning and ( 0 1 ), *<sup>a</sup>*

*O H A L H dV*

12 (85)

 

*c* and *B* **0**, 33), the

. In the vector

( , , , , ). *x y z spin etc* In this

, corresponding to

, we


34 BookTitle


1 [30] Bychkov, Y. A. and Rashba, E. I., Jour. Phys. C. 17, 6039 (1984).

34 BookTitle

3 [16] Choy, T. S., 3D Fermi surface Site,

232 Selected Topics in Applications of Quantum Mechanics

13 Wiley-Interscience, USA.

21 Vol. 1.

24 1, pp22-27.

4 http://www.phys.ufl.edu/~tschoy/r2d2/Fermi/Fermi.html

11 [19] Hossenfelder, S., Anti-gravitation, *Elsevier Science*, 2006.

18 *Spintronics*, Vol. 2, no. 4, USA, 2013, pp19-29.

26 Gas".*Physical Review* 104 (4): 1189–1190.

29 analysis). New York: Academic Press, 1972.

35 the-space-time-through-path-integrals-and-t

15 [22] Sun, B. Y., et al. Phys Lett. B683 (2010)pp134-139, arXiv:0911.2559 16 [23] Verkelov, I., Goborov, R, and Bulnes, F., "Fermionic Fock Space in

1 [15] Bulnes, F (2013) Mathematical Nanotechnology: Quantum Field Intentionality. 2 *Journal of Applied Mathematics and Physics*, 1, 25-44. doi: 10.4236/jamp.2013.15005.

5 [17] Llano, M., "Unificación de la Condensación de Bose-Einstein con la Teoría BSC de 6 Superconductores," Rev. Ciencias Exactas y Naturais, 2003;vol. 5, no. 1, pp. 9-21. 7 [18] F. Bulnes and A. Álvarez, "Homological Electromagnetism and Electromagnetic 8 Demonstrations on the Existence of Superconducting Effects Necessaries to 9 Magnetic Levitation/Suspension," *Journal of Electromagnetic Analysis and*  10 *Applications*, Vol. 5 No. 6, 2013, pp. 255-263. doi: 10.4236/jemaa.2013.56041.

12 [20] Emch, G. (1972), *Algebraic Methods in Statistical Mechanics and Quantum Field Theory*,

14 [21] Dixmier, J. (1969), *Les C\*-algèbres et leurs représentations*, Gauthier-Villars, France.

17 Superconducting Phenomena and their Applications," *Journal on Photonics and* 

19 [24] Bulnes, F., *Analysis of prospective and development of effective technologies through*  20 *integral synergic operators of the mechanics,* CCIA 2008, CIMM 2008, C. CIMM, ed.,

25 [26] Cooper, L (November 1956). "Bound Electron Pairs in a Degenerate Fermi

27 [27] Bogoliubov, N., *On the theory of superfluidity*, J. Phys. (USSR), 11, p. 23 (1947). 28 [28] Simon, B. and Reed, M. Mathematical methods for physics, Vol. I (functional

30 [29] F. Bulnes (2013). Quantum Intentionality and Determination of Realities in the 31 Space-Time Through Path Integrals and Their Integral Transforms, Advances in 32 Quantum Mechanics, Prof. Paul Bracken (Ed.), ISBN: 978-953-51-1089-7, InTech, 33 DOI: 10.5772/53439. Available from: http://www.intechopen.com/books/advances-34 in-quantum-mechanics/quantum-intentionality-and-determination-of-realities-in-

22 [25] Mahmoud, J, "Spintronics in Devices: A Quantum Multi-Physics Simulation of the 23 Hall Effect in Superconductors," Journal on Photonics and Spintronics, Vol. 2. No.


**Selected Topics in Applications of Quantum Mechanics**

## **The Nuclear Mean Field Theory and Its Application to Nuclear Physics**

M.R. Pahlavani

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60517

#### **1. Introduction**

Our universe consists of substance. Atoms and molecules are basic components of material. Each atom contains a nucleus which is spread in a small area of atom, and electrons. Also, a nucleus contains Neutrons and protons. It is well known today that electrons in atom and Neutrons and protons in the nucleus are interacting together through different forces. It is clear today that the source of different interactions are composed of four basic forces of the universe, namely gravitational, coulomb, strong and weak nuclear interactions.

In quantum mechanics, to study a particle, it is necessary to have knowledge about its interaction with the surrounding media. The Schrödinger equation is a second-order differ‐ ential equation that is solved to obtain energy spectrum and wave functions of a particle in quantum mechanics. For a many-body system such as atom or nucleus, it is not possible to solve a set of Schrödinger equations to obtain energy spectrum and wave functions analyti‐ cally. Therefore in such situations, it is necessary to use an average potential which is a mean potential of all interacting forces acting upon a single particle. Then the Schrödinger equation is should be solve for a single particle. This procedure is called the mean field method [1, 2, 3].

To review this method consider a system consisting of N identical interacting particles. The Hamiltonian of system composed of kinetic energy, *T* , and potential energy , *V* , is defined as

$$H = T + V = \sum\_{i=1}^{N} t\left(r\_i\right) + \sum\_{i,j<1 \atop i$$

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

where *mN* is the mass of each particle, and *ri* denotes the coordinates of particle *i*. A summed single particle potential energy, so far undefined, can be added and subtracted of the Hamil‐ tonian to obtain the following relation,

$$\mathbf{H} = \left[\mathbf{T} + \sum\_{l=1}^{N} \mathbf{v}(\mathbf{r}\_{l})\right] + \left[\mathbf{V} - \sum\_{l=1}^{N} \mathbf{v}(\mathbf{r}\_{l})\right] = \left. H\_{\text{MF}} + V\_{\text{RES}} \tag{2}$$

where

$$\begin{aligned} \boldsymbol{H}\_{\text{MF}} &= \boldsymbol{T} + \sum\_{l=1}^{N} \mathbf{v} \left( \mathbf{r}\_{i} \right) \equiv \boldsymbol{T} + \boldsymbol{V}\_{\text{MF}} = \\ \sum\_{l=1}^{N} \left[ \boldsymbol{t} \left( \boldsymbol{r}\_{i} \right) + \mathbf{v} \left( \mathbf{r}\_{i} \right) \right] &= \sum\_{i=1}^{N} \boldsymbol{h} \left( \boldsymbol{r}\_{i} \right) \end{aligned} \tag{3}$$

is the mean field Hamiltonian of the system and

$$\mathbf{V}\_{\text{RES}} = \mathbf{V} - \sum\_{l=1}^{N} \mathbf{v}(\mathbf{r}\_{l}) = \sum\_{\substack{l, \boldsymbol{\cdot} \in \mathbf{1} \\ \boldsymbol{\cdot} \neq \boldsymbol{\cdot}}}^{N} \mathbf{v}(\mathbf{r}\_{l}, \boldsymbol{r}\_{\boldsymbol{\cdot}}) - \sum\_{l=1}^{N} \mathbf{v}\left(\mathbf{r}\_{l}\right), \tag{4}$$

is the mean residual interaction. It should be noted that the residual interaction is related to the strength of the actual interaction and can be reduced if the mean field potential is close to the actual potential of the system.

Actually, the mean field method is an approximation in which each particle of system moves under an external field generated by the remaining *N* −1 particles. This mean potential, *VMF* , can be considered as an average of all possible interactions of nucleons during the short time interval *ΔT* , between the selected nucleon and its surrenders,

$$\mathcal{V}\_{\rm MF} = \sum\_{l=1}^{N} \mathbf{v}(\mathbf{r}\_{l})\_{\prime} \cdot \mathbf{v}\left(\mathbf{r}\_{l}\right) = \frac{1}{\Delta T} \int\_{T}^{T \ast \Delta T} dt \sum\_{j=l \atop j \neq l}^{N} v\left(r\_{l}(t) \cdot r\_{j}(t)\right). \tag{5}$$

It is important to know that the time average idea was considered only for clearance of the subject and not applicable in practice unless one studies the thermo-dynamical behavior of nucleus.

Therefore the idea of using mean field theory capable of reducing many particles interacting system in to a system of non-interacting (quasi-particles) considered in an external field , *VMF* which is the mean potential of possible forces of interaction. The mean field potential is considered such that the stationary Schrödinger equation is solved simply to obtain single particle states and their related energy spectrum. These single-particle states are used to construct the *N* particle wave function as follows.

where *mN* is the mass of each particle, and *ri*

tonian to obtain the following relation,

238 Selected Topics in Applications of Quantum Mechanics

where

nucleus.

denotes the coordinates of particle *i*. A summed

(3)

single particle potential energy, so far undefined, can be added and subtracted of the Hamil‐

() () *HMF*

( )

*i*

=+ º+ =

*MF MF*

*v r*

1

=

*i*

*N*

*i*

interval *ΔT* , between the selected nucleon and its surrenders,

1

*MF i i i*

is the mean field Hamiltonian of the system and

the actual potential of the system.

å

*N*

*H T TV*

1 1 ( )

*v r*

= =

*V vr r*

é ù + º ë û

å å

( ) ( )

*t r h r*

*i*

*i i i*

1 1 , 1 ( ) ( , ) , *N*

is the mean residual interaction. It should be noted that the residual interaction is related to the strength of the actual interaction and can be reduced if the mean field potential is close to

Actually, the mean field method is an approximation in which each particle of system moves under an external field generated by the remaining *N* −1 particles. This mean potential, *VMF* , can be considered as an average of all possible interactions of nucleons during the short time

( ) ( ( ) )

¹

*T j i j i* *i j*

*dt v r t r t*

*V vr v r* (5)

<sup>1</sup> ( ), , ( ) . *T T N*

+D

It is important to know that the time average idea was considered only for clearance of the subject and not applicable in practice unless one studies the thermo-dynamical behavior of

Therefore the idea of using mean field theory capable of reducing many particles interacting system in to a system of non-interacting (quasi-particles) considered in an external field , *VMF* which is the mean potential of possible forces of interaction. The mean field potential is

*T*

= =

= = å å <sup>D</sup> <sup>ò</sup> *<sup>N</sup>*

= = < <

*i j i j*

=- = - åå å *N N RES i i i i*

*i j*

*N*

*i i RES*

*H T vr V vr V* (2)

( )

*V vr v r* (4)

1 1

= = é ùé ù =+ +- = + ê úê ú ë ûë û å å *N N*

*i i*

The corresponding *N* -particle Schrödinger equation is used to obtain solutions of the meanfield Hamiltonian *HMF*

$$H\_{\rm MF} \Psi\_0 \left( r\_{1'} r\_{2'} ... r\_N \right) = E \Psi\_0 \left( r\_{1'} r\_{2'} ... r\_N \right). \tag{6}$$

The wave function Ψ0( *r*1, *r*2, ....*rN* ) can be separated by using the ansatz single particle wave functions

$$
\Psi\_0\left(r\_1, r\_2, \ldots, r\_N\right) = \left.\phi\_{a\_1}\left(r\_1\right)\phi\_{a\_2}\left(r\_2\right)\ldots\phi\_{a\_N}\left(r\_N\right)\right.\tag{7}
$$

Substituting this ansatz in to the Schrödinger equation (6) yields *N* identical one-particle Schrödinger equations

$$h\left(r\right)\phi\_{\alpha}\left(r\right) = \varepsilon\_{\alpha}\phi\_{\alpha}\left(r\right),$$

$$h\left(r\right) = t\left(r\right) + v\left(r\right) = -\frac{\hbar^2}{2m\_N}\nabla^2 + v\left(r\right). \tag{8}$$

With the quasi-particle energy, *εα*, that is satisfies the following condition

$$E = \sum\_{i=1}^{N} \epsilon\_a.\tag{9}$$

The wave function of the many- body system is thus an anti symmetric product of singleparticle wave functions which are one-particle wave functions of an external potential well. In summary the mean field theory reduces the complicated many-body problem in to a simple one-particle system.

The main idea in this approach is to determine the mean field potential or in particular, an appropriate mean field potential in which the residual interactions between the quasi-particles should be optimal. To do so, one may seek an optimal set {*ϕα* (*r*)} of one-quasi-particle states. This is a Rayleigh-Ritz variational approximation in which the variation *ϕα* (*r*)→*ϕα* (*r*) + *δϕα* (*r*) of the single-particle orbital is minimized

$$\begin{aligned} E\_{\rm gs} &= \left\langle \Psi\_0 \middle| H \middle| \Psi\_0 \right\rangle \\ H &= T + \left. V\_{\rm MF} + \left. V\_{\rm RES} \right| \right. \end{aligned} \tag{10}$$

As a starting point, one may construct an ansatz wave function. It is customary to use a product of single particle wave functions as Eigen function of the system,

$$\Psi\_0\left(r\_1, r\_2, \dots, r\_N\right) = \prod\_{i=1}^N \Phi\_{a\_i}\left(r\_i\right) \tag{11}$$

It is an anti-symmetrized product ansatz wave function following the Hartree-Fock method and is called the Slater determinant of the given single particle states

$$\Psi\_{\boldsymbol{0}}\left(\boldsymbol{r}\_{1},\boldsymbol{r}\_{2},...,\boldsymbol{r}\_{N}\right) = \mathbb{C}\left[\prod\_{i=1}^{N}\phi\_{a\_{i}}\left(\boldsymbol{r}\_{i}\right)\right].\tag{12}$$

Here Ψ0( *r*1, *r*2, ....*rN* ) is an anti-symmetric wave function. Also *C* is the normalization constant. For instance, consider a three-particles system with its single-particle Eigen states labeled 1, 2, and 3. Then the normalized anti-symmetric state, or the Slater determinant, is

$$
\Psi\_{\boldsymbol{\alpha}}\left(\boldsymbol{r}\_{1},\boldsymbol{r}\_{2},\ldots,\boldsymbol{r}\_{N}\right) = \frac{1}{\sqrt{6}} \begin{vmatrix}
\boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{1}}\left(\boldsymbol{r}\_{1}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{1}}\left(\boldsymbol{r}\_{2}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{1}}\left(\boldsymbol{r}\_{3}\right) \\
\boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{2}}\left(\boldsymbol{r}\_{1}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{2}}\left(\boldsymbol{r}\_{2}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{2}}\left(\boldsymbol{r}\_{3}\right) \\
\boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{3}}\left(\boldsymbol{r}\_{1}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{3}}\left(\boldsymbol{r}\_{2}\right) & \boldsymbol{\phi}\_{\boldsymbol{\alpha}\_{3}}\left(\boldsymbol{r}\_{3}\right) \\
\end{vmatrix}.\tag{13}
$$

The energy *E* of the system has to be varied under the constraint that the normalization of Ψ<sup>0</sup> is preserved, i.e. Ψ0|Ψ<sup>0</sup> =1. This leads to the constrained variational problem,

$$\delta \left( \frac{\left\langle \Psi\_0 \middle| H \middle| \Psi\_0 \right\rangle}{\Psi\_0 \middle| \Psi\_0} \right) = 0\_\prime$$

which can be transformed in to an unconstrained one by minimizing the energy for normalized wave function, Ψ0( *r*1, *r*2, ....*rN* ). After performing the variation, the single-particle energy, *εα* , is can also be obtained.

One powerful method to address such uncertainties is the following Hartree consistent equation [4,5],

$$\begin{aligned} -\frac{\hbar^2}{2m\_N} \nabla^2 \phi\_a \left( r \right) + V\_{H(F)} \left( \left< \phi\_i \right> \right) \phi\_a \left( r \right) &= \epsilon\_a \phi\_a \left( r \right), \\ \mathbf{i} &= \mathbf{1}, \mathbf{2}, \dots \mathbf{N}, \ \mathbf{a} = \mathbf{1}, \mathbf{2}, \dots \mathbf{z} \end{aligned} \tag{14}$$

This equation is like the Schrödinger equation except that the simple potential term, V (*r*), is replaced with a function of unknown wave function,

$$V\left(r\right) = V\_{H(F)}\left(\left\{\phi\_i(r)\right\}\right)$$

As a starting point, one may construct an ansatz wave function. It is customary to use a product

1

1

= é ù <sup>=</sup> ê ú ë û

Here Ψ0( *r*1, *r*2, ....*rN* ) is an anti-symmetric wave function. Also *C* is the normalization constant. For instance, consider a three-particles system with its single-particle Eigen states labeled 1, 2,

a f

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

*rrr*

123

= (13)

123

*rrr*

11 1 22 2 33 3

aaa

fff

fff

fff

aaa

aaa

*N N i i*

=

It is an anti-symmetrized product ansatz wave function following the Hartree-Fock method

a f

<sup>=</sup> Õ (11)

Õ (12)

*N N i i rr r r*

0 12 ( ) ( )

0 12 ( ) ( )

*rr r C r*

Ψ , ,.... . *<sup>i</sup>*

0 12 123

<sup>1</sup> Ψ , ,.... . <sup>6</sup> *<sup>N</sup>*

*rr r r r r*

The energy *E* of the system has to be varied under the constraint that the normalization of Ψ<sup>0</sup>

0 0 0 0

which can be transformed in to an unconstrained one by minimizing the energy for normalized wave function, Ψ0( *r*1, *r*2, ....*rN* ). After performing the variation, the single-particle energy, *εα* ,

One powerful method to address such uncertainties is the following Hartree consistent

*rV r r*

a

 aa

ò

 ¥  f

(14)

( ) ({ }) ( ) ( ) <sup>2</sup>

 ff

a

This equation is like the Schrödinger equation except that the simple potential term, V (*r*), is

( ) , 2 1, 2, ...... , 1, 2, ..... *HF i*

0, Ψ |Ψ

Ψ Ψ

*H*

æ ö ç ÷ <sup>=</sup> ç ÷ è ø

and 3. Then the normalized anti-symmetric state, or the Slater determinant, is

is preserved, i.e. Ψ0|Ψ<sup>0</sup> =1. This leads to the constrained variational problem,

d

( )

2

a

f

*i N*


= =

*N*

replaced with a function of unknown wave function,

*m*

h

is can also be obtained.

equation [4,5],

Ψ , ,.... *<sup>i</sup>*

of single particle wave functions as Eigen function of the system,

240 Selected Topics in Applications of Quantum Mechanics

and is called the Slater determinant of the given single particle states

Here, the Hartree mean field potential, *V <sup>H</sup>* (*<sup>F</sup>* ) , is different from, *V HF* , Hartree-Fock mean field.

The differential equation (14) is nonlinear and therefore, much more difficult to solve than the regular Schrödinger equation. The solution can only be carried using consistent iteration method. In this procedure, one can start using a complete set of guessed single-particle states {*ϕi* <sup>0</sup> (*r*)}, *<sup>i</sup>* =1, ....*<sup>N</sup>* to calculate the initial potential term, *<sup>V</sup> <sup>H</sup>* (*<sup>F</sup>* ) (0) . In the next step, the equation for a complete set of new wave functions {*ϕα* (1) (*r*)} *α* =1, .....*∞* is solved to obtain Eigen energies *εα* (1) . The procedure is then repeated with new Eigen function *ϕα* (1) (*r*) to obtain the new potential , *V <sup>H</sup>* (*<sup>F</sup>* ) (1) . This approach can be depicted through the following schematic diagram,

$$\begin{array}{c} \phi\_l^0 \left( r \right) \to \ V\_{H(\mathcal{F})}^{\left( 0 \right)} \to \epsilon\_a^{\left( 1 \right)} \\\\ \epsilon\_a^{\left( 1 \right)} \xrightarrow{\phi\_l^{\left( 1 \right)}} V\_{H(\mathcal{F})}^{\left( 1 \right)} \to \dots \to \phi\_a^{\left( \begin{smallmatrix} 1 \end{smallmatrix} \right)} , \epsilon\_a^{\left( \begin{smallmatrix} 1 \end{smallmatrix} \right)} . \end{array} \tag{15}$$

This procedure is repeated to achieve self-consistency for wave functions (or Eigen energies). This means that after each loop the resultant wave function or Eigen energies compared with the starting wave function or Eigen energies and when their difference becomes less than a given preset limit, i.e.

$$\left\| \phi\_a^{(n-1)} - \phi\_a^{(n)} \right\| < present \text{ limit}\_{\omega}$$

the procedure is repeated, otherwise, it will be automatically terminated. Where the ... denotes the norm.

The results of each run, contain a self-consistent mean field, *V <sup>H</sup>* (*<sup>F</sup>* ) (*r*), the Eigen state, *ϕα* (*r*), and its associated Eigen energies , *εα*, are all simultaneously generated. We may also note that for a finite potential-well, there will be, in addition to the bound states, an infinite number of unbound states.

In our discussions, the generated mean-field potential is a central one, that is only a function of *r*. Central mean field potentials describe only systems with spherical symmetry such as spherical nuclei or atoms. This is because of natural real forces that are conservative and satisfy the conservation of energy.

In some convenient way to avoid self-consistency loops, a phenomenological potential like a simple square well with finite depth, simple harmonic oscillator well and complicated Woods-Saxon with considerable parameters that can be determined using the fit of potential with experimental data, is introduced.

#### **2. Applications of mean-field theory in nuclear physics**

Over the years after Rutherford's valuable experiments that suggest nuclei for atom, many theoretical and experimental attempts have been done to obtain knowledge about the stability of nuclei. It is clear today that a nucleus of mass number *A*, Neutron number *N* and proton number (atomic number) *Z*, consists of *A* strongly interacting nucleons (protons and neutrons in the nucleus without considering their different properties called nucleon.). In addition to the strong nuclear force that is responsible for nuclear stability, the protons also sense the attractive coulomb potential because of their charge. In regular nuclear physics, the protons and neutrons are considered the point particles without any internal structure. This is an excellent approximation when the aim is to study nuclear structure at low energies. In such approach, the nuclear forces are considered a central attractive force with proper specifications like independence of charge and low range. Note that in advance models of nuclear physics such as the Yukawa Meson exchange model, it is believed that nucleons constructed quarks and interact together through the meson exchange mechanism in the base of the particle physics lows. The lightest nucleus is Deuterium with one neutron and one proton. The interaction of nucleons in the nucleus can be studied both theoretically and practically using simple Deuteron nucleus. This two-nucleon system is described by two-body interaction matrix elements, without a detailed account of the methods used to obtain them. On the other hand, the *A* - nucleons nucleus in quantum mechanics using the Schrödinger equation is not a solvable problem analytically at least for *A*>10. Therefore, one has to look for a reasonable approximate method to solve this many-body problem consisting of strongly interacting nucleons. A powerful approximation is to convert such many-body system in to a noninteracting system of quasi-particles using a suitable external mean field potential. The remaining interactions, called residual interaction, can be treated as a perturbation potential in the base of perturbation approximation. As discussed earlier, the transformation of system of particles in to quasi-particles is not simple, and its success depends on the nuclear system under consideration.

As mentioned above, a conventional approach is to select a particular type of mean field potential to avoid the steps leading to self-consistency. The selected mean field potential and considered remaining residual interactions as approximations produce the preciseness of the obtained results. The simplest custom potential is the three-dimensional harmonic oscillator potential well

$$\mathbf{V}\_{HO}(\mathbf{r}) = \mathbf{-V}\_1 + \mathbf{k}\mathbf{r}^2 = \mathbf{-V}\_1 + \frac{1}{2}m o\rho^2 r^2 \tag{16}$$

where *V*1 and *k* are the parameters to be fitted to the practical data for best result. A common, more realistic choice is the Woods–Saxon potential [6]

$$v\_{\rm WS}(r) = \frac{-V\_0}{1 + e^{(r - R)}\zeta\_a}.$$

where *V*0, *R* and *a* are the nuclear potential depth, the nuclear radius, and the surface diffuse‐ ness, respectively. They are parameterized as follows,

$$V\_0 = \left(51 \pm 33 \frac{N-Z}{A}\right) \text{MeV}, \ R = r\_0 A^{\frac{1}{3}} = 1.27 A^{\frac{1}{3}} \text{ fm}, a = 0.67 \text{ fm}.$$

The + and – signs are considered for protons and neutrons, respectively. In the case when there is no distinction between protons and neutrons a suitable average value of *V*<sup>0</sup> = 57 *MeV* can be used for nucleons.

The Woods–Saxon potential, *vWS* , is a suitable choice for the mean field potential however it is a complicated function of, *r*, and it is not an analytically solvable one. To overcome this problem, it is possible to select the proper three- dimensional oscillator potential with energy quantum, ℏ*ω* and depth, *V*1. The energy difference of levels, ℏ*ω*, and depth, *V*1, can be obtained with a best fit to the Woods–Saxon potential, *vWS* , as a function of, *V*0, *R* and *a* as the nuclear potential depth, nuclear radius, and surface diffuseness of the Woods–Saxon potential, respectively. The wave functions and energy spectrum of equivalent harmonic oscillator potential agreed well with the Woods–Saxon potential ones especially near the bottom of the wells in low energies. The difference of these potentials increases when the potential ap‐ proaches zero. Actually the major difference of these potentials is that the harmonic oscillator potential varies more sharply than the Woods–Saxon one near the surface of the nucleus.

#### **2.1. The spin–orbit interaction**

**2. Applications of mean-field theory in nuclear physics**

242 Selected Topics in Applications of Quantum Mechanics

under consideration.

potential well

Over the years after Rutherford's valuable experiments that suggest nuclei for atom, many theoretical and experimental attempts have been done to obtain knowledge about the stability of nuclei. It is clear today that a nucleus of mass number *A*, Neutron number *N* and proton number (atomic number) *Z*, consists of *A* strongly interacting nucleons (protons and neutrons in the nucleus without considering their different properties called nucleon.). In addition to the strong nuclear force that is responsible for nuclear stability, the protons also sense the attractive coulomb potential because of their charge. In regular nuclear physics, the protons and neutrons are considered the point particles without any internal structure. This is an excellent approximation when the aim is to study nuclear structure at low energies. In such approach, the nuclear forces are considered a central attractive force with proper specifications like independence of charge and low range. Note that in advance models of nuclear physics such as the Yukawa Meson exchange model, it is believed that nucleons constructed quarks and interact together through the meson exchange mechanism in the base of the particle physics lows. The lightest nucleus is Deuterium with one neutron and one proton. The interaction of nucleons in the nucleus can be studied both theoretically and practically using simple Deuteron nucleus. This two-nucleon system is described by two-body interaction matrix elements, without a detailed account of the methods used to obtain them. On the other hand, the *A* - nucleons nucleus in quantum mechanics using the Schrödinger equation is not a solvable problem analytically at least for *A*>10. Therefore, one has to look for a reasonable approximate method to solve this many-body problem consisting of strongly interacting nucleons. A powerful approximation is to convert such many-body system in to a noninteracting system of quasi-particles using a suitable external mean field potential. The remaining interactions, called residual interaction, can be treated as a perturbation potential in the base of perturbation approximation. As discussed earlier, the transformation of system of particles in to quasi-particles is not simple, and its success depends on the nuclear system

As mentioned above, a conventional approach is to select a particular type of mean field potential to avoid the steps leading to self-consistency. The selected mean field potential and considered remaining residual interactions as approximations produce the preciseness of the obtained results. The simplest custom potential is the three-dimensional harmonic oscillator

> 1 1 <sup>1</sup> V (r) = -V + kr = -V + 2 *HO m r*

> > *v r*

more realistic choice is the Woods–Saxon potential [6]

where *V*1 and *k* are the parameters to be fitted to the practical data for best result. A common,


2 2 2

0 ( ) ( ) . 1 *WS r R*

*V*

*e* -

*a*

w

(16)

Sometimes in 1949, Meyer and independently, Haxel, Jensen, and Swees showed that if in addition to mean field central potential, *VMF* , a non-central potential is included in the Schrödinger equation, all closed shell nucleon numbers can be obtained successfully. These numbers 2, 8, 20, 28, 50, 82, and 126 are called magic numbers because the origin of these numbers was not known at that time. The Woods-Saxon or its equivalent harmonic oscillator central potential is not able to reproduce experimentally observed precise data of the singleparticle structure energies of the nucleus using the mean field approach.

The non-central potential due to the interaction between the spin of nucleons with the angular momentum of orbital that nucleons located on it, is called spin-orbit interaction. As a result of spin-orbit interaction [7, 8], the nuclear energy level for a given *l* (except for *l* =0) is split in to two sublevels. The sublevels are characterized by total angular momentum numbers equal to (*<sup>l</sup>* <sup>+</sup> <sup>1</sup> <sup>2</sup> ) and (*<sup>l</sup>* <sup>−</sup> <sup>1</sup> <sup>2</sup> ) corresponding to whether the spin is parallel or anti-parallel to the orbital angular momentum. Each sublevel with spin *j* accommodates ( 2 *j* + 1 ) neutrons or protons. The same interaction with a different structure is observed in atoms with a different sign as in the nucleus.

Consider that the harmonic oscillator central potential is produced only for the first three observed magic numbers 2, 8, and 20. To obtain the remaining numbers 28, 50, 82 and 126, it is necessary to add a spin-orbit interaction potential to the Schrödinger equation.

The origin of the spin–orbit interaction is not the same in atoms and nucleus. The atomic spin– orbit force is due to a well-known electromagnetic interaction, and the scale of energy separation is in the order of milli-electronvolts, while the energy difference of sublevels separated because of the nuclear spin-orbit interaction is in the order of million electronvolts and its origin is not well understood yet. In most cases, this force is considered phenomeno‐ logically. For the spin–orbit term, we use [9]

$$\left| \upsilon\_{LS}(r) = \upsilon\_{LS}^{0}\left(\frac{r\_0}{\hbar}\right)^2 \frac{1}{r} \frac{d}{dr}\right| \frac{1}{1 + e^{-(r-R)}\bigvee\_a} \tag{17}$$

The second pair of parentheses guarantees that the derivative does not operate on the wave function when substituted in the radial Schrödinger equation. The *r* dependence of this interaction arises from its central nature.

The derivative part of this potential is often neglected for simplicity and *vLS* (*r*) is replaced by a constant; however, to obtain precise results, the radial part should be considered. We have

$$\upsilon\_{LS}^{0} = 0.44V0.$$

To obtain the strength of the spin-orbit part, we use

$$\begin{aligned} J^2 &= \left(L + S\right)^2\\ J^2 &= L^2 + S^2 + 2LS\_{\prime\prime} \end{aligned}$$

and its expectation value for the nuclear wave equation made

$$\begin{aligned} &< LS> = \frac{\hbar^2}{2} \left[ j(j+1) - l(l+1) - \frac{1}{2} (\frac{1}{2} + 1) \right], \\ &< LS> = \frac{\hbar^2}{2} l \quad \text{for} \quad j=l+\frac{1}{2}, \\ &< LS> = \frac{\hbar^2}{2} (-l-1) \quad \text{for} \quad j=l-\frac{1}{2}. \end{aligned}$$

In addition to the mean field plus spin–orbit interaction, protons in nuclei interact together via the coulomb force, which is defined by the following relation, considering nuclei as a sphere with a constant charge density [10]

$$V\_{\mathbb{C}}(r) = \frac{Ze^2}{4\pi\varepsilon\_0} \left\{ \frac{3 - \left(\frac{r}{R}\right)^2}{2R} \right. \qquad r \le R \quad , \quad \frac{1}{r} \quad \quad r \ge R \tag{18}$$

To obtain the energy spectrum and wave functions for neutrons, one needs to solve the radial Schrödinger equation for the Woods-Saxon and spin-orbit potentials. Such second-order differential equation cannot be solved analytically. To solve this complicated differential equation, it is necessary to introduce some new variables and use reasonable approximations. By introducing new variable [11] *<sup>y</sup>* <sup>=</sup> <sup>1</sup> <sup>1</sup> <sup>+</sup> exp (*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*) *a* , the Woods–Saxon potential reduces to its simple form *VWS* <sup>=</sup> *<sup>V</sup>*0*<sup>y</sup>* while the spin-orbit term changes to *VLS* <sup>=</sup> (*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>* 2) *<sup>R</sup>*<sup>0</sup> <sup>+</sup> *<sup>a</sup>*ln( <sup>1</sup> *<sup>y</sup>* <sup>−</sup> <sup>1</sup>) . For orbits with small *l*, the Taylor expansion of the <sup>1</sup> *<sup>r</sup>* near *r* =*rm*, is reasonable. According to the definition of variable *y* we have, *<sup>f</sup>* (*y*) <sup>≡</sup> <sup>1</sup> <sup>1</sup> <sup>+</sup> exp (*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*) *a* hence by expanding *<sup>f</sup>* (*y*) around *ym* <sup>≡</sup> <sup>1</sup> 1 + exp (*r <sup>m</sup>* <sup>−</sup> *<sup>R</sup>*0) *a* with 0< *ym* <1, since 0< *<sup>y</sup>* <1, *<sup>y</sup>* <sup>3</sup> and the higher terms are negligible, the radial part of the spin-orbit term can be approximated using [12],

$$\left| -\frac{a}{r} \frac{d}{dr} \middle| \frac{1}{1 + \exp\frac{(r - R)}{a}} \right| = \frac{1}{r\_w} \left( \mathcal{C}\_0 + \mathcal{C}\_1 y + \mathcal{C}\_2 y^2 \right) , \tag{19}$$

where *C*0, *C*1, and *C*2 are dimensionless coefficients and evaluated as

$$\begin{aligned} C\_0 &= \frac{-ar\_m y\_m + 2a^2 y\_m}{2r\_m^2 (1 - y\_m)}, \\ C\_1 &= 1 + \frac{ar\_m y\_m + 2a^2}{r\_m^2 (1 - y\_m)}, \\ C\_2 &= -1 + \frac{ar\_m (1 + 2y\_m) + 2a^2}{r\_m^2 (1 - y\_m) 2y\_m}. \end{aligned}$$

Likewise, the Taylor expansion is applicable for <sup>1</sup> *r* 2 ,

The origin of the spin–orbit interaction is not the same in atoms and nucleus. The atomic spin– orbit force is due to a well-known electromagnetic interaction, and the scale of energy separation is in the order of milli-electronvolts, while the energy difference of sublevels separated because of the nuclear spin-orbit interaction is in the order of million electronvolts and its origin is not well understood yet. In most cases, this force is considered phenomeno‐

( )

*e* -

*a*

é ù æ ö ê ú <sup>=</sup> ç ÷ ê ú è ø ê ú <sup>+</sup> ë û <sup>h</sup> (17)

1

2 0 0

*r d*

*r dr*

The second pair of parentheses guarantees that the derivative does not operate on the wave function when substituted in the radial Schrödinger equation. The *r* dependence of this

The derivative part of this potential is often neglected for simplicity and *vLS* (*r*) is replaced by a constant; however, to obtain precise results, the radial part should be considered. We have

0 0.44 0. *LS v V* =

( ) 2 2

22 2 2., *J LS J L S LS* = + =+ +

1 1 . ( 1) ( 1) ( 1) <sup>2</sup> 2 2

< >= + - + - + ê ú

é ù

ë û

(18)

1 1 ( )

*vrv*

*LS LS r R*

logically. For the spin–orbit term, we use [9]

244 Selected Topics in Applications of Quantum Mechanics

interaction arises from its central nature.

with a constant charge density [10]

To obtain the strength of the spin-orbit part, we use

and its expectation value for the nuclear wave equation made

2

3

0

ï ï î

pe

2

h

*LS j j l l*

*L S l for j l*

< >= = +

2

h

<sup>1</sup> . , 2 2

*L S l for j l*

< >= - - = -

2

*Ze <sup>R</sup> V r rR rR*

 *R r* <sup>ì</sup> æ ö <sup>ï</sup> - ç ÷ <sup>ï</sup> è ø = £³ <sup>í</sup>

<sup>1</sup> ( ) , 4 2 *<sup>C</sup>*

*r*

<sup>1</sup> . ( 1) . 2 2

In addition to the mean field plus spin–orbit interaction, protons in nuclei interact together via the coulomb force, which is defined by the following relation, considering nuclei as a sphere

2

h

$$\frac{1}{r^2} \approx \frac{1}{r\_w^2} \left( D\_0 + D\_1 y + D\_2 y^2 \right) / r$$

where *Di* 's, similar to the *Ci* ', are obtained through

$$\begin{aligned} D\_0 &= 1 - \frac{4a}{r\_m \left(1 - \mathcal{Y}\_m\right)} + \frac{3a^2 + ar\_w}{r\_m^2 \left(1 - \mathcal{Y}\_m\right)^2}, \\ D\_1 &= \frac{6a}{r\_m \mathcal{Y}\_m \left(1 - \mathcal{Y}\_m\right)} - \frac{6a^2 + 2ar\_w}{r\_m^2 \mathcal{Y}\_m \left(1 - \mathcal{Y}\_m\right)^2}, \\ D\_2 &= \frac{3a^2 - ar\_m \left(1 - 2\mathcal{Y}\_m\right)}{r\_m^2 \left(1 - \mathcal{Y}\_m\right) 2\mathcal{Y}\_m^2}. \end{aligned}$$

This type of expansion has been widely used for differential equations resulting from the Schrödinger equation with different potentials [12].

By means of these expansions, the spin-orbit term transforms into

$$\text{VLS} = \left(\text{C0} + \text{C1y} + \text{C2y2}\right)\text{V}$$

and the centrifugal term is obtained

$$\text{Vcf:} \newline \text{=(D0+D1y+D2y2)} \newline \text{.}$$

By using these expansions, the spin-orbit term transforms into *VLS* <sup>∝</sup>(*Co* <sup>+</sup> *<sup>C</sup>*1*<sup>y</sup>* <sup>+</sup> *<sup>C</sup>*2*<sup>y</sup>* <sup>2</sup> ) and the centrifugal term is changed to the favorable type *VCF* <sup>∝</sup>(*Do* <sup>+</sup> *<sup>D</sup>*1*<sup>y</sup>* <sup>+</sup> *<sup>D</sup>*2*<sup>y</sup>* <sup>2</sup> ). The substi‐ tution of *VLS* and *VCF* as a function of variable y into the Schrödinger equation transforms this equation in to the following analytically solvable differential equation

$$y\left(1-y\right)\left[y\left(1-y\right)\frac{d^2R\left(y\right)}{d\left y^2}+\left(1-2y\right)\frac{dR\left(y\right)}{d\left y^2}\right]+\cdots\right]$$

$$+\frac{2ma^2}{\hbar^2}\left[V\_{0y}+\left(V\_{LS}^{(0)}\right)\left(C\_o+\left(C\_1y+\left(C\_2y\right)^2\right)\right)\right]$$

$$-\frac{\hbar^2}{2m}\frac{l\left(l+1\right)}{r\_w^2}\left(D\_o+\left(D\_1y+\left(D\_2y\right)^2\right)R\left(y\right)\right)=0$$

$$V\_{LS}^{(0)}=\frac{r\_0^2V\_{LS}^{(0)}}{2ar\_m}\left(j\left(j+1\right)-l\left(l+1\right)-3/4\right)$$

This equation can be transformed into the following simple form,

$$\frac{d}{dy}(1-y)\frac{d^2R(y)}{dy^2} + \left(1-2y\right)\frac{dR(y)}{dy} + \frac{-\varepsilon^2 + \beta^2 y - \gamma^2 y^2}{y\left(1-y\right)}R\left(y\right) = 0\tag{20}$$

Equation (20) can be transformed into the well-known form of hypergeometric differential equation or, alternatively Nikiforo-Avorono (NU) type [13]. The obtained results using the NU method are

#### The Nuclear Mean Field Theory and Its Application to Nuclear Physics http://dx.doi.org/10.5772/60517 247

$$-\varepsilon^{2} = \frac{2ma^{2}}{\hbar^{2}} \left( E + V\_{LS}^{(0)} \right) - \frac{l(l+1)}{r\_{m}^{2}} D\_{0}$$

$$\beta^{2} = \frac{2ma^{2}}{\hbar^{2}} \left( V\_{0} + V\_{LS}^{(0)} C\_{1} - \frac{l(l+1)}{r\_{m}^{2}} D\_{1} \right.$$

$$\gamma^{2} = -\frac{2ma^{2}}{\hbar^{2}} \left( E + V\_{LS}^{(0)} C\_{2} \right) - \frac{l(l+1)}{r\_{m}^{2}} D\_{2}$$

$$R = C \frac{\Gamma\left(2\varepsilon - 1\right)\Gamma\left(-2i\lambda\right)}{\Gamma\left(\varepsilon + \eta^{+} + 1 - i\lambda\right)\Gamma\left(\varepsilon - \eta^{-} - i\lambda\right)} \left[ \left(1 - y\right)^{i\lambda} + \frac{\Gamma\left(2i\lambda\right)\Gamma\left(\varepsilon - \eta^{+} + 1 - i\lambda\right)\Gamma\left(\varepsilon - \eta^{-} - i\lambda\right)}{\Gamma\left(-2i\lambda\right)\Gamma\left(\varepsilon - \eta^{+} + 1 + i\lambda\right)\Gamma\left(\varepsilon - \eta^{+} + i\lambda\right)} \left(1 - y\right)^{-i\lambda} \right]$$

where Γ is the well-known gamma function, and C is the normalization constant. λ, μ, and η' are defined as follows

$$
\vec{\lambda} = \sqrt{\beta^2 + \varepsilon^2 + \gamma^2}
$$

$$
\mu = i\vec{\lambda}
$$

$$
\eta^\dagger = \sqrt{\gamma^2 - 1/4} - 1/2
$$

Note that λ is valid only for the *β* <sup>2</sup> >*ε* <sup>2</sup> + *γ* <sup>2</sup> condition. In a special case where *l=0,* the solution reduces to its simple form. Also, the energy eigenvalues are obtained as a function of z satisfying the following relation

$$-\frac{z}{\sqrt{1-z^2}} = \tan\mathcal{O}\left(z\right),\ z = \sqrt{1-\frac{|E|}{V\_o}}$$

which *Ф* (z) can be evaluated using a graphical method.

$$\begin{aligned} \mathcal{L}\{z\} &= k\_0 R\_0 z + \\ &+ \tan^{-1}\left[\frac{z}{\sqrt{1-z^2 + \left(\frac{1}{k\_0 a}\right)}}\right] - \sum\_{s=0}^{\nu} \left[\tan^{-1}\left(\frac{2k\_0 a}{n} z\right) - \tan^{-1}\left(\frac{z}{\sqrt{1-z^2 + \left(\frac{n}{k\_0 a}\right)}}\right) - \tan^{-1}\left(\frac{z}{\sqrt{1-z^2 + \left(\frac{n+1}{k\_0 a}\right)}}\right)\right] \end{aligned}$$

Note that

This type of expansion has been widely used for differential equations resulting from the

VLS C0 C1y C2y2 , =+ + ( )

Vc.f= D0+D1y+D2 ( y2).

tution of *VLS* and *VCF* as a function of variable y into the Schrödinger equation transforms this

( ) ( ) ( ) ( ) ( )

<sup>2</sup> ' 0 <sup>2</sup> 2 0 1 2

2 1 2


= +- +-

( ) ( ) <sup>2</sup> 2 2 22 <sup>2</sup> 1 1 2 0

Equation (20) can be transformed into the well-known form of hypergeometric differential equation or, alternatively Nikiforo-Avorono (NU) type [13]. The obtained results using the NU

*d R y dR y y y y y <sup>y</sup> R y dy dy y y*

2

*y yy y <sup>y</sup> d y d y*

*ma V V C Cy Cy*

( )

*d R y dR y*

é ù - - +- + ê ú ë û

2 2

2

1

 g- +- + <sup>=</sup> - (20)


( ) ( )

*D Dy Dy R y*

0

( ) ( )

( 1 1 3 / 4)

( )

*o*

*y LS o*

+ + ++

1 1 1 2

) and

). The substi‐

By using these expansions, the spin-orbit term transforms into *VLS* <sup>∝</sup>(*Co* <sup>+</sup> *<sup>C</sup>*1*<sup>y</sup>* <sup>+</sup> *<sup>C</sup>*2*<sup>y</sup>* <sup>2</sup>

the centrifugal term is changed to the favorable type *VCF* <sup>∝</sup>(*Do* <sup>+</sup> *<sup>D</sup>*1*<sup>y</sup>* <sup>+</sup> *<sup>D</sup>*2*<sup>y</sup>* <sup>2</sup>

equation in to the following analytically solvable differential equation

( )

+

1

2

*LS*

2

h h

<sup>2</sup> [

( ) ( )

*m*

2

' 0 0

This equation can be transformed into the following simple form,

( ) ( ) ( ) ( )

method are

*l l*

*m r*

2 0

*LS*

*r V <sup>V</sup> jj ll ar*

*m*

Schrödinger equation with different potentials [12].

and the centrifugal term is obtained

246 Selected Topics in Applications of Quantum Mechanics

By means of these expansions, the spin-orbit term transforms into

$$k = \frac{\mathcal{A}}{a} = \left[\frac{2m}{\hbar^2} \left(V\_0^\cdot - \left|E\right|\right)^{1/2}\right]^{1/2}$$

and

$$k\_0 = \frac{2m}{\hbar^2} \left( V\_0^{'} \right)^{1/2} \text{ . } z = \frac{k}{k\_0} \text{ .}$$

Finally,

$$\left| E \right| = V\_0 \left( 1 - Z^2 \right)$$

The results obtained in this special case are in agreement with the results obtained using other methods [14].

#### **3. Conclusions**

In this chapter we briefly discussed the idea of mean field theory as an improvable approxi‐ mation method for many-body problems of identical particles like atoms and nucleus that cannot be solved analytically. We have shown that for a system of A - nucleons nucleus by considering a suitable potential using this model, one is able to obtain energy spectrum and wave equations. However, the obtained results cannot reproduce the measured nuclear spectroscopy, but one may hope to become successful by considering an accurate potential in the Schrödinger equation.

#### **Author details**

M.R. Pahlavani

Address all correspondence to: m.pahlavani@umz.ac.ir

Department of Nuclear Physics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran

#### **References**


and

248 Selected Topics in Applications of Quantum Mechanics

Finally,

methods [14].

**3. Conclusions**

the Schrödinger equation.

Address all correspondence to: m.pahlavani@umz.ac.ir

delberg Berlin, pages 126, 314 and 438 (1980).

**Author details**

M.R. Pahlavani

**References**

Iran

' 1/2

( ) <sup>2</sup> <sup>0</sup> *EV Z* = -1

The results obtained in this special case are in agreement with the results obtained using other

In this chapter we briefly discussed the idea of mean field theory as an improvable approxi‐ mation method for many-body problems of identical particles like atoms and nucleus that cannot be solved analytically. We have shown that for a system of A - nucleons nucleus by considering a suitable potential using this model, one is able to obtain energy spectrum and wave equations. However, the obtained results cannot reproduce the measured nuclear spectroscopy, but one may hope to become successful by considering an accurate potential in

Department of Nuclear Physics, Faculty of Basic Science, University of Mazandaran, Babolsar,

[1] Suhonen Jouni. From nucleon to nucleus. Springer-Verlag Berlin Heidelberg (2007).

[2] Ring P, Schuck P. The nuclear many-body problem. Springer-Verlag New York Hei‐

2 ( ) , . *m k k Vz <sup>k</sup>* = = <sup>h</sup>

0

0 0 2


#### **Non-Extensive Entropies on Atoms, Molecules and Chemical Processes Non-Extensive Entropies on Atoms, Molecules and Chemical Processes**

N. Flores-Gallegos, I. Guillén-Escamilla and J.C. Mixteco-Sánchez N. Flores-Gallegos, I. Guillén-Escamilla and J.C. Mixteco-Sánchez

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

#### **1. Introduction**

10.5772/59139

During the last decade, information theory [1] as applied to the basic sciences has taken two routes in the study of physical and chemical systems, considering both extensivity and non-extensivity - these concepts are fundamental to the development of new physical theories that try to describe the behaviour of natural systems. In this sense, non-extensivity is an important concept that it is necessary to incorporate into the description of atoms, molecules and chemical processes.

At present, one of the ways to incorporate the concept of non-extensivity is by using deformed entropies, or Tsallis entropy [2]. This entropy is a generalization of Shannon entropy and has a dependency of a parameter, usually denoted by "*q*" and generally called a 'non-extensivity parameter', that permits us to perform a modulation between extensive and non-extensive behaviour. These new kinds of entropies are built using a new area of mathematics called "*q*-algebra" , or "deformed algebra" [3–5]. One important aspect to the use and application of deformed entropy is that the original definition of the entropy used for building deformed entropy needs to be strictly positive over all space and dimensionless. As such, in this work we propose a definition that fulfils this. This entropy uses the electron density obtained by the methods of quantum mechanics - this is an important point because the electron density is an observable, and so this permits us to establish a gate between the non-extensivity of classic entropies and the non-extensivity of quantum entropies. Consequently, this entropy permits us to incorporate the important concept of non-extensivity in quantum theory. In the same way, it is known that the chemistry interpretation of the same behaviour of these systems can be enriched by quantum information theory.

As we will show in this work, it is trivial to obtain some important functionals using deformed entropies. In this sense, we show how, with simple mathematical manipulations, it is possible to obtain two of the most important functionals of physics - the kinetic energy

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons distribution, and reproduction in any medium, provided the original work is properly cited. Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

©2012 Author(s), licensee InTech. This is an open access chapter distributed under the terms of the Creative

functional of the Thomas-Fermi [6, 7] model, and the exchange energy functional of Dirac from deformed entropy. In our opinion, this opens the door to exploring the possibility of the generation of density functionals based in entropic criteria. Moreover, we will present a simple chemical process where we show the effect of the non-extensive parameter, and in the same way we present the general trends of the "*q*" parameter for the first 54 atoms of the periodic table. Finally, we might raise a general question that motivates this work, that is, when does a natural system become extensive or non-extensive? As we mentioned above, this parameter "*q*" has a strong relation with deformed algebra - in this algebra, all the operations have a dependence upon a parameter, "*q*", and in general when "*q*" is different to that of the unit, this implies that their basic properties are not completely separable and do not necessary commute. This causes us to raise a general question, namely, can nature be represented by deformed algebra? In this context, it is necessary to incorporate the concept of non-extensivity to rewrite many expressions in terms of deformed algebra and investigate their new properties.

#### **2. Theoretical background**

Since the 1980s, when the first applications of Shannon entropy to chemical systems were made, we might observe two basic definitions of it, namely

$$S = -\int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r},\tag{1}$$

and

$$S = -\int \frac{\rho(\mathbf{r})}{N} \ln \frac{\rho(\mathbf{r})}{N} d\mathbf{r},\tag{2}$$

where *ρ*(**r**) is the electron density subject to *ρ*(**r**)*d***r** = *N*, and *N* it is the electron number of the system. However, if we perform a dimensional analysis, we immediately note that neither definition of Shannon entropy is dimensionless; in addition, and in our opinion, the more serious deficiency is that neither definition is strictly positive over all space1. Given this situation, it is evident that we cannot apply these two initial definitions to chemical systems. Accordingly, we propose a redefinition of Shannon entropy [8], such that

$$S = -\int \frac{\rho(\mathbf{r})}{N} \ln \frac{\rho(\mathbf{r})}{\rho\_{\text{max}}} d\mathbf{r},\tag{3}$$

where *ρ*max is the electron density in the nuclei position. In the case of a molecular system *ρ*max, it is necessary to take the higher value of the electron density of all the atoms that constitute the molecule. This definition fulfils the following: it is dimensionless and strictly positive over all space. In this sense, we suggest the use of the definition (3) for entropy calculations of chemical systems.

<sup>1</sup> A more detailed study of this aspect will be presented elsewhere.

On the other hand, in general the entropy of a composed system is very often equal to the sum of all its parts. This is fulfilled only when the energy is the sum of the parts and if the work performed by all the parts is the sum of the work performed by the system. That is,

$$S(A,B) = S(A) + S(B)\_\prime \tag{4}$$

or in general,

2 ime knjige

their new properties.

and

**2. Theoretical background**

calculations of chemical systems.

<sup>1</sup> A more detailed study of this aspect will be presented elsewhere.

made, we might observe two basic definitions of it, namely

*S* = − 

*S* = −

Accordingly, we propose a redefinition of Shannon entropy [8], such that

*S* = −

functional of the Thomas-Fermi [6, 7] model, and the exchange energy functional of Dirac from deformed entropy. In our opinion, this opens the door to exploring the possibility of the generation of density functionals based in entropic criteria. Moreover, we will present a simple chemical process where we show the effect of the non-extensive parameter, and in the same way we present the general trends of the "*q*" parameter for the first 54 atoms of the periodic table. Finally, we might raise a general question that motivates this work, that is, when does a natural system become extensive or non-extensive? As we mentioned above, this parameter "*q*" has a strong relation with deformed algebra - in this algebra, all the operations have a dependence upon a parameter, "*q*", and in general when "*q*" is different to that of the unit, this implies that their basic properties are not completely separable and do not necessary commute. This causes us to raise a general question, namely, can nature be represented by deformed algebra? In this context, it is necessary to incorporate the concept of non-extensivity to rewrite many expressions in terms of deformed algebra and investigate

Since the 1980s, when the first applications of Shannon entropy to chemical systems were

*ρ*(**r**)

*ρ*(**r**)

where *ρ*max is the electron density in the nuclei position. In the case of a molecular system *ρ*max, it is necessary to take the higher value of the electron density of all the atoms that constitute the molecule. This definition fulfils the following: it is dimensionless and strictly positive over all space. In this sense, we suggest the use of the definition (3) for entropy

where *ρ*(**r**) is the electron density subject to *ρ*(**r**)*d***r** = *N*, and *N* it is the electron number of the system. However, if we perform a dimensional analysis, we immediately note that neither definition of Shannon entropy is dimensionless; in addition, and in our opinion, the more serious deficiency is that neither definition is strictly positive over all space1. Given this situation, it is evident that we cannot apply these two initial definitions to chemical systems.

*<sup>N</sup>* ln *<sup>ρ</sup>*(**r**)

*<sup>N</sup>* ln *<sup>ρ</sup>*(**r**) *ρ*max

*ρ*(**r**)ln *ρ*(**r**)*d***r**, (1)

*<sup>N</sup> <sup>d</sup>***r**, (2)

*d***r**, (3)

$$S(A, B, \mathbb{C}, \dots) = S(A) + S(B) + S(\mathbb{C}) + \dotsb \,\, \} \tag{5}$$

However, this not quite obvious, and in some cases this may not be fulfilled. For example, consider a system composed of two different homogeneous substances - in this case, it is only possible to express the energy as the sum of the individual energies if, and only if, we neglect the interaction energy of the substances or subsystems. However, this energy plays an important role in the description of natural systems; unfortunately, the mathematical development of it is, frequently, complicated. One interesting aspect of entropies involves entropic balances [9], in which is possible to write the joint entropy in terms of the subsystems' entropy and conditional entropy,

$$\mathcal{S}\_{\boldsymbol{q}}(A+\mathcal{B}) = \mathcal{S}\_{\boldsymbol{q}}(A) + \mathcal{S}\_{\boldsymbol{q}}(\mathcal{B}|A) + (1-\boldsymbol{q})\mathcal{S}\_{\boldsymbol{q}}(A)\mathcal{S}\_{\boldsymbol{q}}(\mathcal{B}|A) \tag{6}$$

where

$$S\_{q}(A) = -\iint \frac{\rho(\mathbf{a}, \mathbf{b})}{N\_{AB}(q-1)} \left[ 1 - \left\{ \int \frac{\rho(\mathbf{a}, \mathbf{b})}{\rho\_{\text{max}}^{A} N\_{B}} d\mathbf{b} \right\}^{q-1} \right] d\mathbf{a} d\mathbf{b} \tag{7}$$

$$S\_{q}(\mathcal{B}|A) = -\iint \frac{\rho(\mathbf{a}, \mathbf{b})}{N\_{AB}(q-1)} \left[ 1 - \left\{ \iint \frac{\frac{\rho(\mathbf{a}, \mathbf{b})}{N\_{A} + N\_{B}}}{\int \frac{\rho(\mathbf{a}, \mathbf{b})}{\rho\_{\text{max}}^{A} N\_{B}} d\mathbf{b}} d\mathbf{a} d\mathbf{b} \right\}^{q-1} \right] d\mathbf{a} d\mathbf{b},\tag{8}$$

where *ρ<sup>A</sup>* max and *<sup>ρ</sup><sup>B</sup>* max are the maximum density values of the fragments2 *A* and *B*, respectively. *NAB* is the total electron number and *NA* and *NB* are the electron numbers of the fragments *A* and *B* respectively. The marginal densities of the probabilities are defined as

$$
\int \rho(\mathbf{a}, \mathbf{b}) d\mathbf{a} = \rho(\mathbf{b}) \,\prime \,\tag{9}
$$

$$
\int \rho(\mathbf{a}, \mathbf{b}) d\mathbf{b} = \rho(\mathbf{a}) \,\prime \tag{10}
$$

#### and these densities fulfil

<sup>2</sup> This implies that it is necessary to select an electron density partition scheme subject to the rules of information theory. In chemistry, the scheme that fulfils this is the Stock-Holder partition scheme [10, 11].

$$
\int \int \rho(\mathbf{a}, \mathbf{b}) d\mathbf{a} d\mathbf{b} = 1,\tag{11}
$$

$$\int \rho(\mathbf{a})d\mathbf{a} = 1,\tag{12}$$

$$
\int \rho(\mathbf{b})d\mathbf{b} = 1,\tag{13}
$$

In all cases, *Sq* satisfies the following properties,


From the last paragraph, it is possible to think in terms of the use of linear description; in this sense, Tsallis proposes a generalization of Boltzmann-Gibbs entropy, using the so-called 'deformed functions', and substituting the original definitions by the deformed definitions. In general, two definitions are used, namely the deformed logarithm (or *q*-logarithm)

$$\ln\_q \ge := \frac{1 - \varkappa^{q-1}}{q - 1},\tag{14}$$

and the deformed exponential (or *q*-exponential)

$$\exp\_q^x := \left[1 + (1 - q)x\right]^{\frac{1}{1-q}},\tag{15}$$

These definitions can be obtained by solving the differential equation *dy dx* <sup>=</sup> *<sup>y</sup>q*, *<sup>y</sup>*(0) = 1; *<sup>q</sup>* <sup>∈</sup> ℜ, see [12].

Using the deformed logarithm, we can obtain the deformed entropy, which has the following explicit form

$$\begin{split} S\_{\boldsymbol{q}} &= -\int \frac{\rho(\mathbf{r})}{N} \ln\_{\boldsymbol{q}} \frac{\rho(\mathbf{r})}{\rho\_{\text{max}}} d\mathbf{r}, \\ &= -\int \frac{\rho(\mathbf{r})}{N} \left[ \frac{1 - \left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^{q-1}}{1 - q} \right] d\mathbf{r}, \\ &= \frac{1}{1 - q} + \frac{1}{N(q - 1)\rho\_{\text{max}}^{q-1}} \int \rho(\mathbf{r})^{q} d\mathbf{r}, \end{split} \tag{16}$$

where *q* ∈ ℜ, and for a composed system, this entropy is

$$\mathcal{S}\_{\emptyset}(A+B) = \mathcal{S}\_{\emptyset}(A) + \mathcal{S}\_{\emptyset}(B) + (1-q)\mathcal{S}\_{\emptyset}(A)\mathcal{S}\_{\emptyset}(B),\tag{17}$$

This implies that the subsystems are correlated, and immediately we can note that for *q* �= 1 the entropy of a composed system is non-extensive, though if we select *q* → 1, then the definition (16) becomes the definition (3), that is,

$$\lim\_{q \to 1} \left\{ \frac{1}{1 - q} + \frac{1}{N(q - 1)\rho\_{\text{max}}^{q - 1}} \int \rho(\mathbf{r})^q d\mathbf{r} \right\} = -\int \frac{\rho(\mathbf{r})}{N} \ln \frac{\rho(\mathbf{r})}{\rho\_{\text{max}}} d\mathbf{r},\tag{18}$$

and recover the extensive behaviour.

4 ime knjige

*i*) *Sq* ≥ 0;

ℜ, see [12].

explicit form

��

�

�

From the last paragraph, it is possible to think in terms of the use of linear description; in this sense, Tsallis proposes a generalization of Boltzmann-Gibbs entropy, using the so-called 'deformed functions', and substituting the original definitions by the deformed definitions. In general, two definitions are used, namely the deformed logarithm (or *q*-logarithm)

ln*<sup>q</sup> <sup>x</sup>* :<sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup>q*−<sup>1</sup>

*<sup>q</sup>* :<sup>=</sup> [<sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)*x*]

Using the deformed logarithm, we can obtain the deformed entropy, which has the following

*ρ*(**r**) *ρ*max *d***r**,

1 *N*(*q* − 1)*ρ*

1 − *q*

*q*−1 max �

1

�*q*−<sup>1</sup>

 *d***r**,

In all cases, *Sq* satisfies the following properties,

and the deformed exponential (or *q*-exponential)

exp*<sup>x</sup>*

These definitions can be obtained by solving the differential equation *dy*

� *ρ*(**r**) *<sup>N</sup>* ln*<sup>q</sup>*

� *ρ*(**r**) *N*

  1 − � *<sup>ρ</sup>*(**r**) *ρ*max

*Sq* = −

= −

= 1 1 − *q* +

*ii*) *Sq* is a continuous function of *ρ*(**a**, **b**), *ρ*(**a**) or *ρ*(**b**); *iii*) *Sq* increases monotonically with the particle number; *iv*) *Sq*(*A*, *B*) = *Sq*(*A*) + *Sq*(*B*)+(1 − *q*)*Sq*(*A*)*Sq*(*B*);

*ρ*(**a**, **b**)*d***a***d***b** = 1, (11)

*ρ*(**a**)*d***a** = 1, (12)

*ρ*(**b**)*d***b** = 1, (13)

*<sup>q</sup>* <sup>−</sup> <sup>1</sup> , (14)

<sup>1</sup>−*<sup>q</sup>* , (15)

*ρ*(**r**)*qd***r**, (16)

*dx* <sup>=</sup> *<sup>y</sup>q*, *<sup>y</sup>*(0) = 1; *<sup>q</sup>* <sup>∈</sup>

Now, using Eq. (16), if we select that *q* = <sup>4</sup> <sup>3</sup> , we obtain

$$S\_{q=4/3} = -3 + \frac{3}{N\rho\_{\text{max}}^{1/3}} \int \rho(\mathbf{r})^{4/3} d\mathbf{r},\tag{19}$$

In this expression, we can note immediately that the integral has the same form as that of the exchange functional of Dirac, and if we select that *q* = <sup>5</sup> 3 ,

$$S\_{q=5/3} = -\frac{3}{2} + \frac{3}{2N\rho\_{\text{max}}^{2/3}} \int \rho(\mathbf{r})^{5/3} d\mathbf{r},\tag{20}$$

this integral corresponds to the Thomas-Fermi kinetic energy functional. Consequently, it is trivial to obtain any density functional that involves some of the powers of electron density. Naturally, it is a simple matter to rewrite both functionals in terms of the deformed entropy this allows us to hypothesize that the electronic energy of a system can be rewritten as a linear combination of deformed entropies. This, of course, implies that the local density functionals are a particular case of the deformed entropy. This allows us to raise the following question: for any electron density, to what does the value of *q* correspond?

The definition (16) can be simplified if we perform a series expansion of the term *<sup>ρ</sup>*(**r**) *<sup>ρ</sup>*max *q*−<sup>1</sup> ,

$$\begin{split} \left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^{q-1} &= \frac{\rho\_{\text{max}}}{\rho(\mathbf{r})} + \frac{\rho\_{\text{max}}}{\rho(\mathbf{r})} \log\left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right) + \\ &\frac{\rho\_{\text{max}}}{2\rho(\mathbf{r})} \log\left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^2 + \frac{\rho\_{\text{max}}}{6\rho(\mathbf{r})} \log\left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^3 + \dots + \\ &\frac{\rho\_{\text{max}}}{n!\rho(\mathbf{r})} \log\left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^n, \end{split} \tag{21}$$

and if we suppose that only the first term contributes to the general behaviour and that it is the most important term,

$$\left(\frac{\rho(\mathbf{r})}{\rho\_{\text{max}}}\right)^{q-1} \sim \left(\frac{\rho\_{\text{max}}}{\rho(\mathbf{r})}\right),\tag{22}$$

Replacing this in Eq. (16), namely

$$S\_q^{approx} = -\int \frac{\rho(\mathbf{r})}{N} \left[ \frac{1 - \left(\frac{\rho\_{\text{max}}}{\rho(\mathbf{r})}\right)}{1 - q} \right] d\mathbf{r}\_\prime$$

$$= \left[ \frac{1}{q - 1} + \frac{\rho\_{\text{max}}}{N(1 - q)} \right] \int d\mathbf{r}\_\prime \tag{23}$$

this is a good result, because it is possible to perform a simple computational implementation of *Sapprox <sup>q</sup>* and explore the behaviour of those very large systems for which the *ab initio* calculations of the electron density are very expensive (for example, for systems constituted by more than 10<sup>4</sup> atoms). This definition satisfies the condition of the dimensionless of the entropy.

#### **3. Characterization of atoms in the basal state**

One of the principal questions that emerges in this study is concerned with the parameter "*q*", namely, for an atomic system in a basal state, what is its *q* value? For this, we calculated the electron density in the position space using the following functionals, B3LYP, BHandH, M062x, MP2, MP3 and TPSS, and with the following *ab initio* methods, CCS, CCSD, CISD, using a standard quantum chemistry program, Gaussian 09 [13], with the basis set DGDZVP [14, 15], to obtain the energy value and the corresponding wave function. The electron density in the position space was calculated with the DGrid program [16] using the wave function obtained through several methodologies, and for the entropy calculations we used the integration algorithm designed by Pérez-Jordá *et al.* [17] with a precision of 1 × <sup>10</sup><sup>−</sup>5.

In the Figures (1(a) - 1(f)), we show the results of *Sq* using *q* = 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, in which we note that the entropy has no dependence upon the methods, and it is possible to recognize the periodicity of the elements in the periodic table. Naturally, when the *q* parameter changes, the difference is magnified between the different periods of the periodic table; however, with the atoms that involve *d*-orbitals, we note a small disruption in this tendency. In general, we can observe that the Shannon entropy increases with respect to the atomic number in a natural way, if we appeal to the interpretation of this information measure, we can specify that the content of the information tends to increase. This assumption is based on the follow interpretation: if we consider an ideal gas, this system has a uniform particle distribution; therefore, the entropy is maximum. As such, we expect that the entropy increases in proportion to the electron number -in principle- and, considering the physics of the system, when the number of particles increases the Shannon entropy tends to the Thomas-Fermi limit as a consequence of a decrease in the Wigner-Seitz radii [18]. Thus, the definition proposed in this work permits us to recover the original idea of the content of

the information of a system in relation to the physical interactions between the electrons of systems, such that when the number or particles, *N*, tends to infinity, *q* will tend to the unit.

6 ime knjige

of *Sapprox*

entropy.

the most important term,

Replacing this in Eq. (16), namely

and if we suppose that only the first term contributes to the general behaviour and that it is

� *ρ*max *ρ*(**r**) �

, (22)

*d***r**, (23)

�*q*−<sup>1</sup> ∼

� *ρ*(**r**) *N*

 

*<sup>q</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>ρ</sup>*max

this is a good result, because it is possible to perform a simple computational implementation

One of the principal questions that emerges in this study is concerned with the parameter "*q*", namely, for an atomic system in a basal state, what is its *q* value? For this, we calculated the electron density in the position space using the following functionals, B3LYP, BHandH, M062x, MP2, MP3 and TPSS, and with the following *ab initio* methods, CCS, CCSD, CISD, using a standard quantum chemistry program, Gaussian 09 [13], with the basis set DGDZVP [14, 15], to obtain the energy value and the corresponding wave function. The electron density in the position space was calculated with the DGrid program [16] using the wave function obtained through several methodologies, and for the entropy calculations we used the integration algorithm designed by Pérez-Jordá *et al.* [17] with a precision of 1 × <sup>10</sup><sup>−</sup>5.

In the Figures (1(a) - 1(f)), we show the results of *Sq* using *q* = 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, in which we note that the entropy has no dependence upon the methods, and it is possible to recognize the periodicity of the elements in the periodic table. Naturally, when the *q* parameter changes, the difference is magnified between the different periods of the periodic table; however, with the atoms that involve *d*-orbitals, we note a small disruption in this tendency. In general, we can observe that the Shannon entropy increases with respect to the atomic number in a natural way, if we appeal to the interpretation of this information measure, we can specify that the content of the information tends to increase. This assumption is based on the follow interpretation: if we consider an ideal gas, this system has a uniform particle distribution; therefore, the entropy is maximum. As such, we expect that the entropy increases in proportion to the electron number -in principle- and, considering the physics of the system, when the number of particles increases the Shannon entropy tends to the Thomas-Fermi limit as a consequence of a decrease in the Wigner-Seitz radii [18]. Thus, the definition proposed in this work permits us to recover the original idea of the content of

*<sup>q</sup>* and explore the behaviour of those very large systems for which the *ab initio* calculations of the electron density are very expensive (for example, for systems constituted by more than 10<sup>4</sup> atoms). This definition satisfies the condition of the dimensionless of the

1 −

*N*(1 − *q*)

� *<sup>ρ</sup>*max *ρ*(**r**) �

� �

 *<sup>d</sup>***r**,

1 − *q*

� *ρ*(**r**) *ρ*max

> = � 1

*Sapprox <sup>q</sup>* = −

**3. Characterization of atoms in the basal state**

On the other hand, if we consider a system in a basal state and in equilibrium, what is the value of the parameter *q*? To address this question, we propose a computational form to find this value,

$$q\_{i+1} = q\_i + \frac{S\_{q\_i}}{S\_{q\_{i+1}}},\tag{24}$$

This approximation requires that *Sqi* ≈ *Sqi*<sup>+</sup><sup>1</sup> - when this condition occurs, it also satisfies that *<sup>∂</sup>Sqi ∂qi* ≈ *Sqi*<sup>+</sup><sup>1</sup> *∂qi*<sup>+</sup><sup>1</sup> . With this assumption, the slope of *Sq* tends to zero, and this corresponds to the zone where *q* does not change. This will be the *q* value for the system; moreover, we can fix the precision with 1 × 103 and the error was calculated as

$$\%Error = \left| \frac{S\_{q\_{l+1}} - S\_{q\_l}}{S\_{q\_{l+1}}} \right| \,\,\,\tag{25}$$

Given this consideration, we obtain the trends shown in Figure (2), where the general trend for *q* was calculated using the functionals B3LYP, TPSS and M062x, and with the wave function methods CCS, CCSD, MP2, MP3 and CISD with the basis set DGDZVP. In this figure, we note that only for the block *d* of the periodic table the tendency of *q* parameter, has a breaking of the tendency, this would be attributed to the basis set, but in respect to the methodologies the general trend of *q* permanence without considerable changes, this permit us establish that the *q* values has not a dependency of the methodologies. It is important to note that, according to the physics of the system, we expect that when the system sees a considerable increase in the number of electrons, the general behaviour will be like that of a Fermi gas, and consequently the system becomes an extensive system. This implies that the entropy becomes extensive, that is *q* → 1, and that Eq. (16) becomes as in Eq. (3). From these results, we also obtain that

$$\lim\_{q \to q^{\circ p}} S\_q = 1.\tag{26}$$

In the Figure (3) we present the general trend of *q* with CCSD/DGDZVP with dotted-crosses, and the following polynomial,

$$f(q) = \mathbb{C}\_1 + \mathbb{C}\_2 \exp\{-\mathbb{C}\_3 Z + \mathbb{C}\_4\} + \mathbb{C}\_5 \exp\{-\mathbb{C}\_6 Z^2 + \mathbb{C}\_7\},\tag{27}$$

with a continuous line, where the coefficients have the values listed in Table (1).

*Z* denotes the atomic number, and in Table (2) we show the *qopt* values for the first 54 atoms of the periodic table. Analysing the values of this table, we note that in all cases the characteristic value of each atom in the basal state is close to the unit when the electron number increases.

**Figure 1.** Effect of the variation of the parameter *q* for the trends of entropy, using several methodologies of quantum chemistry with the DGDZVP basis set.


**Table 1.** Values of the constants of Eq. (27).

8 ime knjige

 2 2.5 3 3.5 4 4.5 5 5.5 6

> B3LYP BHandH

chemistry with the DGDZVP basis set.

Sq, q = 1.7

B3LYP BHandH

Sq, q = 1.3

B3LYP BHandH

Sq, q = 0.9

0 10 20 30 40 50 60

Z

0 10 20 30 40 50 60

Z

(c) *Sq*, *q* = 1.3

CISD M062x

0 10 20 30 40 50 60

Z

(e) *Sq*, *q* = 1.7

CISD M062x MP2 MP3

CCS CCSD

CCS CCSD

(a) *Sq*, *q* = 0.9

CISD M062x MP2 MP3

MP2 MP3

> B3LYP BHandH

 0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06 3.5e+06 4e+06 4.5e+06 5e+06

> B3LYP BHandH

Sq, q = 1.5

Sq, q = 1.9

**Figure 1.** Effect of the variation of the parameter *q* for the trends of entropy, using several methodologies of quantum

B3LYP BHandH

Sq, q = 1.1

0 10 20 30 40 50 60

Z

0 10 20 30 40 50 60

Z

(d) *Sq*, *q* = 1.5

CISD M062x

0 10 20 30 40 50 60

Z

CISD M062x MP2 MP3

CCS CCSD

(f) *Sq*, *q* = 1.9

CCS CCSD

(b) *Sq*, *q* = 1.1

CISD M062x MP2 MP3

> MP2 MP3

CCS CCSD

CCS CCSD


**Table 2.** Values of *qopt* using CCSD(full)/DGDZVP.

#### **4. Characterization of a Simple Chemical process**

One of the interests of this work is in the study of the effect of the parameter "*q*" in a dissociation process. The idea is to study the effect of small interactions when a homonuclear system is dissociated.

In this case, we select the dissociation of the *H*<sup>2</sup> molecule,

$$H\_2 \longrightarrow H + H.$$

The calculations were performed with Gaussian 03 [19] with CCSD(full) and the basis set cc-pVTZ [20]. For the entropy calculations, we used the wave function generated by Gaussian 03 to generate the electron density, while *ρ*max was calculated in the position of the nuclei of each atom of the molecular system, in this case by the symmetry *ρ*max(*A*) = *ρ*max(*B*) = *ρ*max. The electron density was calculated with DGrid and the algorithm of integration that we used was designed by Peréz-Jodá *et al.* with a precision of 1 × <sup>10</sup><sup>−</sup>5.

In Figure (4), we present the general trend of this simple chemical process, where we can note that the internuclear equilibrium distance is 0.754 Å, which corresponds to the minimum electronic energy; the dissociation process was carried out more than two times the van der

**Figure 2.** Trends of the parameter *q* for the atoms 1 < *Z* < 54 using several methodologies with the basis set DGZVP; all calculations were performed in Gaussian 09.

**Figure 3.** Trends of the parameter *q* for the atoms 1 < *Z* < 54, with CCSD(full)/DGDZVP in Gaussian 09.

Waals radii of the hydrogen atom in the basal state (1.2 Å) to ensure that no weak chemical interactions were present. With this in mind, in this case it is natural to think that, for an internuclear distance of 3.0 Å, the electronic energy will be twice that of the electronic energy of the hydrogen atom in the basal state; however, this does not occur, and the value obtained with CCSD(full)/cc-pVTZ is -1.0007258069 a.u. Consequently, there exists a difference of 0.0007258069 a.u. (1.9056 kJ/mol) - this energy value is closer to that of hydrogen bonding. In principle, the explanation of this anomaly can be addressed in the following way: by definition, the wave function is extended over all space, and by construction the wave function used in a quantum mechanics calculation is a finite superposition of the basis set functions, *<sup>ψ</sup>* = <sup>∑</sup>*<sup>n</sup> <sup>i</sup> χiφi*. However, notwithstanding that, in a limit this function will be exact, the correct description obtained with this wave function will be correct only in the equilibrium. This condition is not obvious, and how we see is not fulfil, this probable

permit us talk about of a necessity of a different statistical ensemble for the more adequate description of the systems, and with this new ensembles possible we can describe of a more appropriate some phenomena present in the quantum world. The real justification for proposing (and postulating) the existence of this new set of definitions will reside in their implications, namely the incompleteness of the descriptions obtained by the actual tools and theories.

10 ime knjige

 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16

 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 B3LYP CCS

q

all calculations were performed in Gaussian 09.

q

set functions, *<sup>ψ</sup>* = <sup>∑</sup>*<sup>n</sup>*

0 10 20 30 40 50 60

Z

0 10 20 30 40 50 60

Z CCSD f(q)

*<sup>i</sup> χiφi*. However, notwithstanding that, in a limit this function will

Waals radii of the hydrogen atom in the basal state (1.2 Å) to ensure that no weak chemical interactions were present. With this in mind, in this case it is natural to think that, for an internuclear distance of 3.0 Å, the electronic energy will be twice that of the electronic energy of the hydrogen atom in the basal state; however, this does not occur, and the value obtained with CCSD(full)/cc-pVTZ is -1.0007258069 a.u. Consequently, there exists a difference of 0.0007258069 a.u. (1.9056 kJ/mol) - this energy value is closer to that of hydrogen bonding. In principle, the explanation of this anomaly can be addressed in the following way: by definition, the wave function is extended over all space, and by construction the wave function used in a quantum mechanics calculation is a finite superposition of the basis

be exact, the correct description obtained with this wave function will be correct only in the equilibrium. This condition is not obvious, and how we see is not fulfil, this probable

**Figure 3.** Trends of the parameter *q* for the atoms 1 < *Z* < 54, with CCSD(full)/DGDZVP in Gaussian 09.

MP3

TPSS CISD

M062x MP2

**Figure 2.** Trends of the parameter *q* for the atoms 1 < *Z* < 54 using several methodologies with the basis set DGZVP;

CCSD

In Table (3), we present the values of the electron energy for the hydrogen molecule at an internuclear distance of 3.0 Åusing several methodologies of quantum chemistry, and with the basis sets cc-pvDZ, cc-pvTZ, cc-pvQZ and cc-pv5Z. The basis sets are designed to converge systematically on the complete basis set, such that this basis set permits us to analyse the improvement of the electron energy, and we note that the best result that we can obtain corresponds to CCSD(full)/cc-pvTZ. However, even using this sophisticated methodology and basis set, there exists an excess energy of 0.0012563951 a.u. (3.2986 kJ/mol). Here, it is convenient to observe that it is not necessary to make use of a bigger basis set corresponding to a better description (again, this is in reference to Tables (3) and (4), in which the best energy value corresponds to CISD/cc-pvTZ and not to cc-pv5Z, which is the more complete basis set of all those used in this work). In all cases, all the methodologies and basis sets overestimate the energy over large distances, but in the case of MP2, PBE and B3LYP, the overestimation it of the order of the energy of a simple covalent bond, such as an oxygen molecule (145 kJ/mol), or a simple bond of a nitrogen molecule (170 kJ/mol).

On the other side, and continuing our discussion, in Figure (4) we observe that the system can be additive but not necessarily extensive. To explain this in the Figures (6(a)-6(f)), we present a comparison of the electron energy and the deformed entropy using several *q* values, where we note that for different *q* values the minimum of the entropy change of the position, for *q* = −4.1, −2.8, 0.3, 1.5, 1.8 and a distance greater than of 2.0 Å, the slope of tendency it is zero, that is, the entropy is constant and consequently it is additive, but not extensive, because for a this values, *Sq* is constant, now if we consider that this system its constituted by identical subsystems we have, *Sq*(*A*, *B*) = *Sq*(*A*) + *Sq*(*B*)+(1 − *q*)*Sq*(*A*)*Sq*(*B*) and by the system characteristics *Sq*(*A*) = *Sq*(*B*) so we have *Sq*(*A*, *B*) = 2*Sq*(*A*)+(1 − *q*)[*Sq*(*A*)]2. This opens the door to an interesting question: in physical systems, it is the same additive that extensive? In our opinion, they are different concepts and it is probable that the use of these concepts as synonyms is a result of the historical background. It is interesting to note that, for *q* = −2.8 = −14/5, the minimum of the entropy corresponds to the minimum of the energy (see Figure (6(b))); consequently, if we find the appropriate *q* value for the system, it is possible to reproduce the electron energy behaviour. The interesting aspect of this is that the deformed entropy, that it is a local functional (because has not dependency of external potential), that we can found the same tendencies of the energy in which are present the effects of an external potential. Another notable characteristic of this tendency is presented in Figure (6(c)), where in the final state of the system the slope again tends to zero but the total entropy is greater than the initial content. This implies that the term (1 − *q*)*Sq*(*A*)<sup>2</sup> is greater than 2*Sq*(*A*), and if we retake the interpretation of the term (1 − *q*)*Sq*(*A*)<sup>2</sup> then, like the degree of non-separability, we can conclude that the representation of the system with *q* = 0.3 is non-extensive over all processes, even if the internuclear distance implies that the system has no physical interactions. This is consistent with the interpretation that the wave function is extended over all space - if we accept this, probably we can establish a link between non-extensivity and quantum entanglement [21]. Consequently, it is possible to build a bridge between quantum information theory and non-extensive statistical mechanics and reclaim the idea that we can improve our understanding of nature by not only analysing behaviour exclusively in terms of matter and energy (even at the level of elementary particles), but also that study using the techniques and methods of modern physics and chemistry integrate concepts and tools that allow us to comprehensively investigate the behaviour of natural systems in order to deepen our understanding of them to incorporate information measures that take into account concepts such as entanglement, known since the early days of Quantum Mechanics, for which, however, there are no measures in many modern theories, at a more fundamental level, it has become clear that an Information Theory based on the principles of Quantum Mechanics, expands and complements the Classical Information Theory [22]. In addition to the quantum generalizations of classical notions such as sources, channels and codes, this new theory includes two complementary types of quantifiable data: classical information and quantum entanglement.

**Figure 4.** Trends of the electron energy of the dissociation process of *H*2, with CCSD(full)/cc-pvTZ.

**Figure 5.** Comparison between the electron energy and the *qopt* parameter for the dissociation process of *H*<sup>2</sup> with CCSD(full)/cc-pvTZ.


12 ime knjige

and reclaim the idea that we can improve our understanding of nature by not only analysing behaviour exclusively in terms of matter and energy (even at the level of elementary particles), but also that study using the techniques and methods of modern physics and chemistry integrate concepts and tools that allow us to comprehensively investigate the behaviour of natural systems in order to deepen our understanding of them to incorporate information measures that take into account concepts such as entanglement, known since the early days of Quantum Mechanics, for which, however, there are no measures in many modern theories, at a more fundamental level, it has become clear that an Information Theory based on the principles of Quantum Mechanics, expands and complements the Classical Information Theory [22]. In addition to the quantum generalizations of classical notions such as sources, channels and codes, this new theory includes two complementary types of

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms]

0 0.5 1 1.5 2 2.5 3 -1.2

**Figure 5.** Comparison between the electron energy and the *qopt* parameter for the dissociation process of *H*<sup>2</sup> with

R: H-H [Angstroms] q E


E [a.u.]

**Figure 4.** Trends of the electron energy of the dissociation process of *H*2, with CCSD(full)/cc-pvTZ.

quantifiable data: classical information and quantum entanglement.


 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36

q

CCSD(full)/cc-pvTZ.

E [a.u.]

**Table 3.** Values of the electron energy for *H*<sup>2</sup> at an internuclear distance of 3Å(in a.u.). The energy calculations were performed in Gaussian 09.


**Table 4.** Absolute difference of values between the electron energy of *H*<sup>2</sup> at 3.0 Å, and 2*H* with several methodologies and with the cc-pvTZ basis set. All values are kJ/mol.

In the Figure (5), we show the general trend of *qopt* compared with the electron energy, in which the tendency of *qopt* has a maximum in approximately 0.529 Å, plotted as a vertical continuous black line, this value correspond at the first Bohr radii for the Hydrogen atom, this is an interesting point because it is possible talk about a non-extensive radii of the systems, where the non-extensivity it is maximum and the point of this is that the we can associate the *q* parameter at a physical property like the distances between the subsystems, so we suspect that the non-extensive behaviour is closely related at two characteristics; the distance and the particle number.

**Figure 6.** Comparison between the trends of the *Sq* entropy using several *q* values and the electron energy for the dissociation process of a *H*<sup>2</sup> molecule.

### **5. Characterization of the Chemical Reaction** *<sup>H</sup>*<sup>2</sup> <sup>+</sup> *<sup>H</sup>*<sup>−</sup>

In this section, we present the results of the reaction *<sup>H</sup>*<sup>2</sup> <sup>+</sup> *<sup>H</sup>*<sup>−</sup> <sup>→</sup> *<sup>H</sup>*<sup>2</sup> <sup>+</sup> *<sup>H</sup>*−. This reaction is one of the more studied reactions and it is very well-characterized [23–25]. The IRC calculation was performed with MP2(full)/6-311G, and the singles points with CISD/6-311++G\*\*, both in Gaussian 03. This reaction is symmetric, has a maximum in the transition state (which has an energy of -1.6501559031 a.u.) and an internuclear distance of 0.93236 Åbetween each hydrogen atom. Naturally, the electron energy in the reactants is the same as in their products (-1.6680093713 a.u). In the Figures (7(a))-(8(f)), we present the tendency of the deformed entropy using several values of the *q* parameter, with *q* = −10.0, Figure (7(a)). The entropy has a maximum value at *RX* = 0.0 and a possible local minimum in *RX* = −2 and *RX* = 2. However, is not very clear how to determine whether this *q* value is associated with the changes of the entropy for some other parameters related at changes physical or chemical, when we use the *q* = −4.6, Figure (7(b)), the entropy tendency has a maximum value at *RX* = 0.0 and it is similar at the tendency of the energy. The more interesting aspects of the changes in the entropy are in the interval −1.1 ≤ *q* ≤ 1.6, in Figure (7(c)); with *q* = −1.1, the entropy has a minimum in *RX* = −1 and *RX* = 1, and these minimums are associated at a zone where the process of the breaking and forming of the chemical bonds occurs. This zone corresponds to a zone where the normal modes of vibration have negative frequencies. In Figure (9(a)), we show this comparison, and in the same way we compare this tendency with the distances of the hydrogen's involved in the process, we labeled the atoms like like *Hin* for the Hydrogen that will be form the new bond and *Hout* for the Hydrogen that gonna be break the bond, in this case, the critic region where the physical changes occurs is −0.5 ≤ *RX* ≤ 0.5, see the Figure (9(b)), this only can be observed in the same zone of the entropy where has a small change in their slope, in this sense, it is possible that changing the value of *q* or increase the precision we can observe with more detail the changes that occurs in this zone. Figure (9(c)) presents a comparison of *Sq*, *q* = −0.7 with the Dipolar Moment, how occurs in the case of the frequencies this parameter has a maximums in *RX* = −0.85 and *RX* = 0.85, is it in this zone where the most important changes of the electron density occurs. In general, we can say that the changes in the deformed entropy permit us to discover some zones where the most important changes of the electron density of a system occurs; however, it is not yet known how to select the appropriate value of *q*, for example, when we use a value of *q* = −0.7, the tendency of the entropy has minimums in *RX* ∼ −0.9 and *RX* ∼ 0.9. We can say that this tendency is related to the change of the electron density, but in the case of *Sq*, *q* = −0.1 the entropy behaves like a specular image of the energy (see Figures (8(a)) and (8(c))). With this evidence, we believe that it is possible to derive some density functionals in which a combination of different entropic terms can be expressed, not only the deformed entropy with the form of Eq. (16), but also a contribution of a deformed Fisher entropy (for this, it will be necessary to write the gradient of the electron density in terms of deformed algebra). That is,

$$E[\rho] = \sum\_{i} \mathbf{x}\_{i} \mathbf{S}\_{q} + \mathbf{x}\_{i} I\_{q\_{i}} \tag{28}$$

where

14 ime knjige

 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2 0.202 0.204 0.206

 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24

Sq

Sq

Sq

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = -4.1 E (a) *Sq*, *q* = −4.1

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = 0.3 E (c) *Sq*, *q* = 0.3

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = 1.8 E (e) *Sq*, *q* = 1.8

dissociation process of a *H*<sup>2</sup> molecule.




E [a.u.]

Sq

**Figure 6.** Comparison between the trends of the *Sq* entropy using several *q* values and the electron energy for the

E [a.u.]

Sq

E [a.u.]

Sq

 0.245 0.25 0.255 0.26 0.265 0.27 0.275

 0 1e+16 2e+16 3e+16 4e+16 5e+16 6e+16 7e+16 8e+16 9e+16 0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = -2.8 E (b) *Sq*, *q* = −2.8

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = 1.5 E (d) *Sq*, *q* = 1.5

0 0.5 1 1.5 2 2.5 3

R: H-H [Angstroms] q = 2.2 E (f) *Sq*, *q* = 2.2 -1.2 -1.18 -1.16 -1.14 -1.12 -1.1 -1.08 -1.06 -1.04 -1.02 -1



E [a.u.]

E [a.u.]

E [a.u.]

$$I\_q = \int \varrho(\mathbf{r}) |\nabla \ln\_q \varrho(\mathbf{r})|^2 d\mathbf{r},\tag{29}$$

**Figure 7.** Comparison between the trends of the *Sq* entropy using several *q* values and the electron energy for the reaction *<sup>H</sup>*<sup>2</sup> + *<sup>H</sup>*−.

and *̺*(**r**) is the shape factor, defined as *̺*(**r**) = *<sup>ρ</sup>*(**r**) *N* .

16 ime knjige

Sq

 0.09068 0.0907 0.09072 0.09074 0.09076 0.09078 0.0908

 0.4668 0.467 0.4672 0.4674 0.4676 0.4678 0.468 0.4682 0.4684

 0.5118 0.512 0.5122 0.5124 0.5126 0.5128 0.513 0.5132

reaction *<sup>H</sup>*<sup>2</sup> + *<sup>H</sup>*−.

Sq

Sq


RX q = -10.0 E (a) *Sq*, *q* = −10.0


RX q = -1.1 E (c) *Sq*, *q* = −1.1


and *̺*(**r**) is the shape factor, defined as *̺*(**r**) = *<sup>ρ</sup>*(**r**)

RX q = -0.9 E (e) *Sq*, *q* = −0.9 -1.645 -1.64 -1.635 -1.63 -1.625 -1.62 -1.615



E [a.u.]

Sq

**Figure 7.** Comparison between the trends of the *Sq* entropy using several *q* values and the electron energy for the

*N* .

E [a.u.]

Sq

E [a.u.]

Sq

 0.1799 0.18 0.1801 0.1802 0.1803 0.1804 0.1805 0.1806

 0.4884 0.4886 0.4888 0.489 0.4892 0.4894 0.4896 0.4898 0.49

 0.5376 0.5378 0.538 0.5382 0.5384 0.5386 0.5388 -4 -3 -2 -1 0 1 2 3 4 -1.65

RX q = -4.6 E (b) *Sq*, *q* = −4.6


RX q = -1.0 E (d) *Sq*, *q* = −1.0


RX q = -0.8 E (f) *Sq*, *q* = −0.8 -1.645 -1.64 -1.635 -1.63 -1.625 -1.62 -1.615



E [a.u.]

E [a.u.]

E [a.u.]

**Figure 8.** Comparison between the trends of the *Sq* entropy using several *q* values and the electron energy for the reaction *<sup>H</sup>*<sup>2</sup> + *<sup>H</sup>*−.

(c) *Sq*, *q* = −0.7 and Dipolar Moment

**Figure 9.** Comparison between the trends of the *Sq* entropy using several *q* values, frequencies, distances and Dipolar moment for the reaction *<sup>H</sup>*<sup>2</sup> + *<sup>H</sup>*−.

#### **6. Entropic profiles of atoms**

18 ime knjige


0.5

 0 0.02 0.04 0.06 0.08 0.1 0.12

Dipolar Moment

moment for the reaction *<sup>H</sup>*<sup>2</sup> + *<sup>H</sup>*−.

1

1.5

R: H-H2 [Angstroms]

2

2.5

3

Frequencies [cm^-1]


RX Freq. Sq

(a) *Sq*, *q* = −1.1*andFrequencies*


(b) *Sq*, *q* = −1.1 and Distances

RX Hin-H2 Hout-H2 Sq


RX Dipolar Moment Sq

(c) *Sq*, *q* = −0.7 and Dipolar Moment

**Figure 9.** Comparison between the trends of the *Sq* entropy using several *q* values, frequencies, distances and Dipolar

 0.447 0.4472 0.4474 0.4476 0.4478 0.448 0.4482 0.4484 0.4486 0.4488

 0.4472 0.4474 0.4476 0.4478 0.448 0.4482 0.4484 0.4486 0.4488

 0.5118 0.512 0.5122 0.5124 0.5126 0.5128 0.513 0.5132

Sq, q = -0.7

Sq, q = -1.1

Sq, q = -1.1

In this section, we present the results of the entropic profiles from hydrogen to neon. The density was calculated with a precision of 1 × <sup>10</sup>−<sup>5</sup> with CCSD(full)/DGDZVP obtained in Gaussian 09. It is our particular interest to characterize the hydrogen atom, because the simplicity of this system permits us to determine and - as far as possible - try to find some periodic properties. In this sense, in the Figure (10(a)), we present the entropic profile for this system, in which we note that the maximum present is in *R* ≈ 1*a*.*u*., that is, the first Bohr radii. It is possible to speculate that it is at this distance when the system exhibits the maximum degree of non-extensivity. In the case of He, the maximum value of the entropy is displaced close to the nucleus, *R* = 0.545*a*.*u*., Figure (10(b)). For the second period (lithium to neon, Figures (11(a)-12(b))), the maximum of the entropy coincides with the first maximum of the electron density, and the minimum of the electron density coincides with the inflection point of the entropy trend; subsequently, in the region of the maximum density the entropy there is a small change in their slope. Finally, the electron density and the entropy tends to zero. In general, we also propose verifying the changes of the entropy tendency using several values of *q* in the interval 0 ≤ *q* ≤ 10 with a step size of 0.1, though we cannot observe significant changes. In this sense, it is possible that the critical points of the deformed entropy are in relation to the chemical reactivity parameters, such as the Fukui function [26–28], hardness, softness [29], [30], chemical potential, *inter alia*, [31]. However, it will be necessary to perform studies of the relation between the *q* parameter and these chemical descriptors. We consider that it will be important to carry this concepts into the field of deformed algebra.

**Figure 10.** Comparison between the trends of the *Sq* entropy with *q* = 0.2 and the electron density for hydrogen and helium.

**Figure 11.** Comparison between the trends of the *Sq* entropy with *q* = 0.2 and the electron density for lithium to oxygen.

**Figure 12.** Comparison between the trends of the *Sq* entropy with *q* = 0.2 and the electron density for fluorine and neon.

#### **7. Conclusion**

20 ime knjige

 0 0.5 1 1.5 2 2.5 3

oxygen.

Sq, q = 0.2

Sq, q = 0.2

Sq, q = 0.2

0 1 2 3 4 5 6 7<sup>0</sup>

R [a.u.] Sq Density (a) *Sq*, *q* = 0.2, Li

0 1 2 3 4 5 6 7 8

R [a.u.] Sq Density (c) *Sq*, *q* = 0.2, B

0 1 2 3 4 5 6 7<sup>0</sup>

R [a.u.] Sq Density (e) *Sq*, *q* = 0.2, N 0.005

 0 0.01 0.02 0.03 0.04 0.05 0.06

 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Density

Density

Sq, q = 0.2

0.01

0.015

Density

Sq, q = 0.2

0

5

10

15

Sq, q = 0.2

**Figure 11.** Comparison between the trends of the *Sq* entropy with *q* = 0.2 and the electron density for lithium to

20

25

0 1 2 3 4 5 6 7 8 9<sup>0</sup>

R [a.u.] Sq Density (b) *Sq*, *q* = 0.2, Be

0 1 2 3 4 5 6 7

R [a.u.] Sq Density (d) *Sq*, *q* = 0.2, C

0 1 2 3 4 5 6<sup>0</sup>

R [a.u.] Sq Density (f) *Sq*, *q* = 0.2, O  0.005 0.01 0.015 0.02 0.025 0.03

 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

> 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Density

Density

Density

0.02

0.025

In this work, we applied the fundamental idea of Tsallis of entropy generalization and we propose a definition of deformed entropy that is applied to the description of the first 54 atoms in the basal state of the periodic table, showing that each chemical system has a characteristic *q* value. Furthermore, we show that the characteristic *q* value can be related to the first Bohr radii, in which we suppose that there corresponds a distance where the non-extensive behaviour of the system is dominant. In the same way, we show the numerical tendencies of the deformed entropy compared to the variation of the electron density for the first 10 atoms in the basal state and observe that the changes of the deformed entropy are in relation to the significant changes in the electron density.

This has allowed us to start a new line of investigation, and with some of these results we continue with the study of the formalisms for the construction of a functional based in a principles of physics and information theory. In addition, we intend to continue with the development of models that permit us to find a direct relation between electron energy and chemical reactivity concepts with deformed entropy.

On the other hand, the application of the concepts of information theory permit us form a description that is more accurate than that based on energetic criteria alone; we speculate that it is possible to define or find a form that derives the Density Functional Theory from some fundamental expression.

Finally, with these examples we have tried to link information from a system that is subjected to a process with physical and chemical changes. Thus, we have linked the concept of *information*, which is an epistemological concept completely with ontological concepts and the interpretation of the results allows us feedback these concepts in ontological terms, according to the authors, is probable that today do not exist a orthodoxical definition of what actually is the *information*, beyond that presented by Shannon and its guidelines, criteria, characterization of it, among other things, the interpretation and the relationship with other concepts, such as energy, electron density, chemical reactivity parameters and many others need be discussed to try of establisha formal relation between concepts.

Is it clear that information concept and the model itself is interdisciplinary or transdisciplinary. The concept of information and -moreover- the model itself promote a systematic relation with causal analogies and parallelism with scientific knowledge, which transcends the framework of the source domain and extends in various directions, thus making the knowledge acquire an unusual resonance. Accordingly, we believe it is feasible to complement the explanations of natural processes and natural systems.

#### **8. Acknowledgements**

N. Flores-Gallegos wishes to thank the CONACyT for financial support through the programme Apoyos Complementarios para la Consolidación Institucional de Grupos de Investigación (Repatriación, Retención y Estancias de Consolidación) and Centro Universitario de los Valles, of Benemérita Universidad de Guadalajara.

#### **Author details**

N. Flores-Gallegos, I. Guillén-Escamilla and J.C. Mixteco-Sánchez

Centro Universitario de los Valles Benemérita Universidad de Guadalajara, Ameca, Jalisco, México

#### **References**



22 ime knjige

**8. Acknowledgements**

**Author details**

México

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[4] V. Kac, and P. Cheing, Quantum Calculus. Springer; 2000.

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concepts, such as energy, electron density, chemical reactivity parameters and many others

Is it clear that information concept and the model itself is interdisciplinary or transdisciplinary. The concept of information and -moreover- the model itself promote a systematic relation with causal analogies and parallelism with scientific knowledge, which transcends the framework of the source domain and extends in various directions, thus making the knowledge acquire an unusual resonance. Accordingly, we believe it is feasible

N. Flores-Gallegos wishes to thank the CONACyT for financial support through the programme Apoyos Complementarios para la Consolidación Institucional de Grupos de Investigación (Repatriación, Retención y Estancias de Consolidación) and Centro

Centro Universitario de los Valles Benemérita Universidad de Guadalajara, Ameca, Jalisco,

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### Karlheinz Schwarz

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59108

#### **1. Introduction**

#### **1.1. Inorganic solids**

Many inorganic solid materials are of great interest from a fundamental point of view or for technologic applications. The challenge to theory and computation is that they are governed by different length scales, where from meters (m) down to micrometers (μm) classical mechanics and continuum models provide proper descriptions (for example using finite element methods). However, if the length scale is down to nanometers (nm) or atomic dimensions measured in Å, such as for modern devices in the electronic industry (for example in magnetic recording) or surface science and catalysis, the properties are determined (or critically influenced) by the electronic structure and thus quantum mechanics. Understanding the properties at the atomic scale is often essential for improving or designing modern materials in a systematic way. Computation has become a key element in this process, since it allows one to analyze and interpret sophisticated measurements or to plan future experiments in a rational way, replacing the old trial and error scheme. Instead of trying all kinds of elements to improve a material by preparation, characterization and functional analysis, a simulation with computers is often much more efficient and allows one to "narrow the design space". Why should one prepare or measure a sample that is not promising based on modern computation? Some facilities (for example those providing synchrotron radiation) have already adapted this concept, since the beam time for measurements is limited and thus they should be used for promising investigations only.

There is a classical treatment at the atomic scale that is often based on atomic force fields. In this case the interactions between the atoms are specified with forces which are parameterized usually in a way to reproduce a set of experimental data such as equilibrium geometries, bulk muduli or vibrational (phonon) frequencies. For a class of materials, in which good parameters are known, this can be a useful approach to answer certain questions, since force-field

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

calculations require little computational effort. The main drawback is that no information can be provided about the electronic structure, since force fields do not explicitly contain electrons. Therefore these approaches will not be covered here, although they have reached a high level of sophistication.

Here we focus on the electronic structure of solids (metals, insulators, minerals, etc.) and surfaces (or interfaces) which require a quantum mechanical treatment. In recent review articles [1-5] one possible approach has been described, which is based on the WIEN2k program package [6, 7] that has been developed in my group during the last 35 years (see www.wien2k.at). Several other computer codes are available which will not be covered here. Each of them may have a different focus in terms of efficiency, accuracy, sophistication, capabilities (properties), user friendliness, parallelization, documentation etc. It should be stressed that the large variety of computer codes is very beneficial for this growing field of computational material sciences, since these codes all have advantages and disadvantages compared to others. For obvious reasons we will focus on our own code to illustrate important aspects of this field.

Some basic concepts will be described below as summarized in Figure 1. As a very first step one needs to represent the atomic structure of a solid material as is outlined in Section 2. Idealized assumptions must be made which one should keep in mind when theoretical results are compared with experiments. In order to describe the electronic structure of a system of interest by means of quantum mechanics (Section 3) we first briefly mention some fundamental concepts of solid state physics (like symmetry). Next we sketch how chemists handle quantum mechanics and then focus on the most important theory for solids, namely density functional theory. Before we can describe how to solve the corresponding Kohn-Sham equations (Section 4) it is necessary to select which electrons should be included in the calculations: all of them or only the valence electrons. The form of the potential must also be chosen, where we discuss such methods as pseudo-potential, muffin-tin or full-potential approximations. This choice is important for deciding which basis set can best describe the wave functions of the electrons. A relativistic treatment will be important if the system contains heavier elements. For magnetic systems spin-polarization is essential and thus must be included. The main ideas of the allelectron full-potential linearized-augmented-plane-wave (LAPW) with local orbitals method as implemented in WIEN2k are briefly described. This should be sufficient, since there is a good documentation of the underling details.

Section 5 lists selected results which can be obtained with WIEN2k. Since about 2500 groupsboth from academic and industrial institutions – are using this code there are many details available in literature (see the link papers at www.wien2k.at). Based on the experience of developing WIEN2k, it is appropriate to make some general comments about the computer code development (Section 6). As more and more experimental scientists in this field rely on computer simulations, it is useful to raise some critical questions and summarize several points, which can cause deviations between theoretical results and experiment, a topic covered in Section 7. In Section 8 concluding remarks are made and then the very appropriate ac‐ knowledgments.

**Figure 1.** Possible choices for calculating the electronic structure of a material at the atomic scale (the focus is highlight‐ ed in red).

#### **2. Atomic structure**

calculations require little computational effort. The main drawback is that no information can be provided about the electronic structure, since force fields do not explicitly contain electrons. Therefore these approaches will not be covered here, although they have reached a high level

Here we focus on the electronic structure of solids (metals, insulators, minerals, etc.) and surfaces (or interfaces) which require a quantum mechanical treatment. In recent review articles [1-5] one possible approach has been described, which is based on the WIEN2k program package [6, 7] that has been developed in my group during the last 35 years (see www.wien2k.at). Several other computer codes are available which will not be covered here. Each of them may have a different focus in terms of efficiency, accuracy, sophistication, capabilities (properties), user friendliness, parallelization, documentation etc. It should be stressed that the large variety of computer codes is very beneficial for this growing field of computational material sciences, since these codes all have advantages and disadvantages compared to others. For obvious reasons we will focus on our own code to illustrate important

Some basic concepts will be described below as summarized in Figure 1. As a very first step one needs to represent the atomic structure of a solid material as is outlined in Section 2. Idealized assumptions must be made which one should keep in mind when theoretical results are compared with experiments. In order to describe the electronic structure of a system of interest by means of quantum mechanics (Section 3) we first briefly mention some fundamental concepts of solid state physics (like symmetry). Next we sketch how chemists handle quantum mechanics and then focus on the most important theory for solids, namely density functional theory. Before we can describe how to solve the corresponding Kohn-Sham equations (Section 4) it is necessary to select which electrons should be included in the calculations: all of them or only the valence electrons. The form of the potential must also be chosen, where we discuss such methods as pseudo-potential, muffin-tin or full-potential approximations. This choice is important for deciding which basis set can best describe the wave functions of the electrons. A relativistic treatment will be important if the system contains heavier elements. For magnetic systems spin-polarization is essential and thus must be included. The main ideas of the allelectron full-potential linearized-augmented-plane-wave (LAPW) with local orbitals method as implemented in WIEN2k are briefly described. This should be sufficient, since there is a

Section 5 lists selected results which can be obtained with WIEN2k. Since about 2500 groupsboth from academic and industrial institutions – are using this code there are many details available in literature (see the link papers at www.wien2k.at). Based on the experience of developing WIEN2k, it is appropriate to make some general comments about the computer code development (Section 6). As more and more experimental scientists in this field rely on computer simulations, it is useful to raise some critical questions and summarize several points, which can cause deviations between theoretical results and experiment, a topic covered in Section 7. In Section 8 concluding remarks are made and then the very appropriate ac‐

of sophistication.

276 Selected Topics in Applications of Quantum Mechanics

aspects of this field.

knowledgments.

good documentation of the underling details.

The properties of materials at the nanometer (nm) scale or of atomic dimensions (measured in Å) are essentially determined by the electronic structure. In such a case one tries to represent the material of interest (such as a solid, a surface or a molecule) as a collection of atoms, which play the role of building blocks. It is important to realize that in practice one is forced to assume an idealized atomic structure in theoretical work, which deviates from the real structure that may be studied experimentally.

A few examples will illustrate this important point: Let us first consider a molecule which in theory is studied in vacuum, whereas in experiment it is often measured on a support (surface) or in a solution. The latter may be simulated by surrounding the molecule by a few solvent molecules or by using an embedding scheme with a dielectric constant (simulating the solvent). Recently the combination of a quantum mechanical (QM) treatment of the molecule with a cruder mechanical mechanics (MM) representation of the environment is used in QM/MM schemes. Such treatments will necessarily be approximate.

As a second example we focus on a solid. In early days one modeled a solid as a cluster of atoms, but due to size limitations this cannot represent bulk properties. With the increase in computer power we assume ‒ especially in theory ‒ that a solid is a perfect crystal and can be characterized by a unit cell that is repeated to infinity in all three dimensions. This means that one assumes periodic boundary conditions. A real crystal, however, is certainly finite. For experimental studies a crystalline sample is often available in the form of a powder consisting of small crystal domains. Even if experiments are carried out on a single crystal, it still has a surface and imperfections (such as defects or impurities, etc.). In compounds there are additional uncertainties, such as the stoichiometry, which may not be perfect, or the atomic arrangement, which may deviate from the underlying idealized order.

The situation of a perfect single crystal is illustrated for TiO2 crystallizing in the rutile structure (Figure 2). The symmetry belongs to one of the 230 space groups that are tabulated in the International Tables for Crystallography [8]. Nowadays this information is also available from the Bilbao Crystallographic Server (*www.cryst.ehu.es/cryst/*). For a given crystal the unit cell must be specified with the three lattice parameters (a, b, c) and the corresponding angles (α, β, γ). For the atomic positions the Wyckoff positions must be defined, where for each type of atoms only one of the equivalent atoms needs to be specified, while all others are generated by symmetry. For known structures this type of information is available for example from the inorganic crystal structure data base [9] or can be taken from standardized CIF files, which are often available directly from literature.

**Figure 2.** The crystal structure of TiO2 in the rutile structure: It is characterized by the space group 136 which is speci‐ fied in the International Table of Crystallography [8]. The two types of atoms occupy the Wyckoff positions 2a (Ti) and 4f (O) where for the latter the parameter x is not given by symmetry and thus needs to be specified. The equivalent positions are defined by symmetry. In this tetragonal structure the angles of the unit cell are all 90 degrees and for the lattice constants only a and c need to be specified, since a=b.

In Figure 3 the unit cell of a borocarbide is shown, which contains main group elements (B and C), transition metals (Ni), and rare earth elements (Nd). Often interesting materials have such complex compositions. In this case the crystal structure is well known but for an understanding

**Figure 3.** The unit cell of NdNi2B2C

computer power we assume ‒ especially in theory ‒ that a solid is a perfect crystal and can be characterized by a unit cell that is repeated to infinity in all three dimensions. This means that one assumes periodic boundary conditions. A real crystal, however, is certainly finite. For experimental studies a crystalline sample is often available in the form of a powder consisting of small crystal domains. Even if experiments are carried out on a single crystal, it still has a surface and imperfections (such as defects or impurities, etc.). In compounds there are additional uncertainties, such as the stoichiometry, which may not be perfect, or the atomic

The situation of a perfect single crystal is illustrated for TiO2 crystallizing in the rutile structure (Figure 2). The symmetry belongs to one of the 230 space groups that are tabulated in the International Tables for Crystallography [8]. Nowadays this information is also available from the Bilbao Crystallographic Server (*www.cryst.ehu.es/cryst/*). For a given crystal the unit cell must be specified with the three lattice parameters (a, b, c) and the corresponding angles (α, β, γ). For the atomic positions the Wyckoff positions must be defined, where for each type of atoms only one of the equivalent atoms needs to be specified, while all others are generated by symmetry. For known structures this type of information is available for example from the inorganic crystal structure data base [9] or can be taken from standardized CIF files, which are

**Figure 2.** The crystal structure of TiO2 in the rutile structure: It is characterized by the space group 136 which is speci‐ fied in the International Table of Crystallography [8]. The two types of atoms occupy the Wyckoff positions 2a (Ti) and 4f (O) where for the latter the parameter x is not given by symmetry and thus needs to be specified. The equivalent positions are defined by symmetry. In this tetragonal structure the angles of the unit cell are all 90 degrees and for the

In Figure 3 the unit cell of a borocarbide is shown, which contains main group elements (B and C), transition metals (Ni), and rare earth elements (Nd). Often interesting materials have such complex compositions. In this case the crystal structure is well known but for an understanding

arrangement, which may deviate from the underlying idealized order.

often available directly from literature.

278 Selected Topics in Applications of Quantum Mechanics

lattice constants only a and c need to be specified, since a=b.

of the properties of this compound it is essential that all these atom types (from light to heavy) can be properly treated. Details of this compound can be found in [10].

In modern material science it is often mentioned that "nano materials" have significantly different properties than their bulk analogs. A simple explanation can be provided by estimating the ratio of atoms on the surface with respect to those in the bulk of the material. The atoms on the surface have a different coordination number than the atoms in the bulk and thus have a different bonding environment. Consequently these atoms may move to a relaxed atomic position with respect to the ideal bulk crystal structure. In a nano particle a significant fraction of atoms are on the surface, whereas in a large single crystal this fraction is rather small and thus can (to a good approximation) be neglected, so that periodic boundary conditions may be used for calculating many properties.

In all the cases mentioned above the assumed atomic structure is idealized and differs from the real structure of a material that is investigated experimentally. These aspects should be kept in mind when theoretical results are compared with experimental data (as will be discussed in Section 7). This idealization is an advantage for theory with respect to experi‐ ments, since in computations the atomic structure is clearly defined as input. This is in contrast with experimental studies in which the material is often not so well characterized, in terms of stoichiometry, defects, impurities, surfaces, disorder etc. When theory simulates a structure, which is not a good representation of the real system, deviations between theory and experi‐ ment must be expected irrespective of the accuracy of the theoretical method.

With the concept of a supercell one can approximately simulate some aspects of a real system. For example one can artificially enlarge a unit cell by forming a 2 x 2 x 2 supercell, containing eight times as many atoms as the original. Figure 4 shows this case for a simple cubic case. In such a supercell one can, for example, remove an atom (representing a defect) or substitute one atom by another one (simulating a substitution) or add vacuum (about 10-15 Å) on one side of the cell (to represent a surface).

**Figure 4.** Schematic diagram to generate a 2 x 2 x 2 supercell from a simple cubic structure.

The possibilities of such supercells are schematically shown in Figure 5. A two-dimensional array of atoms is called slab. In a multi-layer slab the central layer approximately represents the bulk of a system, whereas the top (and bottom) layer can be used to represent a surface provided the distance to the next layer is sufficiently far away (due to the periodic boundary). On such an artificial surface one can place molecules and study for example catalytic reactions. In all such supercells one still has introduced an artificial order, since for example a defect will have a periodic image in the neighboring cells. The larger one can make the supercell the less critical the interaction between each periodic images become but this requires higher compu‐ tational effort. Therefore one must make compromises.

**Figure 5.** The construction of a supercells is schematically shown, first with three cubic cells (body-centered-cubic) plus vacuum, second a schematic supercell forming a 3-layer slab; and third a 5-layer slab (formed of metal atoms) with molecules on top.

The use of supercells is steadily increasing, since a more realistic modeling of real structures becomes attractive. In large supercells (with a few hundred atoms) one can even approximately model disorder as was recently illustrated for so called misfit layer compounds, in which the layers of PbS and TaS2 can be stabilized by occasionally substituting Pb by Ta [11]. Such aspects will be mentioned in Section 5.

#### **3. Quantum mechanics**

#### **3.1. DFT Fundamentals**

#### *3.1.1. Symmetry*

eight times as many atoms as the original. Figure 4 shows this case for a simple cubic case. In such a supercell one can, for example, remove an atom (representing a defect) or substitute one atom by another one (simulating a substitution) or add vacuum (about 10-15 Å) on one

The possibilities of such supercells are schematically shown in Figure 5. A two-dimensional array of atoms is called slab. In a multi-layer slab the central layer approximately represents the bulk of a system, whereas the top (and bottom) layer can be used to represent a surface provided the distance to the next layer is sufficiently far away (due to the periodic boundary). On such an artificial surface one can place molecules and study for example catalytic reactions. In all such supercells one still has introduced an artificial order, since for example a defect will have a periodic image in the neighboring cells. The larger one can make the supercell the less critical the interaction between each periodic images become but this requires higher compu‐

**Figure 5.** The construction of a supercells is schematically shown, first with three cubic cells (body-centered-cubic) plus vacuum, second a schematic supercell forming a 3-layer slab; and third a 5-layer slab (formed of metal atoms) with

**Figure 4.** Schematic diagram to generate a 2 x 2 x 2 supercell from a simple cubic structure.

tational effort. Therefore one must make compromises.

molecules on top.

side of the cell (to represent a surface).

280 Selected Topics in Applications of Quantum Mechanics

As mentioned in the previous section we focus on materials at the atomic scale. Here we use periodic boundary conditions and start with an ideal crystal structure that is studied at zero temperature. Our unit cell (or supercell) may contain several atoms. With present computer technology unit cells with around one thousand atoms can still be simulated within a reason‐ able time.

Quantum mechanics (QM) governs the electronic structure which is responsible for the properties of the system, such as the relative stability, chemical bonding, relaxation of the atoms, phase transitions, electrical, mechanical or magnetic behavior, etc. In addition many quantities related to experimental data (such as spectra) are determined by QM principles.

Several basic concepts from solid state physics and group theory are needed to characterize the electronic structure of solids as summarized for example in [12]. Here we just briefly mention some of these concepts such as the Born-Oppenheimer approximation, according to which the electrons can move independent form the nuclei (which can be assumed to be at fixed positions), or the direct lattice and the Wigner-Seitz cell. Owing to the translational symmetry of a crystal, it is convenient to define a reciprocal lattice with the Brillouin zone as the unit cell. The symmetry is defined by operators for translation, rotation, reflection or inversion and leads to group theory with the space group and point group. The electronic structure of an infinite solid looks so complicated that it would seem impossible to calculate it. Two important steps make the problem feasible. The single particle approach, in which each electron moves in an average potential V(**r**) that is translational invariant V(**r**+**T**)=V(**r**) under the translation **T**. The second important concept is the Bloch theorem, which defines how the wave function (which is not translational invariant) changes under **T**, namely by a phase factor, called the Bloch factor eikT

$$\Psi\_k\left(r+T\right) \ = \ e^{ikT}\psi\_k\left(r\right) \tag{1}$$

where **k** is a vector in reciprocal space that plays the role of a quantum number in solids. The **k** vector can be chosen in the first Brillouin zone, because any **k**' that differs from **k** by just a lattice vector **K** of the reciprocal lattice has the same Bloch factor and the corresponding wave function satisfies the Bloch condition again.

#### *3.1.2. Quantum chemistry and ab initio methods*

The quantum mechanical treatment of a system on the atomic scale has been discussed in many papers (for example in [5, 12]) and thus it is sufficient to summarize a few basic concepts here. According to the Pauli principle, because electrons are indistinguishable Fermions, their wave functions must be antisymmetric when two electrons are interchanged leading to the phenom‐ enon of exchange. In a variational wave-function description this can be enforced by forming one Slater determinant (set up from one-electron wave functions), representing the wellknownHartree-Fock(HF)approximation.TheHFequationshavethecomputationaldisadvant‐ age that each electron moves in a different potential (becoming orbital dependent). In HF the exchange is treated exactly but correlation effects, caused by the specific Coulomb interaction between the electrons are omitted by definition. Correlation can be included by more sophisti‐ cated approaches such as configuration interaction (CI) in which additional Slater determi‐ nants (including single, double or triple excitations into unoccupied states) are added in order to increase the variationalflexibility ofthe basis set[13].Anothertreatment of correlationeffects is the coupled cluster (CC) scheme that is often used in quantum chemistry [14]. Such schemes are labeled *ab initio* (or first principles) methods and are highly accurate refinements that can reach an almost exact solution. Unfortunately the corresponding computational effort dramat‐ ically increases with N7 , where the system size is proportional to N, the number of electrons. Such nearly exact solutions would be desirable but in practice they can only be obtained for relatively small systems (atoms or smallmolecules).Whenthe systemsize is significantly larger (as in condensed matter applications) approximations are unavoidable.

In quantum mechanics the term *ab inito* means that for a simulation of a material it is sufficient to know its constituent atoms (or isotopes) but the rest is governed by quantum mechanics. One does not need to know whether a material is insulating, metallic, magnetic, or has any other specific property. In principle an *ab initio* calculation should determine these properties from the atomic structure alone. A different situation occurs for calculations that are based on parameters that had been fitted to known properties of other systems that are similar to the material of interest. The latter type of calculations is often less demanding (in terms of computer resources) but is necessarily biased towards the related class of materials for which the parameters had been determined. Consequently one cannot find an unconventional behavior. In practice, however, it helps to know something about the system in order to choose proper approximations in the complicated quantum mechanical calculations. For example, why should one perform a spin-polarized calculation knowing that the system is not magnetic.

#### *3.1.3. Density Functional Theory*

The well-established scheme to calculate electronic properties of solids is based on density functional theory (DFT), for which Walter Kohn has received the Nobel Prize in chemistry in 1998. Fifty years ago, in 1964, Hohenberg and Kohn [15] have shown that the total energy E of an interacting inhomogeneous electron gas (as it appears in atoms, molecules or solids), in the presence of an external potential (coming from the nuclei) is a functional of the electron density ρ which uniquely defines the total energy E of the system, i.e E[ρ].

$$E = T\_o \left[ \rho \right] + \int V\_{ext} \rho(\vec{r}) d\vec{r} + \frac{1}{2} \int \frac{\rho(\vec{r}) \rho(\vec{r}')}{|\vec{r}' - \vec{r}|} d\vec{r} d\vec{r}' + E\_{xc} \left[ \rho \right] \tag{2}$$

The four terms correspond to the kinetic energy (of non-interacting electrons), the nuclearelectronic interaction energy Ene, the Coulomb energy (including the self-interaction) and the exchange correlation energy Exc, which contains all the quantum mechanical contributions. This theorem is still exact. From a numerical point of view one can stress that the first three terms are large numbers while the last is essential but small and thus can be approximated. Thus one does not need to know the many-body electronic wave function. This is an enormous simplification. To clarify this point, consider the very simple case of a system (atom or molecule) with 100 electrons but which is still small. Each electron needs to be described by a wave function which depends on three space coordinates and the spin. Therefore the manybody wave function would depend on 400 coordinates. According to DFT all that is needed is the density ρ(**r**) which only depends on the position **r**, i.e. on three coordinates. Unfortunately the exact form of the functional is not known but the conditions it should satisfy have been formulated, as will be discussed in Section 3.3.

#### **3.2. The Kohn-Sham equations**

lattice vector **K** of the reciprocal lattice has the same Bloch factor and the corresponding wave

The quantum mechanical treatment of a system on the atomic scale has been discussed in many papers (for example in [5, 12]) and thus it is sufficient to summarize a few basic concepts here. According to the Pauli principle, because electrons are indistinguishable Fermions, their wave functions must be antisymmetric when two electrons are interchanged leading to the phenom‐ enon of exchange. In a variational wave-function description this can be enforced by forming one Slater determinant (set up from one-electron wave functions), representing the wellknownHartree-Fock(HF)approximation.TheHFequationshavethecomputationaldisadvant‐ age that each electron moves in a different potential (becoming orbital dependent). In HF the exchange is treated exactly but correlation effects, caused by the specific Coulomb interaction between the electrons are omitted by definition. Correlation can be included by more sophisti‐ cated approaches such as configuration interaction (CI) in which additional Slater determi‐ nants (including single, double or triple excitations into unoccupied states) are added in order to increase the variationalflexibility ofthe basis set[13].Anothertreatment of correlationeffects is the coupled cluster (CC) scheme that is often used in quantum chemistry [14]. Such schemes are labeled *ab initio* (or first principles) methods and are highly accurate refinements that can reach an almost exact solution. Unfortunately the corresponding computational effort dramat‐

Such nearly exact solutions would be desirable but in practice they can only be obtained for relatively small systems (atoms or smallmolecules).Whenthe systemsize is significantly larger

In quantum mechanics the term *ab inito* means that for a simulation of a material it is sufficient to know its constituent atoms (or isotopes) but the rest is governed by quantum mechanics. One does not need to know whether a material is insulating, metallic, magnetic, or has any other specific property. In principle an *ab initio* calculation should determine these properties from the atomic structure alone. A different situation occurs for calculations that are based on parameters that had been fitted to known properties of other systems that are similar to the material of interest. The latter type of calculations is often less demanding (in terms of computer resources) but is necessarily biased towards the related class of materials for which the parameters had been determined. Consequently one cannot find an unconventional behavior. In practice, however, it helps to know something about the system in order to choose proper approximations in the complicated quantum mechanical calculations. For example, why should one perform a spin-polarized calculation knowing that the system is not magnetic.

The well-established scheme to calculate electronic properties of solids is based on density functional theory (DFT), for which Walter Kohn has received the Nobel Prize in chemistry in 1998. Fifty years ago, in 1964, Hohenberg and Kohn [15] have shown that the total energy E of an interacting inhomogeneous electron gas (as it appears in atoms, molecules or solids), in the

(as in condensed matter applications) approximations are unavoidable.

, where the system size is proportional to N, the number of electrons.

function satisfies the Bloch condition again.

282 Selected Topics in Applications of Quantum Mechanics

*3.1.2. Quantum chemistry and ab initio methods*

ically increases with N7

*3.1.3. Density Functional Theory*

From a practical point of view it was essential to formulate DFT in such a way that it could be applied. According to the variational principle a set of effective one-particle Schrödinger equations, the so-called Kohn-Sham (KS) equations [16], must be solved (Equation 3) as highlighted in Figure 1 and Figure 6. In this way DFT is a universal approach to the quantum mechanical many-body problem, where the system of interacting electrons is mapped in a unique manner onto an effective non-interacting system that has the same total density. The non-interacting particles of this auxiliary system move in an effective local one-particle potential, which consists of a classical mean-field (Hartree) part and an exchange-correlation part Vxc (due to quantum mechanics) that, in principle, incorporates all correlation effects exactly. Eqn.3 shows its form (written in Rydberg atomic units) for an atom with the obvious generalization to molecules and solids.

$$\left[ -\frac{1}{2}\nabla^2 + V\_{\text{ext}}(\vec{r}) + V\_{\text{C}}\{\rho(\vec{r})\} + V\_{\text{xc}}\{\rho(\vec{r})\} \right] \Phi\_i(\vec{r}) = \,\_{\mathcal{E}i}\Phi\_i(\vec{r})\tag{3}$$

The four terms represent the kinetic energy operator, the external potential from the nucleus, the Coulomb-, and exchange-correlation potential VC and Vxc. The KS equations must be solved iteratively till self-consistency is reached (as illustrated in Figure 6). The iteration cycles are needed due to the interdependence between orbitals and potential. In the KS scheme the electron density is obtained by summing over all occupied states, i.e. by filling the KS orbitals (with increasing energy) according to the aufbau principle.

$$\rho(\vec{r}) = \sum\_{i}^{occ} \left[\rho\_i(\vec{r})\right]^2\tag{4}$$

A typical computation is illustrated in Figure 6. For a system of interest the unit cell must be specified by the lattice constants a, b, c and the corresponding angles (α, β, γ). In addition each atomic position is defined by the Wyckoff positions (as mentioned in Section 2). For this fixed atomic structure the self consistent field (SCF) cycle starts. As a first guess for the crystalline density one can superimpose atomic densities of neutral atoms placed at their proper positions in the unit cell. With this density one can generate a potential (within DFT). In each iteration i the DFT Kohn-Sham equations must be solved as illustrated on the right hand side.

Instead of using a uniform mesh of **k**-points s in the Brillouin zone (BZ) it is sufficient to restrict the **k**-points s to the irreducible wedge of the BZ by applying symmetry relations present in the system. From each star of equivalent **k**-points s only one must be calculated and its corresponding density is weighted according to the **k**-points symmetry (reducing the compu‐ tational effort). For each **k**-points the Kohn-Sham equations must be solved.

The KS wave functions are expanded in basis sets as will be described in the next section. The expansion coefficients Ckn are determined by the variational method by minimizing the expectation value of the total energy with respect to these coefficients. This procedure leads to the generalized eigenvalue problem, HC=ESC, where H is the Hamiltonian, S the overlap matrix, C contains the coefficients and E the energies. After diagonalization we obtain for each energy Enk the KS orbital ψnk and thus can calculate the corresponding electron density, where n is the band index.

By summing over all occupied states (with E**k** smaller than the Fermi energy) the output density is obtained. This output density can be mixed with the input density of the previous iterations to obtain a new density for the next iteration. In order to reduce the number of iterations to reach self consistency, several schemes have been suggested (see section 5 in [4]). A recent mixing scheme is the multisecant version [17] which includes information from several previous iterations from the SCF cycle as samples of a higher dimensional space to generate the new density, from which the VC (solving Poisson's equation) and Vxc (within DFT) potentials can be generated for the next SCF iteration. The exact functional form of the potential Vxc is not known and thus one needs to make approximations. With these potentials the new KS orbitals can be obtained. This closes the SCF cycle.

The SCF cycles are continued till convergence is reached, for example when the total energy of two successive iterations deviates from each other by less than a convergence criterion ε (e.g. 0.001 Ry). At this stage one can look at forces acting on the atoms in the unit cell. If symmetry allows there can be forces on the atoms which are defined as the negative gradient of the total energy with respect to the position parameters. Take for example the rutile TiO2 structure (Figure 2), in which oxygen sits on Wyckoff position 4f which has the coordinates (x,

**Figure 6.** Major steps in DFT electronic structure calculations: self-consistent field (SCF) cycle; Kohn-Sham equations solved within a **k**-points loop; for example, a face-centered-cubic structure (space group 225) has a body-centered cubic reciprocal lattice (space group 229) as Brillouin zone with its irreducible wedge (1/48th of the BZ).

x, 0) where x is not specified by symmetry. In this case x can be varied to minimize the energy and thus a force can occur on the oxygen which vanishes at the equilibrium geometry. However, Ti is located at the Wyckoff position 2a with the fixed coordinates (0, 0, 0) and (½, ½, ½) and thus these positions are fixed and no force will act on Ti. When all atoms are essentially at their equilibrium positions (with forces around 0) then one can change the volume of the unit cell and minimize the total energy E. This would correspond to the equilibrium geometry of the system in the given structure. After this minimization is com‐ pleted one can, as the last step, calculate various properties for this optimized structure.

#### **3.3. DFT-functionals**

electron density is obtained by summing over all occupied states, i.e. by filling the KS orbitals

<sup>2</sup> ( ) [ ( )] *occ i i*

i the DFT Kohn-Sham equations must be solved as illustrated on the right hand side.

tational effort). For each **k**-points the Kohn-Sham equations must be solved.

KS orbitals can be obtained. This closes the SCF cycle.

n is the band index.

 j

A typical computation is illustrated in Figure 6. For a system of interest the unit cell must be specified by the lattice constants a, b, c and the corresponding angles (α, β, γ). In addition each atomic position is defined by the Wyckoff positions (as mentioned in Section 2). For this fixed atomic structure the self consistent field (SCF) cycle starts. As a first guess for the crystalline density one can superimpose atomic densities of neutral atoms placed at their proper positions in the unit cell. With this density one can generate a potential (within DFT). In each iteration

Instead of using a uniform mesh of **k**-points s in the Brillouin zone (BZ) it is sufficient to restrict the **k**-points s to the irreducible wedge of the BZ by applying symmetry relations present in the system. From each star of equivalent **k**-points s only one must be calculated and its corresponding density is weighted according to the **k**-points symmetry (reducing the compu‐

The KS wave functions are expanded in basis sets as will be described in the next section. The expansion coefficients Ckn are determined by the variational method by minimizing the expectation value of the total energy with respect to these coefficients. This procedure leads to the generalized eigenvalue problem, HC=ESC, where H is the Hamiltonian, S the overlap matrix, C contains the coefficients and E the energies. After diagonalization we obtain for each energy Enk the KS orbital ψnk and thus can calculate the corresponding electron density, where

By summing over all occupied states (with E**k** smaller than the Fermi energy) the output density is obtained. This output density can be mixed with the input density of the previous iterations to obtain a new density for the next iteration. In order to reduce the number of iterations to reach self consistency, several schemes have been suggested (see section 5 in [4]). A recent mixing scheme is the multisecant version [17] which includes information from several previous iterations from the SCF cycle as samples of a higher dimensional space to generate the new density, from which the VC (solving Poisson's equation) and Vxc (within DFT) potentials can be generated for the next SCF iteration. The exact functional form of the potential Vxc is not known and thus one needs to make approximations. With these potentials the new

The SCF cycles are continued till convergence is reached, for example when the total energy of two successive iterations deviates from each other by less than a convergence criterion ε (e.g. 0.001 Ry). At this stage one can look at forces acting on the atoms in the unit cell. If symmetry allows there can be forces on the atoms which are defined as the negative gradient of the total energy with respect to the position parameters. Take for example the rutile TiO2 structure (Figure 2), in which oxygen sits on Wyckoff position 4f which has the coordinates (x,

*r r* <sup>=</sup> <sup>å</sup> r r (4)

(with increasing energy) according to the aufbau principle.

284 Selected Topics in Applications of Quantum Mechanics

r

The treatment of exchange and correlation effects has a long history and is still an active field of research. Some aspects were summarized in the review articles [1-5] but also in many other papers in this field. The reader is encouraged to look at recent developments. An excellent book [18] by Cottenier covers DFT and many aspects around the WIEN2k program package and thus is highly recommend to the reader for finding further details.

For the present presentation it is worth mentioning a few historical aspects: in 1951 Slater [19] proposed the replacement of the non-local Hartree-Fock exchange by the statistical exchange, called Slater's exchange. In the 1970ths this was modified by scaling it with the exchange parameter α (for each atom) called the Xα method [20], which was widely used for solid state calculations. It was designed to approximate Hartree-Fock, which (by construction) treats exchange exactly but neglects correlation effects completely. By making a local approximation for the potential the Xα method indirectly included correlation effects making it better than Hartree-Fock but also less accurate, since exchange is treated only approximately. This type of error cancellation is typical for many DFT functionals.

Early applications of DFT were done by using results from quantum Monte Carlo calculations [21] for the homogeneous electron gas, for which the problem of exchange and correlation can be solved exactly. Although no real system has a constant electron density, one can at each point in space use the homogenous electron gas result to treat exchange and correlation, leading to the original local density approximation (LDA). Surprisingly LDA works reasonably well but has some shortcomings mostly due to the tendency to overbind atoms, which cause e.g. too small lattice constants. The next crucial step in DFT was the implementation of the generalized gradient approximation (GGA), for example the version by Perdew, Burke, Ernzerhof (PBE) [22] which improved LDA by adding gradient terms of the electron density. For several cases this GGA gave better results and thus for a long time PBE has been a standard for many solid state calculations. During recent years, however, several improvements of GGA were proposed, which fall in two categories, both with good justifications:


One criterion for the quality of a calculation is the equilibrium lattice constant of a solid, which can be calculated by minimizing the total energy with respect to volume. By studying a large series of solids (as shown in Figure 7) some general trends can be found ([23]): LDA has the tendency of overbinding, leading to smaller lattice constants than the experiment. GGA in the version of PBE [22] always yield larger lattice constants, which sometimes are above the experimental value. The more recently suggested modifications, as discussed in [23], lead to a clear improvement at least for the lattice parameters. In addition, there are other observables (such as cohesive energy or magnetism, to mention just two), which depend on the functional. The best agreement with experiment may require different functionals for various properties. So far no functional works equally well for all cases and all systems. Therefore one must acknowledge that an optimal DFT functional has not yet been found, which is the reason why this remains an active field of research.

A systematic improvement of the exchange and correlation treatment as in quantum chemistry (section 2.3) starting from Hartree-Fock to (full) configuration interaction (CI) or coupled cluster (CC) approaches did not exist for solids and DFT. In 2005 such a scheme was proposed in [24] and was called Jacob's ladder for DFT which becomes progressively more demanding

For the present presentation it is worth mentioning a few historical aspects: in 1951 Slater [19] proposed the replacement of the non-local Hartree-Fock exchange by the statistical exchange, called Slater's exchange. In the 1970ths this was modified by scaling it with the exchange parameter α (for each atom) called the Xα method [20], which was widely used for solid state calculations. It was designed to approximate Hartree-Fock, which (by construction) treats exchange exactly but neglects correlation effects completely. By making a local approximation for the potential the Xα method indirectly included correlation effects making it better than Hartree-Fock but also less accurate, since exchange is treated only approximately. This type

Early applications of DFT were done by using results from quantum Monte Carlo calculations [21] for the homogeneous electron gas, for which the problem of exchange and correlation can be solved exactly. Although no real system has a constant electron density, one can at each point in space use the homogenous electron gas result to treat exchange and correlation, leading to the original local density approximation (LDA). Surprisingly LDA works reasonably well but has some shortcomings mostly due to the tendency to overbind atoms, which cause e.g. too small lattice constants. The next crucial step in DFT was the implementation of the generalized gradient approximation (GGA), for example the version by Perdew, Burke, Ernzerhof (PBE) [22] which improved LDA by adding gradient terms of the electron density. For several cases this GGA gave better results and thus for a long time PBE has been a standard for many solid state calculations. During recent years, however, several improvements of GGA

**•** Semi-empirical GGA, which contain parameters that are fitted to accurate (e.g. experimental

**•** ab initio GGA, in which all parameters are determined by satisfying fundamental con‐

One criterion for the quality of a calculation is the equilibrium lattice constant of a solid, which can be calculated by minimizing the total energy with respect to volume. By studying a large series of solids (as shown in Figure 7) some general trends can be found ([23]): LDA has the tendency of overbinding, leading to smaller lattice constants than the experiment. GGA in the version of PBE [22] always yield larger lattice constants, which sometimes are above the experimental value. The more recently suggested modifications, as discussed in [23], lead to a clear improvement at least for the lattice parameters. In addition, there are other observables (such as cohesive energy or magnetism, to mention just two), which depend on the functional. The best agreement with experiment may require different functionals for various properties. So far no functional works equally well for all cases and all systems. Therefore one must acknowledge that an optimal DFT functional has not yet been found, which is the reason why

A systematic improvement of the exchange and correlation treatment as in quantum chemistry (section 2.3) starting from Hartree-Fock to (full) configuration interaction (CI) or coupled cluster (CC) approaches did not exist for solids and DFT. In 2005 such a scheme was proposed in [24] and was called Jacob's ladder for DFT which becomes progressively more demanding

were proposed, which fall in two categories, both with good justifications:

straints (mathematical condition) which the exact functional must obey.

of error cancellation is typical for many DFT functionals.

286 Selected Topics in Applications of Quantum Mechanics

or ab initio) data

this remains an active field of research.

**Figure 7.** Comparison of several GGA functionals, showing the relative error in the equilibrium lattice constant of many solids between DFT calculations and experiment (for further details see [23]). The calculations were done with WIEN2k.

in terms of computational requirements. In Figure 8 the five rungs of this ladder ("to heaven") are briefly mentioned which indicate what is needed at each level of theory. In LDA the exchange correlation energy εxc is just a function of the density ρ; in the next rung it depends also on the gradient of the density, in meta-GGA εxc is in addition a function of the Laplacian of the density and the kinetic energy density t (see e.g. [25]). In rung 4 one goes from the simple

$$E\_{\rm xc} = \int \epsilon\_{\rm xc}(\mathbf{r}) \, d^3 r$$


**Figure 8.** Jacob's ladder according to [24] with 5 rungs, demonstrating how to improve the exchange correlation treat‐ ment.

dependence on the density alone, to an orbital description, which (for occupied orbitals) allows a correct description of exchange, like in Hartree-Fock. At this level one limits the computation space to the occupied orbitals but can extend it to the hybrid functions (mixing a fraction of Hartree-Fock with a part in DFT). In the highest rung also unoccupied orbitals are included, as for example in the scheme called random phase approximation (RPA).

There are well documented cases for which conventional DFT calculations (LDA or GGA) disagree even qualitatively with experimental data and lead, for instance, to predict a metal instead of an insulator. One of the reasons can be the presence of localized states (often felectrons or late transition metal d-orbitals) for which correlation is very strong. For these highly correlated systems one must go beyond simple DFT calculations. One simple form of improvement is to treat theses local correlations by means of a Hubbard U (see [26]) but use LDA or GGA for the rest of the electrons. With this parameter U the on-site Coulomb repulsion between the localized orbitals is included, but by introducing a parameter. This approach is generally called LDA+U. In a simple picture, U stands for the energy penalty of moving a localized electron to the neighboring site that is already occupied.

The Kohn-Sham energy eigenvalues ε<sup>i</sup> (in equation 3) should – formally speaking – not be interpreted as excitation energies (except for the highest one). Nevertheless optical excitations are commonly described in the independent particle approximations, using these quasi particle states from DFT in the single-particle picture. One well known case is the energy gap of insulators, which in this crude single-particle picture is typically underestimated by about 50 per cent. This has been well known for some time (see e.g. section 6.7 of [4]), since even the exact Kohn-Sham gap misses the integer discontinuity Δxc between occupied and unoccupied states. It is worth considering that in Hartree Fock the gap found would typically be too large. This is one of the reasons, why hybrid functionals were suggested which mix Hartree Fock with DFT in order to produce the correct gap. Better estimates of the quasi-particle spectrum can be obtained by GW calculations employing many-body perturbation theory, which is significantly more computationally expensive. Recently a modified Becke Johnson (mBJ) potential was proposed [27], which is still a local potential (and thus cheap) but yields energy gaps close to experiment..

When the Coulomb potential is written in terms of the density (third term in equation 2) it contains the unphysical self interaction of an electron with itself. In Hartree-Fock this term is exactly canceled by the exchange term. Due to the approximation in DFT, this cancellation is not complete and thus in some functionals a self-interaction-correction (SIC) is added [28].

The van der Waals (vdW) interaction is not described in the simple DFT approximations like LDA or GGA but can be treated with higher order treatments (rung 4 or 5 in Jacob's ladder) which become computationally rather expensive. A pragmatic solution is to add a vdW correction based on adjustable parameters (see for example Grimme's scheme [29]).

In connection with the term "*ab inito*" ‒ as the quantum chemists define it ‒ it is appropriate to consider the situation for large systems: the strategy differs for schemes based on HF (wave function based) or DFT. In HF based methods (including CI and CC) the Hamiltonian is well defined and can be solved almost exactly for small systems but for large cases only approxi‐ mately (i.e. due to limited basis sets). In DFT, however, one must first choose the functional that is used to represent the exchange and correlation effects (or approximations to them) but then one can solve this effective Hamiltonian almost exactly. Thus in both cases an approxi‐ mation enters either in the first or second step. This perspective illustrates the importance of improving the functionals in DFT calculations, since they define the quality of the calculation. The advantage for DFT is that it can treat relatively large systems.

#### **4. Solving the Kohn-Sham equations with WIEN2k**

#### **4.1. The all electron case**

dependence on the density alone, to an orbital description, which (for occupied orbitals) allows a correct description of exchange, like in Hartree-Fock. At this level one limits the computation space to the occupied orbitals but can extend it to the hybrid functions (mixing a fraction of Hartree-Fock with a part in DFT). In the highest rung also unoccupied orbitals are included,

**Figure 8.** Jacob's ladder according to [24] with 5 rungs, demonstrating how to improve the exchange correlation treat‐

There are well documented cases for which conventional DFT calculations (LDA or GGA) disagree even qualitatively with experimental data and lead, for instance, to predict a metal instead of an insulator. One of the reasons can be the presence of localized states (often felectrons or late transition metal d-orbitals) for which correlation is very strong. For these highly correlated systems one must go beyond simple DFT calculations. One simple form of improvement is to treat theses local correlations by means of a Hubbard U (see [26]) but use LDA or GGA for the rest of the electrons. With this parameter U the on-site Coulomb repulsion between the localized orbitals is included, but by introducing a parameter. This approach is generally called LDA+U. In a simple picture, U stands for the energy penalty of moving a

interpreted as excitation energies (except for the highest one). Nevertheless optical excitations are commonly described in the independent particle approximations, using these quasi particle states from DFT in the single-particle picture. One well known case is the energy gap of insulators, which in this crude single-particle picture is typically underestimated by about 50 per cent. This has been well known for some time (see e.g. section 6.7 of [4]), since even the exact Kohn-Sham gap misses the integer discontinuity Δxc between occupied and unoccupied states. It is worth considering that in Hartree Fock the gap found would typically be too large.

(in equation 3) should – formally speaking – not be

as for example in the scheme called random phase approximation (RPA).

localized electron to the neighboring site that is already occupied.

The Kohn-Sham energy eigenvalues ε<sup>i</sup>

288 Selected Topics in Applications of Quantum Mechanics

ment.

A schematic summary of the main choices one has to make for computations is shown in Figure 1, where our selections are marked in red. We want to represent a solid with a unit cell (or a Supercell) as discussed in section 2 and thus invoke periodic boundary conditions. The system may contain all elements of the periodic table, from light to heavy, main group, transition metals, or rare earth atoms as for example shown in Figure 3. Let us look at Ti (with the atomic number Z=22) as an example. Its electronic configuration is 1s2 2s2 2p6 3s2 3p6 3d2 4s2 (Figure 9).

We surround the Ti nucleus by an atomic sphere with a radius of 2 Bohr (about 1Å). With respect to this sphere, the Ti electronic states can be classified in three categories:


**Figure 9.** The electronic states of titanium: core-semi-core, and valence states.

**•** semi-core states, which are in between (medium energy), have a charge leakage (a few per cent of the charge density lies outside the sphere) and have a principal quantum number that is one less than the valence states (i.e. 3s vs. 4s, or 3p vs. 4p).

Traditionally the electronic properties of a material due to the chemical bonding are associated with only the valence electrons and thus the core electrons are often ignored. A typical scheme is the so called frozen core approximation, in which the electron density from the core electrons does not change during the SCF cycle (see Figure 6). This is often justified but there are cases like hyperfine interactions, where the change of the core electrons can contribute significantly and even more so the semi-core states. An all-electron treatment has therefore the advantage of being able to explore the contribution from all electrons to certain experimental data (e.g. the electric field gradient).

As long as a solid contains only light elements, non-relativistic calculations are well justified, but as soon as a system of interest contains heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) so called scalar relativistic schemes [30] are often used, which properly describe the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass velocity term) but omit spin-orbit coupling. Such schemes are computationally relatively simple and thus recommended for a standard case. The inner electrons can reach a high velocity leading to a mass enhancement. This causes a stronger screening of the nuclear charge by the relativistic core electrons with respect to a non-relativistic treatment and affects the valence electrons. The spin-orbit contri‐ bution can be included in a second–variational treatment [31] and is needed for heavier elements. The core electrons are treated by solving Dirac's equation, whereas the semi-core and valence states are described with the scalar relativistic scheme. In the latter spin remains a good quantum number and thus spin-polarized calculations are valid to treat magnetic systems.

#### **4.2. The choice of the potential**

**•** semi-core states, which are in between (medium energy), have a charge leakage (a few per cent of the charge density lies outside the sphere) and have a principal quantum number

Traditionally the electronic properties of a material due to the chemical bonding are associated with only the valence electrons and thus the core electrons are often ignored. A typical scheme is the so called frozen core approximation, in which the electron density from the core electrons does not change during the SCF cycle (see Figure 6). This is often justified but there are cases like hyperfine interactions, where the change of the core electrons can contribute significantly and even more so the semi-core states. An all-electron treatment has therefore the advantage of being able to explore the contribution from all electrons to certain experimental data (e.g.

As long as a solid contains only light elements, non-relativistic calculations are well justified, but as soon as a system of interest contains heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) so called scalar relativistic schemes [30] are often used, which properly describe the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass velocity term) but omit spin-orbit coupling. Such schemes are computationally relatively simple and thus recommended for a standard case. The inner electrons can reach a high velocity leading to a mass enhancement. This causes a stronger screening of the nuclear charge by the relativistic core electrons with

that is one less than the valence states (i.e. 3s vs. 4s, or 3p vs. 4p).

**Figure 9.** The electronic states of titanium: core-semi-core, and valence states.

290 Selected Topics in Applications of Quantum Mechanics

the electric field gradient).

Figure 1 schematically shows the topics where one needs to make a choice. We want to represent a solid with a unit cell (or a supercell). Relativistic and spin-polarization effects can be included as mentioned above. The next crucial point is the choice of the potential that is closely related to the basis sets. This aspect is extensively explained in [18] discussing the advantages and problems connected with pseudo potentials. The main idea is to eliminate the core electrons and replace the real wave functions of the valence states by pseudo wave functions which are sufficiently smooth so that they can be expanded in a plane wave basis set. In the outer region of an atom, where the chemical bonding occurs they should agree with the real wave function. In principle – in mathematical terms – plane waves form a complete basis set and thus should be able to describe any wave function. However, the nodal structure (for example of a 4s wave function) close to the nucleus would need to be described by extremely many plane waves.

For the all-electron case within DFT the potential looks like the one shown in Figure 10. Near each nucleus it has the form Z/r but between the atoms it is nearly flat. In the muffin-tin approximation the potential is assumed to be spherically symmetric around the atom but constant in between. In the full-potential case the potential (without any approximation of its shape) can be represented as a Fourier series in the interstitial region, but in each atomic sphere (with a radius RMT) it can be expressed as a radial function VLM(r) multiplied by crystal harmonics, which are linear combinations of spherical harmonics having the point group symmetry of the atom α for the proper LM value. In this notation the muffin-tin case is the first term in both cases, namely the 0 0 component for LM (i.e. only the spherical part inside each atomic sphere) and a constant for the Fourier series (for the interstitial region). In the 1970ths the muffin-tin approximation was widely used because it made calculations feasible. For closely packed systems it was acceptable but for more covalently bonded systems like silicon (or even surfaces) it is a very poor approximation. Another drawback of the muffin-tin approximation was that the results depended on the choice of sphere radii, whereas in the fullpotential case this dependence is drastically reduced. Due to the muffin-tin approximation different computer codes obtained results that did not agree with each other. This has changed with the use of full-potential calculations. Nowadays different codes based on the full potential yield nearly identical results provided they are carried out to full convergence and use the same structure and DFT version. This has given theory a much higher credibility and predict‐ ability (see Section 7).

**Figure 10.** The full potential vs. muffin tin approximation is shown for a (110) plane of SrTiO3.

#### **4.3. The choice of the basis sets**

For solving the Kohn Sham equations (see Figure 1) basis sets are needed. A linear combination of such basis functions shall describe the Kohn-Sham orbitals. One can use analytic functionssuch as Slater type orbitals (STO) or Gaussian type orbitals-or just plane waves (for example in connection with pseudo potentials). Already in 1937 Slater [32] proposed the augmented plane wave (APW) method. The development of APW and its linearized version, which led to the WIEN code [6] and later to its present version WIEN2k [7] was described in detail in recent reviews [1-5]. An extensive description including many conceptual and mathematical details is given in [18]. Therefore only the main concepts will be summarized below.

#### **4.4. The APW based method and the WIEN2k code**

In the APW method one partition the unit cell into (non-overlapping) atomic spheres (type I) centered at the atomic sites and the remaining interstitial region (II) (Figure 11). Inside each atomic sphere (region I) the wave functions have nearly an atomic character and thus (assum‐ ingamuffin-tinpotential)canbewrittenasaradialfunctiontimes sphericalharmonics.It should be stressed that the *muffin tin* approximation (MTA) is used only for the construction of the APW basis functions and only for that. The radial Schrödinger equation is solved numerically (and thus highly accurately), but as input the energy must be provided, which makes the basis set energy dependent. In region II the potential varies only slowly and thus the wave func‐ tions can be well expressed in a series of plane waves (PW). Each plane wave is augmented by the atomic partial waves inside each atomic sphere (i.e. the PW is replaced inside the spheres). The corresponding weight Aℓm of each partial wave can be fixed by a matching condition at the sphere boundary (as indicated in Figure 11 and Figure 13).

**Figure 11.** The Augmented Plane Wave (APW) method.

**4.3. The choice of the basis sets**

292 Selected Topics in Applications of Quantum Mechanics

For solving the Kohn Sham equations (see Figure 1) basis sets are needed. A linear combination of such basis functions shall describe the Kohn-Sham orbitals. One can use analytic functionssuch as Slater type orbitals (STO) or Gaussian type orbitals-or just plane waves (for example in connection with pseudo potentials). Already in 1937 Slater [32] proposed the augmented plane wave (APW) method. The development of APW and its linearized version, which led to the WIEN code [6] and later to its present version WIEN2k [7] was described in detail in recent reviews [1-5]. An extensive description including many conceptual and mathematical details

In the APW method one partition the unit cell into (non-overlapping) atomic spheres (type I) centered at the atomic sites and the remaining interstitial region (II) (Figure 11). Inside each atomic sphere (region I) the wave functions have nearly an atomic character and thus (assum‐ ingamuffin-tinpotential)canbewrittenasaradialfunctiontimes sphericalharmonics.It should be stressed that the *muffin tin* approximation (MTA) is used only for the construction of the APW basis functions and only for that. The radial Schrödinger equation is solved numerically (and thus highly accurately), but as input the energy must be provided, which makes the basis set energy dependent. In region II the potential varies only slowly and thus the wave func‐ tions can be well expressed in a series of plane waves (PW). Each plane wave is augmented by the atomic partial waves inside each atomic sphere (i.e. the PW is replaced inside the spheres).

is given in [18]. Therefore only the main concepts will be summarized below.

**Figure 10.** The full potential vs. muffin tin approximation is shown for a (110) plane of SrTiO3.

**4.4. The APW based method and the WIEN2k code**

Three schemes of augmentation (APW, LAPW, APW+lo) have been suggested over the years and illustrate the progress in this development of APW-type calculations that was discussed in [18, 4, 5]. Here only a brief summary will be given. The energy dependence of the atomic radial functions uℓ(r,E) can be treated in different ways. In Slater's APW [32] this was done by choosing a fixed energy E, which leads to a non-linear eigenvalue problem, since the basis functions become energy dependent.

In the linearized APW, called LAPW, O. K. Andersen [33], suggested to linearize (that is treat to linear order) this energy dependence as illustrated in Figure 12. The radial Schrödinger equation is solved for a fixed linearization energy Eℓ (taken at the center of the corresponding energy bands) leading to uℓ(r, Eℓ) but adding an energy derivative of this function (taken at the same energy) in order to retain the variational flexibility. This linearization is a good approximation in a sufficiently small energy range around Eℓ. In LAPW the atomic function inside the sphere α is given by a sum of partial waves, namely radial functions times spherical harmonics labeled with the quantum numbers (ℓ, m).

**Figure 12.** The energy variation of the radial wave function uℓ(Eℓ,r) according to LAPW [33] is schematically shown: i) for the center of the band (taken from a sketched density of states shown on the right), ii) for the energy Ebottom at the bottom of the band (bonding case where the radial wave function has zero slop at the sphere boundary RMT as shown in Figure 13), and iii) for the energy Etop at the top of this band (antibonding case, where the wave function has a node at RMT). In LAPW this energy dependence is linearized and expressed as the radial function and its energy derivative both taken at Eℓ, where the relative weights are determined by matching (in value and slope) to plane waves at RMT (as shown in Figure 13).

The two coefficients Aℓm and Bℓm (weight for function and derivative) – as given in Figure 12 can be chosen so as to match each plane wave (characterized by K) continuously (in value and slope) to the one-center solution inside the atomic sphere at the sphere boundary (for details see e.g. [18]). The main advantage of the LAPW basis set is that it allows finding all needed eigenvalues with a single diagonalization, in contrast to APW, which has the non-linear eigenvalue problem. Historically, the more strict constrain (a matching of both value and slope) had the disadvantage that in LAPW more PWs were needed to reach convergence than in APW. The LAPW basis functions u and it derivative are recalculated in each iteration cycle (see Figure 6) and thus can adjust to the chemical changes (for example due to charge transfer) requiring an expansion or contraction of the radial function. The LAPW method made it computationally attractive to go beyond the muffin-tin approximation and to treat both the crystal potential and the charge density without any shape approximation (called fullpotential) as pioneered by the Freeman group [34].

In section 4.1 the partition of electronic states in core, semi-core and valence states was described and illustrated for Ti in Figure 9. Let us focus on the p-type orbitals. The 2p core state is treated fully relativistic as an atomic core state while the valence 4p state is computed within LAPW using a linearization energy at the corresponding high energy. The 3p semi-core states reside mostly inside the Ti sphere but have a "core-leakage" of a few per cent. The 3p states are separated in energy from the 4p states and thus the linearization (with the lineari‐ zation energy of the 4p state) would not work here. For such a case Singh [35] proposed adding local orbitals (LO) to the LAPW basis set in order to accurately treat states with different principal quantum numbers (e.g. 3p and 4p) while retaining orthogonality. In this example the 3p LOs look very similar to the 3p radial function but are constrained to have zero value and slope at the sphere radius RMT.

The concept of LO fostered another idea, namely the APW plus local orbitals (APW+lo) method by Sjöstedt et al [36]. These local orbitals are labeled in lower case to distinguish them from the semi-core LO. In APW+lo, one returns to the APW basis but with the crucial difference that each radial function is expanded at a fixed energy. The matching is again (as in APW) only made between values (Figure 13). This new scheme is significantly faster while maintaining the convergence of LAPW [37].

**Figure 12.** The energy variation of the radial wave function uℓ(Eℓ,r) according to LAPW [33] is schematically shown: i) for the center of the band (taken from a sketched density of states shown on the right), ii) for the energy Ebottom at the bottom of the band (bonding case where the radial wave function has zero slop at the sphere boundary RMT as shown in Figure 13), and iii) for the energy Etop at the top of this band (antibonding case, where the wave function has a node at RMT). In LAPW this energy dependence is linearized and expressed as the radial function and its energy derivative both taken at Eℓ, where the relative weights are determined by matching (in value and slope) to plane waves at RMT (as

The two coefficients Aℓm and Bℓm (weight for function and derivative) – as given in Figure 12 can be chosen so as to match each plane wave (characterized by K) continuously (in value and slope) to the one-center solution inside the atomic sphere at the sphere boundary (for details see e.g. [18]). The main advantage of the LAPW basis set is that it allows finding all needed eigenvalues with a single diagonalization, in contrast to APW, which has the non-linear eigenvalue problem. Historically, the more strict constrain (a matching of both value and slope) had the disadvantage that in LAPW more PWs were needed to reach convergence than in APW. The LAPW basis functions u and it derivative are recalculated in each iteration cycle (see Figure 6) and thus can adjust to the chemical changes (for example due to charge transfer) requiring an expansion or contraction of the radial function. The LAPW method made it computationally attractive to go beyond the muffin-tin approximation and to treat both the crystal potential and the charge density without any shape approximation (called full-

In section 4.1 the partition of electronic states in core, semi-core and valence states was described and illustrated for Ti in Figure 9. Let us focus on the p-type orbitals. The 2p core state is treated fully relativistic as an atomic core state while the valence 4p state is computed within LAPW using a linearization energy at the corresponding high energy. The 3p semi-core states reside mostly inside the Ti sphere but have a "core-leakage" of a few per cent. The 3p states are separated in energy from the 4p states and thus the linearization (with the lineari‐

shown in Figure 13).

294 Selected Topics in Applications of Quantum Mechanics

potential) as pioneered by the Freeman group [34].

**Figure 13.** The (linearized) augmented plane wave method as implemented in WIEN2k [7] defining i) the different atomic partial waves in LAPW and APW+lo used inside the atomic sphere, ii) the plane waves used in the interstitial region, iii) the matching at the sphere boundary, and iv) illustrating for an Fe-4p orbital how the different matching looks at the sphere boundary for APW and LAPW.

The APW+lo scheme therefore combines the best features of all APW-based methods. It was known that LAPW converges somewhat slower than APW due to the constraint of having differential basis functions and thus it is an improvement to return to APW but only for the orbitals involved in chemical bonding. The energy-independent basis introduced in LAPW is crucial for avoiding the general eigenvalue problem of APW and thus is also used for all higher ℓ components. The local orbitals provide the necessary variational flexibility to make this new scheme efficient but they are added only where needed (to avoid any further increase in basis set). The crystalline wave functions (of Bloch type) are expanded in these APWs leading to a general eigenvalue problem. The size of the matrix is mainly given by the number of plane waves (PWs) but is increased slightly by the additional local orbitals that are used. As a rule one can say that about 50-100 PWs are needed for every atom in the unit cell in order to achieve good convergence.

#### **5. Results with WIEN2k**

The WIEN2k code is widely used and thus there is an enormous literature with many inter‐ esting results which cannot all be covered here. Many of the publications with WIEN2k can be found on the web page www.wien2k.at under the heading papers. A selected list of results, that can be obtained with WIEN2k, is provided below, where references are specified either to the original literature or in some cases to review articles [4, 5].


be decomposed into its components by means of the qtℓm values mention above. This decomposition becomes even more important in complicated cases, for example if one wants to find which state originates from an impurity atom in a supercell.

differential basis functions and thus it is an improvement to return to APW but only for the orbitals involved in chemical bonding. The energy-independent basis introduced in LAPW is crucial for avoiding the general eigenvalue problem of APW and thus is also used for all higher ℓ components. The local orbitals provide the necessary variational flexibility to make this new scheme efficient but they are added only where needed (to avoid any further increase in basis set). The crystalline wave functions (of Bloch type) are expanded in these APWs leading to a general eigenvalue problem. The size of the matrix is mainly given by the number of plane waves (PWs) but is increased slightly by the additional local orbitals that are used. As a rule one can say that about 50-100 PWs are needed for every atom in the unit cell in order to achieve

The WIEN2k code is widely used and thus there is an enormous literature with many inter‐ esting results which cannot all be covered here. Many of the publications with WIEN2k can be found on the web page www.wien2k.at under the heading papers. A selected list of results, that can be obtained with WIEN2k, is provided below, where references are specified either

**•** After the SCF cycle has been completed one can look at various standard results: the Kohn-Sham eigenvalues En**<sup>k</sup>** can be shown along symmetry lines in the Brillouin zone giving the energy band structure. A symmetry analysis can determine the corresponding irreducible representation (see Fig.1 of [4]). For each of these states with En**k** the wave function (a complex function in three dimensions) contains information about how much the various regions of the unit cell contribute. In the APW framework this can be done by using the partial charges qtℓm which define the fraction of the total charge density of this state (normalized in the unit cell) that resides in the atomic sphere t and comes from the orbital characterized by the quantum numbers ℓm. The fraction of the charge that resides in the interstitial region is contained in qout. These numbers, which depend on the choice of sphere radii, help to interpret each state in terms of chemical bonding. This is an advantages of this type of basis set. There is a useful option to show the character of bands. As one example, three options of presenting the band structure are illustrated for the refractory metal titanium carbide TiC shown in Fig.1 of reference [2] showing the Ti-d (e<sup>g</sup> symmetry) and Cp character bands, which dominate the bonding in this case. The crystal field of TiC splits the fivefold degenerate Ti-d orbitals into t2g and eg states (with a degeneracy of 3 and 2 respectively). Another example is the band structure of Cu shown in Fig. 2.2.16.1 of [12]. **•** The Fermi surface in a metal is often crucial for an understanding of properties (for example superconductivity). It can be calculated on a fine **k**-mesh and plotted (for example with

**•** With a calculation for a (sufficiently fine) uniform mesh of **k**-points s in the irreducible Brillouin zone as discussed in connection with Figure 6 one can determine the density of states (DOS), which gives a good description of the electronic structure. The total DOS can

to the original literature or in some cases to review articles [4, 5].

good convergence.

XCrysDen [38]).

**5. Results with WIEN2k**

296 Selected Topics in Applications of Quantum Mechanics


some regions but unfavorable "wires" in others. This corrugated BN surface was found to agree with experimental data (for further details see [41-43]). Another example is the investigation of so called misfit layer compounds [11], in which the bonding between the layers of TaS2 and PbS required that some Pb atoms are replaced by Ta in an disordered fashion. Relatively large supercells were needed in order to represent this cross substitution. After relaxing the atomic positions the more likely arrangements have been determined on the basis of total energy differences.

**Figure 14.** A hexagonal boron nitride (13 x13) is bonded to a Rh(111) surface (12 x 12) forming (a) a nanomesh; (b)N sits on an unfavorable hollow position (between three Rh) and thus BN is far away from Rh called "wire" ; (c) N is on the favorable position on top of Rh called "pores"; (d): B is on top of Rh, called "pores" (see [43]).

**•** In the case of magnetic systems spin-polarized calculations can provide the magnetic moments. In addition to collinear magnetic systems also non collinear magnetism can be handled, which was for example used in a study of UO2 (see [44]. Another example is the Verwey transition that was investigated for double perovskite BaFe2O5. At low temperature this system has a charged-ordered state (with Fe2+and Fe3+at different sites) but above the Verwey transition temperature at about 309 K a valence mixed state with the formal oxidation state Fe2.5+appears. DFT calculations made it possible to interpret this complicated situation, see [45] and section 7.4.1 of [5]. In the latter it was mentioned that it is now possible to use such calculations to look for fine details such as the magneto-crystalline anisotropy. This is defined as the total energy difference between a case, where the magnetic moment is in the y direction (with the lowest energy) or the x direction. In this case the difference in energy is found to be about 0.4 mRy but the total energy is-115,578.24065 Ry. Therefore the quantity of interest is in the tenth decimal illustrating the numerical precision that is needed for such a quantity. Needless to say that extremely well converged calculations were required, in which both cases are treated practically the same. This is necessary to have a cancellation of errors.

some regions but unfavorable "wires" in others. This corrugated BN surface was found to agree with experimental data (for further details see [41-43]). Another example is the investigation of so called misfit layer compounds [11], in which the bonding between the layers of TaS2 and PbS required that some Pb atoms are replaced by Ta in an disordered fashion. Relatively large supercells were needed in order to represent this cross substitution. After relaxing the atomic positions the more likely arrangements have been determined on

**Figure 14.** A hexagonal boron nitride (13 x13) is bonded to a Rh(111) surface (12 x 12) forming (a) a nanomesh; (b)N sits on an unfavorable hollow position (between three Rh) and thus BN is far away from Rh called "wire" ; (c) N is on

**•** In the case of magnetic systems spin-polarized calculations can provide the magnetic moments. In addition to collinear magnetic systems also non collinear magnetism can be handled, which was for example used in a study of UO2 (see [44]. Another example is the Verwey transition that was investigated for double perovskite BaFe2O5. At low temperature this system has a charged-ordered state (with Fe2+and Fe3+at different sites) but above the Verwey transition temperature at about 309 K a valence mixed state with the formal

the favorable position on top of Rh called "pores"; (d): B is on top of Rh, called "pores" (see [43]).

the basis of total energy differences.

298 Selected Topics in Applications of Quantum Mechanics


on all the other atoms. The necessary independent displacements are determined by the symmetry of the cell. By diagonalizing the dynamical matrix the phonon frequencies can be determined. Such information is also useful for example in connection with ferroelectrics, structural stability, thermodynamics or phase transitions.


#### **6. Computer code development**

From the experience of developing the WIEN2k code some general conclusions can be drawn. Some of the historical perspectives have been summarized in section 7 of [4]. During the last three to four decades it was often necessary to port the code to new architectures starting from main-frame computers, vector processors, PCs, PC-clusters, shared-memory machines, to multi-core parallel supercomputers. The power of computers has increased in several areas by many orders of magnitude such as the available memory (from kB to TB), the speed of communication (e.g. infiniband) all the way to the processors (CPU). An efficient implemen‐ tation of a code made it necessary to closely collaborate with mathematicians and computer scientists in order to find the optimal algorithms, which perform well on the available hardware. One example is the idea of using the scheme of iterative diagonalization [59]. A significant portion of the computational effort in the WIEN2k calculations is the solution of the general eigenvalue problem (see Figure 6) which must be solved repeatedly within the SCF cycle. Changes from iteration to iteration are often small and thus one can use the information from the previous iteration to define a preconditioner for the next iteration and thus simplify the diagonalization and speed up the calculation.

Another aspect is the implementation of linear algebra libraries (e.g. SCALAPACK, MKL), which were highly optimized by other groups, who spend a lot of effort on these tasks, and helped us significantly to speed up our code. Simultaneously, increased computer power made it possible to treat much larger systems, especially using massive parallelization. The matrix size that we could handle on the available hardware has increased by about a factor 1000 over the last several decades. Since solving the general eigenvalue problem scales as N3 , the computer power needed to solve a 1000 times bigger system must be about a factor 109 higher, which is available now.

Often our computational strategy had to be changed or extended. For example, to compute a metallic crystal with a small unit cell many **k**-points s in the Brillouin zone were needed to reach a good convergence. In this case **k**-points parallelization was optimal. Nowadays we can treat large unit cells (containing about 1000 atoms). In such a case, the reciprocal space is small and thus only few **k**-points s are needed for a good calculation. This requires new paralleli‐ zation strategies, in which the large matrices must be distributed to many processors, where data locality and reduced communication is essential for achieving good parallel performance. Another aspect is the complexity of the code with the many tasks that need to be solved (see Figure 6). If only a small fraction is not parallelized, it may keep many processors waiting for the result that is calculated on only a single processor. This has often led to new bottle necks, which did not occur for smaller systems and thus were ignored but load balancing is important. Better computer power requires a continuous improvement of the code.

There are completely different ways of distributing a code (giving representative examples):

**•** open source with a free download

on all the other atoms. The necessary independent displacements are determined by the symmetry of the cell. By diagonalizing the dynamical matrix the phonon frequencies can be determined. Such information is also useful for example in connection with ferroelectrics,

**•** For the analysis of phase transitions a fundamental understanding requires a combination of concepts, namely group theory, DFT calculations, frozen phonons, soft modes or bilinear couplings, and Landau theory. This was illustrated, for example for an Aurivillius com‐ pound [54], which shows multiple instabilities and has a phase transition to a ferroelectric state. For high pressure phase transitions a modified Landau theory was proposed and

**•** Maximally localized Wannier functions can be calculated with wien2wannier [56] and provide a good starting point for more sophisticated many body theory. Dynamical mean field theory (DMFT) is one such example as is illustrated in [57]. Another extension of WIEN2k is the calculation of Berry phases with wien2kPI as modern theory of polarization

**•** Computer graphics and visualization (see [38]] can help to analyze the many intermediate results (atomic structure, character of energy bands, Fermi surfaces, electron densities, partial density of states, etc.). The more complex a case is the more support from computer graphics is needed. For an element one can plot all the energy bands, but for systems with over 1000 atoms one would be lost interpreting the band structure without the help of

From the experience of developing the WIEN2k code some general conclusions can be drawn. Some of the historical perspectives have been summarized in section 7 of [4]. During the last three to four decades it was often necessary to port the code to new architectures starting from main-frame computers, vector processors, PCs, PC-clusters, shared-memory machines, to multi-core parallel supercomputers. The power of computers has increased in several areas by many orders of magnitude such as the available memory (from kB to TB), the speed of communication (e.g. infiniband) all the way to the processors (CPU). An efficient implemen‐ tation of a code made it necessary to closely collaborate with mathematicians and computer scientists in order to find the optimal algorithms, which perform well on the available hardware. One example is the idea of using the scheme of iterative diagonalization [59]. A significant portion of the computational effort in the WIEN2k calculations is the solution of the general eigenvalue problem (see Figure 6) which must be solved repeatedly within the SCF cycle. Changes from iteration to iteration are often small and thus one can use the information from the previous iteration to define a preconditioner for the next iteration and thus simplify

Another aspect is the implementation of linear algebra libraries (e.g. SCALAPACK, MKL), which were highly optimized by other groups, who spend a lot of effort on these tasks, and

structural stability, thermodynamics or phase transitions.

applied [55].

visualization.

in a solid (for details see ref [58]).

300 Selected Topics in Applications of Quantum Mechanics

**6. Computer code development**

the diagonalization and speed up the calculation.


From a commercial point to view it is understandable that a company wants to have strict rules and do not make the source code available. From a scientific perspective, the WIEN2k group favors, the source code is made available to the registered users, who pay a small license fee once. This policy has helped to generate a "WIEN2k community", from which many research‐ ers around the world have contributed to the development of the code and can contribute to do so in the future. It has helped in many aspects, such as to find and fix bugs, but also to add new features which are made available to all the WIEN2k users. In addition, several valuable suggestions were made, which allowed improving the documentation as well as implementing requested new features. We have organized more than 20 WIEN2k workshops worldwide, in which users are introduced to important concepts and learn how to run calculations and use kinds of associated tools. It has become a standard to help each other and thus contribute to the development of computations of solids and surfaces. In total this policy has had very positive impacts for WIEN2k and the field.

The user friendliness of WIEN2k has been improved over the years. A graphical user interface w2web was mainly developed by Luitz (see [7]) and is especially useful for novice users or in cases which are not done routinely. Later many default options were implemented, which were based on the experience of many previous calculations. This has made it much simpler to set up a calculation. For novice users or experimentalists this helps one to get started without being an expert. However, there is also a drawback, namely the danger that the code will be more used like a black box: "push a button and receive the result". In the old version the users were forced to think about how to run the calculation and thus had to look at details. This is a common problem, which all codes face.

With all the possibilities mentioned in the previous section it is often useful to combine different theories according to their advantages but keeping in mind their disadvantages. About 20 years ago the fields of quantum chemistry, DFT and many-body theory were completely separated and there was hardly any cooperation between them: this has fortunately changed. The strength and weaknesses of the different approaches are recognized and mutually appreciated. The solution of complex problems can only be found in close collabo‐ ration of the corresponding experts.

#### **7. Theory and simulations**

#### **7.1. Theory compared to experiment**

Independent of which computer code is used for computations some general questions should be asked, when theory and experiment do not agree. Some possible reasons for a disagreement are listed below (Figure 15):


results or predictions. Would a different code yield other results (within a small error bar)? These tests showed that significant improvements of standard pseudo potentials were necessary, in order to reach the required accuracy. It was shown in [60] that the typical deviation (e.g. in total energy) between codes of different accuracy is an order of magnitude smaller than the typical difference with experiment.



**Figure 15.** Key aspects of modeling materials and the idealizations or approximations that must be made.

When a computation agrees with the experimental observations one should be careful because this is not a proof that the issues mentioned above are non-existent. It can be the case that there is a cancellation of errors: something like a crude model that is poorly converged. For a theorist there is always the temptation to stop improving a calculation when it already agrees with the experiments. Sometimes a deviation helps to find a better treatment in terms of atomic models, quantum mechanics (DFT functional), basis sets, temperature or some other factor that may be important in the specific case.

#### **7.2. Results that can be provided by computations**

to set up a calculation. For novice users or experimentalists this helps one to get started without being an expert. However, there is also a drawback, namely the danger that the code will be more used like a black box: "push a button and receive the result". In the old version the users were forced to think about how to run the calculation and thus had to look at details. This is

With all the possibilities mentioned in the previous section it is often useful to combine different theories according to their advantages but keeping in mind their disadvantages. About 20 years ago the fields of quantum chemistry, DFT and many-body theory were completely separated and there was hardly any cooperation between them: this has fortunately changed. The strength and weaknesses of the different approaches are recognized and mutually appreciated. The solution of complex problems can only be found in close collabo‐

Independent of which computer code is used for computations some general questions should be asked, when theory and experiment do not agree. Some possible reasons for a disagreement

**•** Is the atomic structure model that was chosen for the computation adequate for the experimental situation, as already discussed in Section 2. An advantage of theory is that the structure is well defined, because it is taken as input. Experiments may have uncertainties (stoichiometry, defects, impurities, disorder). It can also be that the theory is based on an idealized structure such as infinite crystal, whereas in the experiment surface effects cannot be neglected. If the latter are included in a supercell calculation, one has periodic boundary conditions and thus still assumes an ordered structure, while in the experiment the sample is disordered or contains some defects or impurities. A delicate question for the experimen‐ talist is whether the sample that has been measured is (at least) close to the system that was

**•** Is the chosen quantum mechanical treatment appropriate for the given system? Is a mean field DFT approach adequate? Are more sophisticated treatments (especially for correlation) needed or can the self-interaction within DFT cause the problem? Is there a significant

**•** Is the performed calculation fully converged to the required accuracy, for example in terms of the basis set (for example in the number of plane waves or in other cases the choice of pseudo potential) or the underlying **k**-points mesh? In this context an evaluation of the chosen computer code can be important. Recently error estimates of solid state DFT calculations have been derived [60] in which the WIEN2k code plays the role of providing the standard (i.e. the most accurate calculation). The idea is that different implementations of the same first principles formalism (the same DFT functional) should lead to the same

a common problem, which all codes face.

302 Selected Topics in Applications of Quantum Mechanics

ration of the corresponding experts.

**7. Theory and simulations**

are listed below (Figure 15):

assumed for the simulation.

dependence on the functional chosen within DFT?

**7.1. Theory compared to experiment**

It is appropriate to list aspects, where theory has advantages over experimental work:

**•** On can carry out computer experiments irrespective of the abundance, environmental effects or cost of materials. Even unstable or artificial systems (which cannot be measured) can be computed. One might wish to understand why they cannot be prepared. Sometimes several proposals (based on experiments or intuition) are under discussion: as long as they are not too many in number, theory can explore all of them and hopefully find the one which agrees with what is known about the system.


#### **8. Conclusion**

In this chapter a selection of aspects, which play a role in modern computational theory of solids, has been given. From the atomic structure to the properties of inorganic materials a wide field of disciplines had to be included which are listed below:


**•** In surface science and catalysis the calculation of observable properties are often needed, such as STM images, X-ray spectra, vibrational frequencies, electric field gradients, etc.. In addition, comparing total energies of proposed atomic structure (after a structure optimi‐ zation) are often essential determining which atomic structure is likely to be correct. For an understanding of such material problems (for example at a surface) a combination of theory

**•** With a good calculation and by using all available tools one can gain insight and a funda‐ mental understanding, especially in terms of trends, for which perfect agreement with experiment is not necessary. On such a basis systematic predictions can be made which can

**•** One often wants to know the driving force for a certain change in properties. Is it coming from a substitution, the difference in chemical bonding, or from the related relaxation of the lattice around the impurity? Sometimes it is possible to set up artificially intermediate models which vary only one of these parameters at a time. Then an analysis can provide the

**•** For materials with a clear structure and moderate correlation effects theory can predict experimental results. However, there are many interesting cases in material science, where the details matter. Often the interest comes, because the system is close to a transition (e.g. becoming magnetic, ferroelectric, or close to a metal-insulator transition). In such cases the two phases of the system can be rather close (e.g. in energy) and thus need special attention. Take the perovskite SrTiO3 as an example. In this well known structure Ti is surrounded by six oxygen atoms forming an octahedron. Under pressure (or with temperature) these octahedra tilt in a certain fashion leading to a structural phase transition. If DFT theory yields a lattice constant that deviates by about 1 per cent from experiment, one could call this good agreement. In this case, however, this small deviation causes a difference of about 3 percent in the unit cell volume, which is sufficiently large to determine whether or not the tilt occurs (and thus such a detail matters). In such a case, one can choose another functional (for better agreement in volume) or one can carry out the calculation for the experimental volume, which can be obtained experimentally with high precision. With this choice a calculation

In this chapter a selection of aspects, which play a role in modern computational theory of solids, has been given. From the atomic structure to the properties of inorganic materials a

wide field of disciplines had to be included which are listed below: **•** chemistry: intuition, interpretation, chemical bonding, stability

**•** physics: fundamentals and concepts, quantum mechanics, relativity

**•** crystallography: space groups, symmetry relations, group subgroup relations

replace the trial and error scheme that is often used in materials optimization.

and experiment is essential.

304 Selected Topics in Applications of Quantum Mechanics

can describe the phase transition properly.

answer.

**8. Conclusion**

**•** computer science: data management, data bases, memory, communication, parallelization, load balancing, efficiency

It is clear that many details had to be skipped and only a few references could be given to guide the reader to the corresponding literature. Details of using the computer codes often change and thus it is highly recommended to look at the updated versions on the web (www.wien2k.at) or at the newest literature in this field. Selected results that can be obtained with WIEN2k had been summarized in this chapter. For a given atomic structure the electronic structure (band structure, density of states, and electron density) provides the basis for understanding chemical bonding. The corresponding total energy allows to judge relative stabilities of various phases or modifications. The effects of surfaces or even disordered structures can be simulated with sufficiently large supercells. Properties of insulators, metals, superconductors, or magnets etc. can be explained. Several quantities (such as spectra) can be computed which allow a direct comparison with experimental data. In some cases it is necessary to go beyond conventional DFT in order to reach agreement with experiment but DFT results are often an important and useful starting point.

It shall be stressed again that it is very useful to have a large variety of computer codes in this field. Different codes each have their emphasis on various aspects, such as accuracy, efficiency, user friendliness, robustness, portability with respect to hard-ware, features, properties and more. Some codes are more specialized for certain cases (e.g. treating only insulators) but do not work for other systems. This variety has helped to increase the importance of simulations in this field. For the comparison between theories (simula‐ tions) and experimental data several general considerations are summarized which are important for all kinds of computer codes.

#### **Acknowledgements**

I am very grateful and want to thank the many researchers who have helped developing the WIEN2k code and made many important contributions or suggestions. First of all I want to thank Peter Blaha, who has been part of the WIEN2k team for almost 35 years and who is the main person managing the code development. He has also been involved in many research projects that were carried out with this code. In addition I want to thank S.B. Trickey from the University of Florida, who has initialized the development and publication of the code (see [6], and the review [4]), as well as the other co-authors of the code [6] and the many researchers, who have contributed as acknowledged in a link at www.wien2k.at. I want to thank in particular Eamon McDermott, who has helped improving the English.

#### **Author details**

Karlheinz Schwarz\*

Address all correspondence to: kschwarz@theochem.tuwien.ac.at

Institute of Materials Chemistry, Vienna University of Technology, Vienna, Austria

#### **References**


[12] Schwarz K, Electrons. In: Authier A(ed.) *International Tables for Crystallography, Vol‐ ume D, Physical Properties of Crystals.* Kluwer Academic Publ.; 2003, 294-313.

**Author details**

306 Selected Topics in Applications of Quantum Mechanics

Karlheinz Schwarz\*

**References**

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Address all correspondence to: kschwarz@theochem.tuwien.ac.at

Institute of Materials Chemistry, Vienna University of Technology, Vienna, Austria

[1] Schwarz K, Blaha P, Madsen G K H, Electronic structure calculations of solids using the WIEN2k package for material science, Computer Physics Communication, 2002;

[2] Schwarz K, Blaha P, Solid state calculations using WIEN2k, Computational Material

[3] Schwarz K, DFT calculations of solids with WIEN2k, J. Solid State Chemistry, 2003,

[4] Schwarz K, Blaha P, Trickey SB: Electronic structure of solids with WIEN2k. Molecu‐

[5] Schwarz K, Blaha P. Electronic structure of solids and surfaces with WIEN2k. In: Leszczyncski J, Shukla M K(eds) *Practical Aspects of Computational Chemistry I: An Overview of the Last Two Decades and Current Trends.* Springer Science+Business Media

[6] Blaha P, Schwarz K, Sorantin P, Trickey S B, Full-potential linearized augmented plane wave programs for crystalline solids, Computer Physics Communication. 1990;

[7] Blaha P, Schwarz K, Madsen G K H, Kvasnicka D, Luitz J, *An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties*, Vienna University of

[8] Hahn T (ed.), *International Tables for Crystallography, Volume A,Space-Group Symmetry*

[10] Diviš M, Schwarz K, Blaha P, Hilscher G, Michor M, Khmelevskyi S: Rare earth boro‐ carbides: Electronic structure calculations and the electric field gradients. Physical

[11] Kabliman E, P, Blaha P, Schwarz K, Ab initio study of the misfit layer compound

[9] Inorganic Crystal Structure Database (ICSD). URL: http://www.fiz-karlruhe.de.


[44] Laskowski R, Madsen G K H, Blaha P, Schwarz K, Magnetic structure of electric-field gradients of uranium dioxide: An ab initio study, Physical Review B, 2004; 69, 140408.

[28] Polo V, Kraka, E, Cremer D, Electron correlation and the self-interaction error of den‐

[29] Grimme S, Antony J, Ehrlich S, Krieg H, A consistent and accurate ab initio parame‐ terization of density functional dispersion correction (DFT-D) for 94 elements H-Pu,

[30] Koelling D D, Harmon B N, A technique for relativistic spin polarize calculations,

[31] MacDonnald A H, Picket W E, Koelling D D, A linearized relativistic augmentedplane-wave method utilizing approximate pure spin basis functions, Journal of Phys‐

[34] Weinert M, Wimmer E,Freeman A J, Total-energy all-electron density functional

[35] Singh D J, Ground-state properties of lanthanum: Treatment of extended core-states,

[36] Sjöstedt E, Nordström L, Singh D J, An alternative way of linearizing the augmented

[37] Madsen G H K, Blaha P, Schwarz K, Sjöstedt E, Nordström L, Efficient linearization of the augmented plane-wave method, Physical Review B, 2001; 64, 195134.

[38] Kokaj A, Computer graphics and graphical user interfaces as tools in simulations of

[39] Bader R W F, *Atoms in Molecules: a Quantum Theory*, Oxford university press, New

[40] Madsen G K H, Iversen B B, Blaha P, Schwarz K, The electronic structure of sodium and potassium electro sodalites (Na/K)8(AlSiO4)6, Physical Review B, 2001; 64,

[41] Laskowski R, Blaha P, Gallauner T, Schwarz K, Single layer model for the h-BN nanomesh on the Rh(111) surface, Physical Review Letters, 2007; 98, 106802.

[42] Laskowski R, Blaha P, Unraveling the structure of the h-BN/Rh(111) nanomesh with ab initio calculations, Journal of Physics: Condensed Matter, 2008; 064207.

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## **Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models**

Maricel Agop, Alina Gavriluț, Călin Buzea, Lăcrămioara Ochiuz, Dan Tesloianu, Gabriel Crumpei and Cristina Popa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59203

#### **1. Introduction**

[59] Blaha P, Hofstätter H, Koch O, Laskowski R, Schwarz K, Iterative diagonalization in APW-based methods in electronic structure calculations, Journal of Computational

[60] Lejaeghere K, Van Speaybroeck V, Van Oost G, Cottentier S 2013; Error estimates for solid-state density-functional theory predictions: An overview of the ground-state el‐ emental crystals, Critical Reviews in Solid State and Materials Science 2103; 39(1)

Physics, 2010; 229, 453-460.

310 Selected Topics in Applications of Quantum Mechanics

1-24, DOI; 10.1080/1048436,2013.772503.

Complex systems are a large multidisciplinary research theme that has been studied using a combination of fundamental theory, derived especially from physics and computational modeling. This kind of systems is composed of a large number of elemental units that interact with each other, being called "agents" [1, 2, 62]. Examples of complex systems can be found in human societies, the brain, internet, ecosystems, biological evolution, stock markets, economies and many others.

The manner in which such a system manifests can't be predicted only by the behavior of individual elements or by adding their behavior, but is determined by the way the elements interact in order to influence global behavior. Very important properties of complex systems are those of emergence, self-organization, adaptability etc. [3, 4, 62].

An example of a complex system is represented by polymers. [Their structures present a multitude of organized networks starting from simple, linear chains of identical structural units to very complex sequences of amino acids that are chained together, thus forming the fundamental units of living fields. Probably one of the most interesting biological complex system is DNA that generates cells by employing a simple but very efficient code. It is the striking way in which individual cells organize into complex systems, such as organs and, subsequently, organisms. Research in the field of complex systems could provide new information on the realistic dynamics of polymers, solving troublesome problems such as protein folding. We note that the dynamics of such complex systems implies the quantum formalism] [1-4, 62].

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Correspondingly, the theoretical models that describe the complex systems dynamics become more and more advanced [1-4]. For all that, this problem can be solved by taking into account that the complexity of the interaction process implies various temporal resolution scales, and the pattern evolution implies different degrees of freedom [5].

[In order to develop new theoretical models we must state the fact that the complex systems displaying chaotic behavior are recognized to acquire self-similarity (space-time structures can appear) in association with strong fluctuations at all possible space-time scales [1-4]. After‐ wards, for temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a set of potential trajectories and the concept of definite positions by that of probability density] [62]. An interesting example is the collisions processes in complex systems, where the dynamics of the particles can be described by non-differentiable curves.

Since non-differentiability can be considered a universal property of complex systems, it is mandatory to develop a non-differentiable physics. In this way, by considering that the complexity of the interaction processes is replaced by non-differentiability, using the entire range of quantities from the standard physics (differentiable physics) is no longer required [19].

This topic was developed in the Scale Relativity Theory (SRT) [6, 7] and in the non-standard Scale Relativity Theory (NSSRT) [8-22]. [In the framework of SRT or NSSRT we assume that the movements of complex system entities take place on continuous but non-differentiable curves (fractal curves) so that all physical phenomena involved in the dynamics depend not only on the space-time coordinates but also on the space-time scales resolution. In this conjecture, the physical quantities that describe the dynamics of complex systems can be considered as fractal functions. In addition, the entities of the complex system may be reduced to and identified with their own trajectories. In this way, the complex system's behavior will be identical to the one of a special interaction-less "fluid" by means of its geodesics in a nondifferentiable (fractal) space] [6, 7, 62].

In such context notions as informational entropy, Onicescu informational energy etc become important in the Nature description. These notions will be correlated with the fractal part of the physical quantities that describe the dynamics of complex systems.

#### **2. Informational entropy and energy**

Independently of scale resolution, the motion, either on infragalactic scale (for instance, the planetary motion), or on atomic scale (for instance, the motion of the electron around its nucleus) takes place on conics (ellipses). Such motion in invariant with respect to the SL(2R) group. In what follows, we shall consider this invariance only with respect to the motion on atomic scale.

#### **2.1. SL(2R) invariance and canonic formalism**

SL(2R) group is the group of transformations [23-26]

Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models http://dx.doi.org/10.5772/59203 313

$$\alpha \mathbf{x}' = \alpha \mathbf{x} + \beta y, \quad y' = \gamma \mathbf{x} + \delta y, \quad \alpha \mathbf{\delta} - \beta \gamma = \mathbf{1} \tag{1}$$

which makes invariant the areas in the phase space (*x*, *y*).

Choosing

Correspondingly, the theoretical models that describe the complex systems dynamics become more and more advanced [1-4]. For all that, this problem can be solved by taking into account that the complexity of the interaction process implies various temporal resolution scales, and

[In order to develop new theoretical models we must state the fact that the complex systems displaying chaotic behavior are recognized to acquire self-similarity (space-time structures can appear) in association with strong fluctuations at all possible space-time scales [1-4]. After‐ wards, for temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a set of potential trajectories and the concept of definite positions by that of probability density] [62]. An interesting example is the collisions processes in complex systems, where the dynamics of the particles can be described

Since non-differentiability can be considered a universal property of complex systems, it is mandatory to develop a non-differentiable physics. In this way, by considering that the complexity of the interaction processes is replaced by non-differentiability, using the entire range of quantities from the standard physics (differentiable physics) is no longer required [19]. This topic was developed in the Scale Relativity Theory (SRT) [6, 7] and in the non-standard Scale Relativity Theory (NSSRT) [8-22]. [In the framework of SRT or NSSRT we assume that the movements of complex system entities take place on continuous but non-differentiable curves (fractal curves) so that all physical phenomena involved in the dynamics depend not only on the space-time coordinates but also on the space-time scales resolution. In this conjecture, the physical quantities that describe the dynamics of complex systems can be considered as fractal functions. In addition, the entities of the complex system may be reduced to and identified with their own trajectories. In this way, the complex system's behavior will be identical to the one of a special interaction-less "fluid" by means of its geodesics in a non-

In such context notions as informational entropy, Onicescu informational energy etc become important in the Nature description. These notions will be correlated with the fractal part of

Independently of scale resolution, the motion, either on infragalactic scale (for instance, the planetary motion), or on atomic scale (for instance, the motion of the electron around its nucleus) takes place on conics (ellipses). Such motion in invariant with respect to the SL(2R) group. In what follows, we shall consider this invariance only with respect to the motion on

the physical quantities that describe the dynamics of complex systems.

the pattern evolution implies different degrees of freedom [5].

by non-differentiable curves.

312 Selected Topics in Applications of Quantum Mechanics

differentiable (fractal) space] [6, 7, 62].

**2. Informational entropy and energy**

**2.1. SL(2R) invariance and canonic formalism**

SL(2R) group is the group of transformations [23-26]

atomic scale.

$$
\alpha = 1 + \frac{1}{2} a\_{2'} \quad \text{ @} = a\_{1'} \quad \gamma = -a\_{3'} \quad \text{ @} = 1 - \frac{1}{2} a\_{2'} \tag{2}
$$

the infinitesimal transformations of the group have the expressions

$$\mathbf{x}' = \mathbf{x} + y\mathbf{a}\_1 + \frac{p}{2}\mathbf{a}\_{2'} \quad \mathbf{y}' = \mathbf{y} - \frac{1}{2}y\mathbf{a}\_2 - \mathbf{x}\mathbf{a}\_3 \tag{3}$$

Then the Lie algebra associated to the group becomes

$$\{\hat{L}\_1, \hat{L}\_2\} = \hat{L}\_1; \quad \{\hat{L}\_2, \hat{L}\_3\} = \hat{L}\_3; \quad \{\hat{L}\_3, \hat{L}\_1\} = -2\hat{L}\_2\tag{4}$$

where

$$
\hat{L}\_1 = y \frac{\partial}{\partial \mathbf{x}} \prime \quad \hat{L}\_2 = \left( \mathbf{x} \frac{\partial}{\partial \mathbf{x}} - y \frac{\partial}{\partial y} \right) \prime \quad \hat{L}\_3 = -\mathbf{x} \frac{\partial}{\partial y} \tag{5}
$$

are the vectors of the Lie base.

The general vector of the algebra (4) is given by the linear combination

$$
\hat{L} = c\hat{L}\_1 + 2b\hat{L}\_2 + a\hat{L}\_3, \quad a, b, c = \text{const.} \tag{6}
$$

The hamiltonian *H* results as an invariant function along the tangent trajectories to the vector (6). Precisely, it is a solution of the equation

$$
\hat{L}H = 0\tag{7}
$$

According to (5), relation (7) becomes

$$(bx+cy)\frac{\partial H}{\partial \mathbf{x}} - (ax+by)\frac{\partial H}{\partial y} = 0\tag{8}$$

whence the characteristic differential system

$$\frac{d\mathbf{x}}{dy + cy} = -\frac{dy}{(a\mathbf{x} + by)} = dt\tag{9}$$

admits the integral

$$H(\mathbf{x}, \mathbf{y}) = \frac{1}{2} \left( a\mathbf{x}^2 + 2b\mathbf{x}\mathbf{y} + c\mathbf{y}^2 \right) \tag{10}$$

We notice that the differential system (9) is Hamilton's system of equations [25]

$$
\dot{\alpha} = \frac{\partial H}{\partial y}, \quad \dot{y} = -\frac{\partial H}{\partial \alpha} \tag{11}
$$

associated to the hamiltonian (1), where the symbol " ⋅ " refers to the derivative with respect to the time.

Among the solutions of the equation (1), we have also the Gaussian

$$\mathfrak{g}(\mathbf{x}, y) = A \exp[-H(\mathbf{x}, y)] \mathsf{l} \quad A = \text{const.} \tag{12}$$

In consequence, all invariant functions on the group (7) will be functions of the hamiltonian (10) and particularly, of the Gaussian (12).

If the quadratic form (10) is positive definite, that is, the condition

$$a > 0, \quad \Omega = ac - b^2 > 0 \tag{13}$$

is fulfiled, then, by deriving the relations

$$\begin{array}{rcl} \dot{x} & = & bx + cy \\ \dot{y} & = & -(ax + by) \end{array} \tag{14}$$

with respect to the time and eliminating *p*˙ and *q*˙, based on the relations (14), we obtain the symmetric equations

$$\begin{array}{rcl} \ddot{x} + \Omega x & = & 0\\ \ddot{y} + \Omega y & = & 0 \end{array} \tag{15}$$

These equations are formally equivalent to the equations of two linear oscillators of coordinates *x, y*.

Then the 2-form

whence the characteristic differential system

314 Selected Topics in Applications of Quantum Mechanics

admits the integral

to the time.

( ) *dx dy dt*

We notice that the differential system (9) is Hamilton's system of equations [25]

*x y*

Among the solutions of the equation (1), we have also the Gaussian

If the quadratic form (10) is positive definite, that is, the condition

(10) and particularly, of the Gaussian (12).

is fulfiled, then, by deriving the relations

symmetric equations

, *H H*

*y x*

associated to the hamiltonian (1), where the symbol " ⋅ " refers to the derivative with respect

In consequence, all invariant functions on the group (7) will be functions of the hamiltonian

( ) *bx cy*

with respect to the time and eliminating *p*˙ and *q*˙, based on the relations (14), we obtain the

*by*

0 0

*y ax*

*x x y y* +W = +W =

&&

= + & =- +

*x*

*bx cy ax by* =- = + + (9)

( ) ( ) <sup>1</sup> 2 2 , 2 <sup>2</sup> *H x y ax bxy cy* = ++ (10)

¶ ¶ = = - ¶ ¶ & & (11)

r= - = ( , ) exp[ ( , )], const. *xy A Hxy A* (12)

<sup>2</sup> *a ac b* > W= - > 0, 0 (13)

& (14)

&& (15)

$$
\alpha = d\mathfrak{x} \wedge dy \tag{16}
$$

has the meaning of the elementary surface in phase space (*x*, *y*) and the transformations (1) are canonic because they maintain the 2-form (16) (Liouville's theorem [25]). Simultaneously, the Gaussian (12) can be considered as a probabilistic density in phase space (*x*, *y*). In this situation, the parameters (*a*, *b*, *c*) can get statistical significance (see also [71]).

#### **2.2. Shannon's informational entropy and transitivity manifolds**

[In standard quantum mechanics, the impossibility of determining the variances of the position coordinate *Δyi* and of the conjugate momentum component *Δxi* (*xi* = −*i*ℏ∇*i*) with arbitrary accuracy is widely accepted as being caused by the unavailable perturbation exerted on the particle by the measuring process. Because the measuring apparatus is most often not defined quantitatively and its perturbation can be very large, the uncertainty relation is formulated as a larger-than-or equal to equation

$$
\Delta \mathbf{x}\_i \Delta \mathbf{y}\_i \ge \frac{1}{2} \hbar
$$

Relating to this, it is unusual that the definitive nonzero variances *Δxi* and *Δ pi* can be obtained for quantum system which are not exposed to a measuring device. This has been shown using the so called negative-result experiments. Furthermore, it can be noticed that we could theoretically obtain the nonzero variance *Δxi* and *Δyi* of quantum systems without including in the analysis perturbations from or in presence of a measuring device at all] [65].

In such a conjecture the uncertainty relations result in a quite natural way from the momentum perturbations associated with the fractal potential, i.e. with the Shannon's information.

Indeed, let be the probability density in the phase space, *ρ*(*x*, *y*) with the constraints [27-32, 62]

$$\begin{aligned} \iiint y \rho(\mathbf{x}, y) dx dy &= \langle y \rangle \\ \iiint \mathbf{x} \rho(\mathbf{x}, y) dx dy &= \langle \mathbf{x} \rangle \\ \iiint (y - \langle y \rangle) \rho(\mathbf{x}, y) dx dy &= (\delta y)^2 \\ \iiint (\mathbf{x} - \langle \mathbf{x} \rangle) \rho(\mathbf{x}, y) dx dy &= (\delta \mathbf{x})^2 \\ \iiint (y - \langle y \rangle) (\mathbf{x} - \langle \mathbf{x} \rangle) \rho(\mathbf{x}, y) dx dy &= \text{cov}(\mathbf{x}, y) \end{aligned} \tag{17}$$

where *y* is the mean value of the position, *x* is the mean value of the momentum, *δy* is the position standard deviation, *δx* is the momentum standard deviation and *cov*(*x*, *y*) is the covariance of the random variables (*x*, *y*)] [62].

Now, we introduce Shannon's informational entropy [27]:

$$
\overline{H} = \iint \rho(\mathbf{x}, y) \ln[\rho(\mathbf{x}, y)] dx dy. \tag{18}
$$

Through Shannon's maximum informational entropy principle

$$
\delta \overline{H} = 0 \tag{19}
$$

with constraints (17), we get the normalized Gaussian distribution:

$$\left\{\rho\left(\mathbf{x}-\langle\mathbf{x}\rangle,y-\langle y\rangle\right)=\frac{\sqrt{ac-b^{2}}}{2\pi}\exp[-H\left(\mathbf{x}-\langle\mathbf{x}\rangle,y-\langle y\rangle\right)]\right\}\tag{20}$$

with

$$\begin{aligned} H\left(\mathbf{x} - \langle \mathbf{x} \rangle, \mathbf{y} - \langle \mathbf{y} \rangle \right) &= \frac{1}{2} [\overline{a}(\mathbf{x} - \langle \mathbf{x} \rangle)^2 + \\ &+ 2\overline{b}(\mathbf{x} - \langle \mathbf{x} \rangle)(\mathbf{y} - \langle \mathbf{y} \rangle) + \overline{c}(\mathbf{y} - \langle \mathbf{y} \rangle)^2] \\ \overline{a} = \frac{(\delta y)^2}{D}, \ \overline{b} = -\frac{cov(\mathbf{x}, y)}{D}, \ \overline{c} = \frac{\langle \delta \mathbf{x} \rangle^2}{D} \\ \mathbf{D} = (\delta \mathbf{x})^2 (\delta y)^2 - cov^2(\mathbf{x}, y) \end{aligned} \tag{21}$$

[We must note that the set of parameters (*a*¯, *b* ¯, *c*¯) has statistical significance given by relations (21).

In such context, the statistical hypothesis are specified through a particular choice of the set of parameters (*a*¯, *b* ¯, *c*¯) of the quadratic form the first Eq (21). Their class is given by the restriction [70]:

$$H(\mathbf{x}', \mathbf{y}') = H(\mathbf{x}' - \langle \mathbf{x} \rangle, \mathbf{y}' - \langle \mathbf{y} \rangle) = H(\mathbf{x} - \langle \mathbf{x} \rangle, \mathbf{y} - \langle \mathbf{y} \rangle) \tag{22}$$

where

$$\begin{aligned} H(\mathbf{x'} - \langle \mathbf{x} \rangle, y' - \langle y \rangle) &= \\ &= \frac{1}{2} \Big[ \overline{a}' (\mathbf{x'} - \langle \mathbf{x} \rangle)^2 + 2\overline{b}' (\mathbf{x'} - \langle \mathbf{x} \rangle)(y' - \langle y \rangle) + \overline{c}' (y' - \langle y \rangle)^2 \Big] \end{aligned} \tag{23}$$

If (*x* '− *x* , *y* '− *y* ) and (*x* − *x* , *y* − *y* ) are dependent through the unimodular transformations (1), we get that (22) imposes through (*a*¯, *b* ¯, *c*¯), the group of three parameters] (see [71] for details)

$$\begin{aligned} \overline{a}' &= \quad \delta^2 \overline{a} - 2\gamma \delta \overline{b} + \gamma^2 \overline{c} \\ \overline{b}' &= \quad -\mathbb{R} \delta \overline{a} - (\mathbb{R}\gamma + \alpha \delta) \overline{b} - \alpha \gamma \overline{c} \\ \overline{c}' &= \quad \delta^2 \overline{a} - 2\alpha \mathbb{R} \overline{b} + \alpha^2 \overline{c} \end{aligned} \tag{24}$$

If for the group (24) we choose the same parameterization as the one given by relations (2), then the corresponding infinitesimal transformations

$$\begin{array}{llll}\overline{a}' &=& \overline{a} - \overline{a}a\_2 + 2\overline{b}a\_3\\\overline{b}' &=& \overline{b} - \overline{a}a\_1 + \overline{c}a\_3\\\overline{c}' &=& \overline{c} - 2\overline{b}a\_1 + \overline{c}a\_2\end{array} \tag{25}$$

can be considered as an incompatible algebraic system in the unknowns *a*1, *a*2, *a*3. In conse‐ quence, there cannot exist a transformation able to ensure the correspondence

$$\left(\left(\overline{a}',\overline{b}',\overline{c}'\right)\to\left(\overline{a},\overline{b}',\overline{c}\right)\right) \tag{26}$$

Thus, the action of the group (24) in the space of variables (*a*¯, *b* ¯, *c*¯) is intransitive and, therefore, there exists a relation among the parameters (*a*¯, *b* ¯, *c*¯), which remains invariant to the action of the group (24). This relation is called transitivity manifold [26] (see also [71]).

The Lie algebra associated to the group (24) is

$$[\hat{A}\_1, \hat{A}\_2] = \hat{A}\_1; \quad [\hat{A}\_2, \hat{A}\_3] = \hat{A}\_3; \quad [\hat{A}\_3, \hat{A}\_1] = -2\hat{A}\_1 \tag{27}$$

where

where *y* is the mean value of the position, *x* is the mean value of the momentum, *δy* is the position standard deviation, *δx* is the momentum standard deviation and *cov*(*x*, *y*) is the

*H x y x y dxdy* =r r ( , )ln[ ( , )] . òò (18)

2

¯, *c*¯) has statistical significance given by relations

d = *H* 0 (19)

(20)

(21)

(23)

covariance of the random variables (*x*, *y*)] [62].

316 Selected Topics in Applications of Quantum Mechanics

with

(21).

[70]:

where

parameters (*a*¯, *b*

Now, we introduce Shannon's informational entropy [27]:

Through Shannon's maximum informational entropy principle

with constraints (17), we get the normalized Gaussian distribution:

( ) ( ) <sup>2</sup> , exp[ , ] <sup>2</sup>

*ac b x xy y Hx x y y* - r -á ñ -á ñ = - -á ñ -á ñ p

( ) <sup>2</sup>

*y cov x y <sup>x</sup> ab c D DD*

<sup>d</sup> <sup>d</sup> = = - =

*Hx x y y ax x*

 2 ( )( ) ( ) ] ( ) (,) ( ) , ,

+ -á ñ -á ñ + -á ñ

22 2

*D x y cov x y*

=d d -

[We must note that the set of parameters (*a*¯, *b*

(' ,' )


*Hx x y y*

( )( ) ( , )

2 2

In such context, the statistical hypothesis are specified through a particular choice of the set of

' 2' 2

*ax x bx x y y cy y* ¢

= -á ñ + -á ñ -á ñ + -á ñ é ù ë û

1 ( ' ) 2 ( ' )( ' ) ( ' ) <sup>2</sup>

¯, *c*¯) of the quadratic form the first Eq (21). Their class is given by the restriction

*Hx y Hx x y y Hx x y y* ( ', ') ( ' , ' ) ( , ) = -á ñ -á ñ = -á ñ -á ñ (22)

<sup>1</sup> , [( ) <sup>2</sup>


*bx x y y cy y*

$$\begin{aligned} \hat{A}\_1 &= \begin{array}{rcl} -\overline{a} \frac{\partial}{\partial \overline{b}} - 2\overline{b} \frac{\partial}{\partial \overline{c}} \\ \hat{A}\_2 &=& -\overline{a} \frac{\partial}{\partial \overline{a}} + \overline{c} \frac{\partial}{\partial \overline{c}} \\ \hat{A}\_3 &=& 2\overline{b} \frac{\partial}{\partial \overline{a}} + \overline{c} \frac{\partial}{\partial \overline{b}} \end{array} \tag{28}$$

are the vectors of the base Lie. By the conditions

$$
\hat{A}\_1 F = 0, \quad \hat{A}\_2 F = 0, \quad \hat{A}\_3 F = 0 \tag{29}
$$

[where *F* is an arbitrary function, we can obtain the transitivity manifolds of the group in the form

$$
\overline{a}\,\,\overline{c}\,-\overline{b}\,^2 = \text{const.}\tag{30}
$$

If *H* has energy significance, then condition (30) shows that a representative point from space (*x*, *y*) (which is in motion on a surface of constant energy (22)), can be also found on a surface of constant probabilistic density (ergodic condition) in Stoler's sense [33]:

$$\frac{\sqrt{\overline{a}^{\prime}\overline{c}^{\prime}} - \overline{b}^{\prime 2}}{2\pi} e^{-H(\underline{x}^{\prime}, \underline{y}^{\prime})} = \frac{\sqrt{\overline{a}^{\prime}}\,\overline{c} - \overline{b}^{2}}{2\pi} e^{-H(\underline{x}, \underline{y})} \tag{31}$$

Therefore, the "class" of statistical hypothesis associated to the Gaussians having the same mean, is given by the ergodic condition. This highlights the strong relationship existing among the energetic issues and the probabilistic ones] (see [71]).

#### **2.3. Informational energy in the sense of Onicescu and uncertainty relations**

For the informational energy we shall use Onicescu's relation [34, 62, 71]:

$$E = \int\_{-\alpha}^{\alpha} [\rho^2(\alpha, y) dx dy] \tag{32}$$

Thus, the informational energy corresponding to the normed Gaussians (20), which is subject to conditions (22), becomes

$$E\left(\overline{a}, \overline{b}, \overline{c}\right) = \int\limits\_{-\infty}^{\emptyset} \rho^2(x, y) dx dy\tag{33}$$

where *H* (*x*, *y*)>0 is a condition imposed by the existence of the integral (33). Thus we get

$$E\left(\overline{a}, \overline{b}, \overline{c}\right) = \frac{\sqrt{\overline{a} \cdot \overline{c} - \overline{b}^2}}{2\pi} \tag{34}$$

and therefore, if *H* has energetic significance, it results (see [62] and [71] for details):

**i.** The informational energy indicates the dispersion distribution (20) because the quantity

$$A = \frac{2\pi}{\sqrt{\overline{a} \cdot \overline{c} - \overline{b}^2}}\tag{35}$$

is a measure of the ellipses' areas of equal probability *H* (*p*, *q*)= const., in the manner that the normed Gaussians are even more clustered the more their informational energy is higher;


$$\left(\left(\delta x\right)^{2}\left(\delta y\right)^{2} = \frac{1}{4\pi^{2}E^{2}\left(\overline{a},\overline{b},\overline{c}\right)} + cov^{2}\left(\mathbf{x},\mathbf{y}\right) \tag{36}$$

or the non-egalitarian one

123 ˆˆˆ

of constant probabilistic density (ergodic condition) in Stoler's sense [33]:

the energetic issues and the probabilistic ones] (see [71]).

to conditions (22), becomes

Thus we get

2 2 *ac b Hx y ac b Hxy e e* ¢ ¢ ¢¢ ¢ - - - - <sup>=</sup> p p

**2.3. Informational energy in the sense of Onicescu and uncertainty relations**


<sup>2</sup> *E x y dxdy* (,) ¥

( ) <sup>2</sup> *E a b c x y dxdy* ,, (,) ¥


( ) <sup>2</sup> , , <sup>2</sup> *ac b Eabc* - <sup>=</sup> <sup>p</sup>

and therefore, if *H* has energetic significance, it results (see [62] and [71] for details):

where *H* (*x*, *y*)>0 is a condition imposed by the existence of the integral (33).

Thus, the informational energy corresponding to the normed Gaussians (20), which is subject

For the informational energy we shall use Onicescu's relation [34, 62, 71]:

form

318 Selected Topics in Applications of Quantum Mechanics

[where *F* is an arbitrary function, we can obtain the transitivity manifolds of the group in the

If *H* has energy significance, then condition (30) shows that a representative point from space (*x*, *y*) (which is in motion on a surface of constant energy (22)), can be also found on a surface

Therefore, the "class" of statistical hypothesis associated to the Gaussians having the same mean, is given by the ergodic condition. This highlights the strong relationship existing among

(,) (,)

2 2

*AF AF AF* === 0, 0, 0 (29)

<sup>2</sup> *ac b* const. - = (30)

= r ò ò (32)

= r ò ò (33)

(31)

(34)

$$
\delta x \delta y \geq \frac{1}{2\pi E\{\overline{a}, \overline{b}, \overline{c}\}}.\tag{37}
$$

In such context we can show that the constant value of the Onicescu informational energy implies, in the case of a linear oscillator, the Planck's quantification condition.

#### **2.4. Quantum mechanics and informational energy — Generalized uncertainty relations**

[The original theory of de Broglie on the wave-corpuscle duality was developed using a theorem found in Lorentz's transformation [35]. This theorem interlinks the local horologes cyclic frequency (in each point of a spatial domain) with a progressive wave frequency in phase with the horologes. This wave gives determines the distribution of the oscillators' phases on the respective spatial domain. We desire to show that a distribution of this kind, in a true sense, can be determined without resorting to Lorentz's transformation [67].

The concept imagined by de Broglie, of equal pulsation horologes, can be evidenced by a periodic field, which is described by the local oscillators of equation

$$
\stackrel{\cdot}{Q} + \Omega^2 \mathcal{Q} = 0 \tag{38}
$$

where *Q* = *y* + *ix* / *mΩ* is the relevant complex coordinate of the field and *Ω* is its pulsation. The general solution of (38) can be written as [23]:

$$Q(t) = \mathsf{z}e^{i(\Omega t \circ \circ)} + \overline{\mathsf{z}}e^{-i(\Omega t \circ \circ)} \tag{39}$$

where *z* is a complex amplitude, *z*¯ its complex conjugate and *φ* is a specific phase. The quantities *z* and *z*¯ give the initial conditions, which are not the same for any point from the space. Precisely, at a time, the various oscillators corresponding to the points of the space are in different states and have different phases. A problem arises: can we apriori indicate a rela‐ tionship among the parameters *z*, *z*¯ and *ei*(*Ωt*+*φ*) of the various oscillators at a given momentum? Because (39) is a solution of the equation (38) gives us an affirmative answer to this problem because (38) possesses a "hidden" symmetry that can be expressed by the homographic group: the ratio *τ*(*t*) of two solutions of the equation (38) is a solution of Schwartz's equation] [71] (see also [36, 67]).

$$
\left(\frac{\text{r}^{\circ}}{\text{r}^{\circ}}\right)^{\circ} - \frac{1}{2} \left(\frac{\text{r}^{\circ}}{\text{r}^{\circ}}\right)^{2} = 2\Omega^{2} \tag{40}
$$

[This equation is invariant to the homographic transformation of *τ*(*t*) : any homographic function of *τ* is itself a solution of (40). Since projections on the line can be characterized by the homography, we can assert that the ratio of two solutions of the equation (38) is a projective parameter for the class of the oscillators of the same pulsation from a given spatial region. Thus, one can define with ease a convenient, suitable projective parameter that should be in bi-univocal correspondence with the oscillator] [69, 71]. First, we observe a "universal" projective parameter: the ratio of the fundamental solutions of (38):

$$k = e^{2i(\Omega t \circ \circ)}\tag{41}$$

Any homographic function of this ratio will be again a projective parameter [67]. Among all other, the function

$$
\pi(t) = \frac{z + \overline{z}k}{1 + k} \tag{42}
$$

has primarily the advantage of being specific to each oscillator. But not only that: let be another function

$$
\pi'(t) = \frac{\underline{z}' + \overline{z}'k'}{1 + k'} \tag{43}
$$

which is specific to another oscillator. Since (42) and (43) are solutions of the equation (40), there exists a homographic relation between them:

$$
\pi' = \frac{\vec{a}'\pi + \vec{b}'}{\vec{c}'\pi + \vec{d}'} \tag{44}
$$

which, made explicit, leads to the Barbilian group equations [37]:

where *Q* = *y* + *ix* / *mΩ* is the relevant complex coordinate of the field and *Ω* is its pulsation. The

where *z* is a complex amplitude, *z*¯ its complex conjugate and *φ* is a specific phase. The quantities *z* and *z*¯ give the initial conditions, which are not the same for any point from the space. Precisely, at a time, the various oscillators corresponding to the points of the space are in different states and have different phases. A problem arises: can we apriori indicate a rela‐

Because (39) is a solution of the equation (38) gives us an affirmative answer to this problem because (38) possesses a "hidden" symmetry that can be expressed by the homographic group: the ratio *τ*(*t*) of two solutions of the equation (38) is a solution of Schwartz's equation] [71] (see

> 2 <sup>1</sup> <sup>2</sup> <sup>2</sup>

[This equation is invariant to the homographic transformation of *τ*(*t*) : any homographic function of *τ* is itself a solution of (40). Since projections on the line can be characterized by the homography, we can assert that the ratio of two solutions of the equation (38) is a projective parameter for the class of the oscillators of the same pulsation from a given spatial region. Thus, one can define with ease a convenient, suitable projective parameter that should be in bi-univocal correspondence with the oscillator] [69, 71]. First, we observe a "universal"

Any homographic function of this ratio will be again a projective parameter [67]. Among all

+

has primarily the advantage of being specific to each oscillator. But not only that: let be another

¢ +

+

( ) <sup>1</sup> *z zk <sup>t</sup> k* +

( ) <sup>1</sup> *z zk <sup>t</sup> k* ¢ ¢¢

¢

t =

t =

2 ¢ ¢¢ ¢¢ ¢ ¢ æö æö t t ç÷ ç÷ - =W t t èø èø

projective parameter: the ratio of the fundamental solutions of (38):

() () ( ) *it it Q t ze ze* W +j - W +j = + (39)

of the various oscillators at a given momentum?

2( ) *i t k e* W +j = (41)

(40)

(42)

(43)

general solution of (38) can be written as [23]:

320 Selected Topics in Applications of Quantum Mechanics

tionship among the parameters *z*, *z*¯ and *ei*(*Ωt*+*φ*)

also [36, 67]).

other, the function

function

$$\begin{array}{rcl} z^{'} &=& \frac{a^{1}z + b^{1}}{c^{1}z + d^{1}}\\ k^{'} &=& \frac{c^{1}\overline{z} + d^{1}}{c^{1}z + d^{1}}k \end{array} \tag{45}$$

[The group may be considered as a 'synchronization' group among various oscillators, a process in which the values of each take part, meaning that not only their phases, but also their amplitudes are correlated. The usual synchronization, manifested through the difference among the oscillators' phases as a whole, represents here just a particular case. Indeed, the group is involved for *z*, *z*¯ and *k*, and also for (45), which indicates the fact that, indeed, the phase of *k* is only shifted with a value depending on the oscillator's amplitude, during passage between various members of the assembly and, moreover, the oscillator's amplitude is homographically affected.

When taking into consideration, for the group (45), the parameterization from [23], the following infinitesimal generators of the above-mentioned group will be obtained] [63]:

$$\begin{aligned} \hat{B}\_1 &= \quad \frac{\partial}{\partial z} + \frac{\partial}{\partial \overline{z}} \\ \hat{B}\_2 &= \quad z \frac{\partial}{\partial z} + \overline{z} \frac{\partial}{\partial \overline{z}} \\ \hat{B}\_2 &= \quad z^2 \frac{\partial}{\partial z} + \overline{z}^2 \frac{\partial}{\partial \overline{z}} + (z - \overline{z})k \frac{\partial}{\partial k} \end{aligned} \tag{46}$$

the following commutation relations

$$\mathbb{I}\left[\hat{B}\_{1'}\hat{B}\_2\right] = \hat{B}\_{1'}; \quad \mathbb{I}\left[\hat{B}\_{2'}\hat{B}\_3\right] = \hat{B}\_{3'}; \quad \mathbb{I}\left[\hat{B}\_{3'}\hat{B}\_1\right] = -2\hat{B}\_2\tag{47}$$

being involved. [Thus, a structure near-identical to group SL(2R)'s Lie algebra is shown. In consequence, the Lie algebra of the group (45) is, again, a result of group SL(2R)'s Lie algebra. Actually, as can be easily observed, the group (45) represents only another action of the group SL(2R), performed in variables *z*, *z*¯, *k*] [63].

Once we fulfill the conditions of the theorem [38], the invariant functions can be found, simultaneously to the actions of the groups (5) and (46) as solutions of the equation

$$
\bar{L}\_{\!\!\!-1}F(\mathbf{x},y,z,\overline{z},k) + \bar{B}\_{\!\!\!-1}F(\mathbf{x},y,z,\overline{z},k) = 0, \quad i = 1,2,3\tag{48}
$$

Explaining this equation by means of Equations (5) and (46) leads to their simple solution, by successive reduction, while simultaneously obtaining the invariant functions in the form

$$f(\mu, v) = const.\tag{49}$$

where *μ* and *ν* are expressed as (see [63]):

$$\begin{array}{rcl} \mu &=& \frac{-i(z-\overline{z})}{(x-zy)(x-\overline{z}y)}\\ \mathbf{v} &=& k\frac{x-\overline{z}y}{x-zy} \end{array} \tag{50}$$

*ν* being a unimodular complex and *μ* a real one. A particular class of such invariant functions is represented by linear combinations of the type

$$
\overline{p}\mu = m\left(\mathbf{v} + \frac{\mathbf{1}}{\mathbf{v}}\right) + 2m\tag{51}
$$

where *m,n* and *p*¯ represent three arbitrary real constants.

If considering Eq (50), then Eq (51) takes the form

$$
\hbar m k^{-1} z^2 + 2m\overline{z}\overline{z} + m k \overline{z}^2 \equiv \overline{p} \tag{52}
$$

where the following notation has been used:

$$z = \frac{\chi - \overline{z}\overline{y}}{\sqrt{-i(z - \overline{z})}}\tag{53}$$

We also noticed that

$$-i(z-\overline{z}) > 0\tag{54}$$

Eq (52) represents a family of conical shapes from the phase space (*x*, *y*). They represent ellipses if

$$n\text{ }^2 - n^2 > 0\tag{55}$$

a condition also fulfilled if

Once we fulfill the conditions of the theorem [38], the invariant functions can be found,

Explaining this equation by means of Equations (5) and (46) leads to their simple solution, by successive reduction, while simultaneously obtaining the invariant functions in the form

> ( ) ( )( ) *iz z x zy x zy*

*ν* being a unimodular complex and *μ* a real one. A particular class of such invariant functions

1 *pm n*<sup>2</sup> æ ö m= n+ + ç ÷ è ø n

( )

*x zy <sup>z</sup> iz z*

*x zy <sup>k</sup> x zy* <sup>=</sup>



ˆ ˆ ( , , , , ) ( , , , , ) 0, 1,2,3 *i i LF x y z z k BF x y z z k i* + == (48)

*f v const* (,) . m = (49)

1 2 <sup>2</sup> *mk z nzz mkz p* 2 - ++ º (52)



(50)

(51)

simultaneously to the actions of the groups (5) and (46) as solutions of the equation

where *μ* and *ν* are expressed as (see [63]):

322 Selected Topics in Applications of Quantum Mechanics

is represented by linear combinations of the type

where *m,n* and *p*¯ represent three arbitrary real constants.

If considering Eq (50), then Eq (51) takes the form

where the following notation has been used:

We also noticed that

$$\begin{array}{rcl} m & = & \overline{Q}\sinh(2r) \\ m & = & \overline{Q}\cosh(2r) \end{array} \tag{56}$$

where *Q*¯ is a real constant and *<sup>r</sup>* is a real variable.

Quite an interesting case appears when *z* is completely imaginary, with no restriction con‐ cerning the generality value *z* =*i*. Thus, the quadratic form (52) may be identified with *H* (*p*, *q*) from (10), resulting

$$\begin{array}{ll}\overline{a} &=& \overline{Q} [\cosh(2r) + \sinh(2r)\cos\phi] \\ \overline{b} &=& -\overline{Q}\sinh(2r)\sin\phi \\ \overline{c} &=& \overline{Q}[\cosh(2r) - \sinh(2r)\cos\phi] \end{array} \tag{57}$$

where *φ* is the value of *k*, assumed fixed. The square value of *Q*¯ represents the value of the constant from (30), which determines the transitivity manifolds of the group (1) (see [63]).

The Gaussian distribution value obtained in such a manner represents only a particular case of the distribution that may occur, assuming in addition the obligation of satisfying the maximum principle of informational entropy under quadratic restrictions. The solutions of Eq (48) could be, however, much more general, being possibly selected from criteria involving group theory. In this context, the informational energy becomes

$$E(\overline{a}, \overline{b}, \overline{c}) = \frac{\overline{Q}}{4\pi} = const.\tag{58}$$

while the uncertainty relation (36) is

$$(\delta \mathbf{x})^2 (\delta y)^2 = \frac{1}{\overline{Q}^2} (1 + \sinh^2(2r) \sin^2 \phi) \tag{59}$$

resulting that the concept of uncertainty is minimum only for *φ* =0, i.e., all the oscillators of the assembly possess the same initial phase of zero. Based on this simplified hypothesis, at any moment of time subsequent to the initial one, the uncertainty relation (59) gives up its condition of minimum, along with the assembly's covariance which differs from zero] [63].

When the creation and annihilation operators refer to a harmonic oscillator, the uncertainty relations have the form from [33, 63] with *Q*¯ =2 / <sup>ℏ</sup> and <sup>ℏ</sup> the reduced Planck constant. In this situation, the "synchronization" is achieved through Stoler's group [33] (the parameter *r* is exactly equal to the frequency ratio).

Onicescu informational energy can be correlated with the standard quantum mechanics and the second quantification (which indicates its utility, for instance in NDA-NRA dynamics) [39].

#### **2.5. Gravity and information**

The structure of the group (45) is given by the equations (46) in the manner that the only nonzero structure constants are [26]:

$$\mathbf{C}\_{12}^1 = \mathbf{C}\_{23}^3 = -\mathbf{1}, \mathbf{C}\_{31}^2 = -\mathbf{2} \tag{60}$$

Therefore, the invariant quadratic form is given by the "quadratic" tensor of the group,

$$\mathbf{C}\_{\alpha\emptyset} = \mathbf{C}\_{\alpha\text{v}}^{\mu} \mathbf{C}\_{\beta\mu}^{\text{v}} \tag{61}$$

or, more explicit, by (60),

$$\mathbf{C}\_{\alpha\emptyset} = \begin{pmatrix} \mathbf{0} & \mathbf{0} & -\mathbf{4} \\ \mathbf{0} & \mathbf{2} & \mathbf{0} \\ -\mathbf{4} & \mathbf{0} & \mathbf{0} \end{pmatrix} \tag{62}$$

This yields that the invariant metric of the group is given by the relation [50]

$$\frac{d\mathbf{s}^2}{k\_0^2} = \mathbf{\dot{\mathbf{o}}\_0^2} - \mathbf{4}\mathbf{\dot{o}}\_1\mathbf{\dot{o}}\_2\tag{63}$$

where *k*0 is an arbitrary factor and *ωα*, three differential 1-forms, which are absolutely invariant through the group.

These 1-forms have the expressions:

$$\begin{aligned} \alpha\_0 &= \quad i \left( \frac{dk}{k} - \frac{dz + d\overline{z}}{z - \overline{z}} \right) \\ \alpha\_1 &= \quad \overline{\alpha}\_2 = \frac{dz}{k(z - \overline{z})} \end{aligned} \tag{64}$$

Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models http://dx.doi.org/10.5772/59203 325

in which case the metric (63) becomes

When the creation and annihilation operators refer to a harmonic oscillator, the uncertainty relations have the form from [33, 63] with *Q*¯ =2 / <sup>ℏ</sup> and <sup>ℏ</sup> the reduced Planck constant. In this situation, the "synchronization" is achieved through Stoler's group [33] (the parameter *r* is

Onicescu informational energy can be correlated with the standard quantum mechanics and the second quantification (which indicates its utility, for instance in NDA-NRA dynamics) [39].

The structure of the group (45) is given by the equations (46) in the manner that the only non-

12 23 <sup>31</sup> *CC C* = =- =- 1, 2 (60)

ab an bm = (61)

=w - ww (63)

(62)

(64)

13 2

Therefore, the invariant quadratic form is given by the "quadratic" tensor of the group,

*C CC*m n

*C*ab

This yields that the invariant metric of the group is given by the relation [50]

2

0

*k*

0

2 2 0 12

1 2 ( )

w = w= -

where *k*0 is an arbitrary factor and *ωα*, three differential 1-forms, which are absolutely invariant

*dk dz dz <sup>i</sup> k zz dz kz z*

æ ö + w= - ç ÷ è ø -

<sup>4</sup> *ds*

00 4 020 40 0

ç ÷ -è ø

æ ö - ç ÷ <sup>=</sup> ç ÷

exactly equal to the frequency ratio).

324 Selected Topics in Applications of Quantum Mechanics

**2.5. Gravity and information**

zero structure constants are [26]:

or, more explicit, by (60),

through the group.

These 1-forms have the expressions:

$$\frac{ds^2}{k\_0^2} = -\left(\frac{dk}{k} - \frac{d\overline{z} + d\overline{z}}{z - \overline{z}}\right)^2 + 4\frac{dz d\overline{z}}{\left(z - \overline{z}\right)^2} \tag{65}$$

It should be mentioned here a property related to integral geometry: the group (45) is meas‐ urable. Indeed, it is simply transitive and, since his structure vector *C<sup>α</sup>* =*Cνα <sup>ν</sup>* is identically zero, which can be seen from (60), it means that he possesses the invariant function

$$F(z, \overline{z}, k) = -\frac{1}{k(z - \overline{z})^2} \tag{66}$$

that is, the inverse of the module of the linear system's determinant obtained through the infinitesimal transformations of the group (45). Therefore, in the field variables space (*z*, *z*¯, *k*), one can build an a priori probabilities theory [40], based on the elementary probability

$$dP(z,\overline{z},k) = -\frac{dz\Lambda d\overline{z}\Lambda dk}{(z-\overline{z})^2k} \tag{67}$$

where *Λ* defines the external product of the 1-forms (64).

We now analyze the metric (65): it reduces to the metric of the Lobacevski's plan in Poincaré representation [26]:

$$\frac{d\boldsymbol{s}^2}{k\_0^2} = 4\frac{d\boldsymbol{z}d\overline{\boldsymbol{z}}}{\left(\boldsymbol{z} - \overline{\boldsymbol{z}}\right)^2} \tag{68}$$

for *ω*<sup>0</sup> =0. Specifying *ω*0 from (64) by the aid of the usual relations

$$
\overline{z} = \mu + i\upsilon, \ k = e^{i\phi} \tag{69}
$$

it results

$$\alpha \alpha\_0 = -\left(d\mathfrak{q} + \frac{du}{\upsilon}\right) \tag{70}$$

and so, the condition *ω*<sup>0</sup> =0 becomes

$$d\phi = -\frac{du}{v} \tag{71}$$

Since by this restriction, the metric (68) in the variables (69) reduces to the Lobacevski's one in Beltrami's representation:

$$\frac{ds^2}{k\_0^2} = -\frac{du^2 + dv^2}{v^2} \tag{72}$$

the condition (71) defines a parallel transport of vectors in a Levi-Civita meaning: the appli‐ cation point of the vector moves on the geodesic, the vector always making a constant angle with the tangent to the geodesic in the current point. Truly, using the fact that the plan's metric is conformal Euclidean, the angle between the initial vector and the vector transported through parallelism can be calculated as the integral of the equation (see [36, 68, 71] for details):

$$d\phi = \frac{1}{2} \left( \frac{\partial \ln E}{\partial v} du - \frac{\partial \ln E}{\partial u} dv \right) \tag{73}$$

along the transport curve. *E*(*u*, *v*) denotes here the conformity factor of the respective metric, in our case *E*(*u*, *v*)=1 / *v* <sup>2</sup> . Substituting it in (73), we get (71).

Now the variables (*z*, *z*¯, *k*) can be considered as amplitudes of a gravitational field Thus, let us admit that we describe the gravitational field through the variables *yi* for which we "discovered"' the metric

$$h\_{ij}dy^idy^j\tag{74}$$

in an ambient space of the metric

$$
\gamma\_{\alpha\beta}dx^{\alpha}dx^{\beta}\tag{75}
$$

Then the field equations derive from the variational principle [41]

$$
\delta \int L \chi^{\frac{1}{2}} d^3 \chi = 0 \tag{76}
$$

Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models http://dx.doi.org/10.5772/59203 327

relative to the Lagrange function

*du <sup>d</sup> v*

Beltrami's representation:

326 Selected Topics in Applications of Quantum Mechanics

in our case *E*(*u*, *v*)=1 / *v* <sup>2</sup>

"discovered"' the metric

in an ambient space of the metric

Since by this restriction, the metric (68) in the variables (69) reduces to the Lobacevski's one in

the condition (71) defines a parallel transport of vectors in a Levi-Civita meaning: the appli‐ cation point of the vector moves on the geodesic, the vector always making a constant angle with the tangent to the geodesic in the current point. Truly, using the fact that the plan's metric is conformal Euclidean, the angle between the initial vector and the vector transported through parallelism can be calculated as the integral of the equation (see [36, 68, 71] for details):

2 22 2 2

*ds du dv k v*

1 ln ln

along the transport curve. *E*(*u*, *v*) denotes here the conformity factor of the respective metric,

Now the variables (*z*, *z*¯, *k*) can be considered as amplitudes of a gravitational field Thus, let us admit that we describe the gravitational field through the variables *yi* for which we

*i j*

*dx dx* a b

1

Then the field equations derive from the variational principle [41]

*E E <sup>d</sup> du dv v u* æ ö ¶ ¶

. Substituting it in (73), we get (71).

2

0

j=- (71)

<sup>+</sup> = - (72)

j = ç ÷ - è ø ¶ ¶ (73)

*ij h dy dy* (74)

ab g (75)

<sup>3</sup> <sup>2</sup> dg = *L dx* <sup>0</sup> ò (76)

$$L = \gamma^{a0} h\_{\bar{\eta}} \frac{\partial \underline{\chi}^{j}}{\partial \underline{\chi}^{\alpha}} \frac{\partial \underline{\chi}^{j}}{\partial \underline{\chi}^{0}} = \gamma^{a0} \frac{\frac{\partial \underline{\chi}}{\partial \underline{\chi}^{\alpha}} \cdot \frac{\partial \overline{\Xi}}{\partial \underline{\chi}^{0}}}{\left(\underline{z} - \overline{z}\right)^{2}} = \frac{\text{" } \underline{z}" \text{ } \overline{\Xi}}{\left(\underline{z} - \overline{z}\right)^{2}}\tag{77}$$

In such a context, Einstein's equations with *z* =*iε* become Ernst's ones for the gravitational field of vacuum [42, 43]

$$\begin{array}{llll} -\frac{1}{2} (\mathfrak{c} + \overline{\mathfrak{c}})^{\mathfrak{n}\_{\perp}} \mathfrak{c} &=& \mathfrak{n}^{\mathfrak{n}} \mathfrak{c} & & \\ - (\mathfrak{c} + \overline{\mathfrak{c}})^{2} R\_{\alpha \overline{\mathfrak{c}}} (\chi) &=& \partial\_{\alpha} \mathfrak{c} \partial\_{\mu} \overline{\mathfrak{c}} + \partial\_{\mu} \mathfrak{c} \partial\_{\alpha} \overline{\mathfrak{c}} \\ \end{array} \tag{78}$$

(*Rαβ* is here the Ricci tensor of the three-dimensional metric *γαβ*).

Thus, adopting as a starting point the variational principle (76), the essential goal of the analysis in the gravitational field domain is to produce metrics of Lobacevski's plan or metrics related to them. All these can be directly related to Einstein's equations (78). Moreover, by substituting the principle of independence of the simultaneous actions, in the form of linear composition in a point of various fields intensities (through the apriori invariance of fields' action with respect to a certain group), we may conceive a gravity theory that has none of the contradictions inherently and commonly present in the current theory [44]. We observe that the SL(2R) group parameters can be interpreted as field amplitudes in a supergravitation model [23] (see also [71] for details).

#### **2.6. Extracellular vesicles convection in haptotaxis with hydrodynamical dissipation, a novel mechanism for vesicle migration**

#### *2.6.1. On the vesicle role*

In the field of cell's biology, we call vesicles those small bags wrapped in a membrane forming part of eukaryotic cell organelles. They are involved in proteins or enzymes transport and absorption, or meet other needs of the cell. Inside the membrane bag of a vesicle, there are macromolecules which require the ability to move outside the cell walls. The membrane encircling the bag merges with the outer wall of the cell to allow such macromolecules to penetrate the wall. The vesicles are important parts of the human cells, although they are also found in other multicellular organisms [66].

Cells found in humans, plants and animals use a variety of types of vesicles, depending on the type of cell and its specific intended function. For example, one type of vesicles, lysosomes, are necessary for the process of digestion. Lysosomes contain enzymes that breakdown food cells. With food absorption, a lysosome vesicle bonds to the food holding cell and releases enzymes by a process called phagocytosis. These enzymes break down food cells into smaller parts that can be better absorbed by other cells.

Secretory vesicles are frequently associated with nerve cells in humans or animals. Their membranes sacs contain neurotransmitters. Nervous system through hormonal signals activates these components. Through the process of exocytosis, the secretory vesicle's outer membrane adheres to the nerve terminal and releases neurotransmitters in the area of the nerve endings, named the synaptic cleft. Neurotransmitters transport information from one nerve terminal to the next, across the entire central nervous system, way up to the brain [66].

Vesicles, in their role as cellular mechanism are internally appointed for transport, uptake and storage of numerous imperative bodily functions. Without these tiny bags wrapped in membranes, cells could not make the exchange of materials necessary to maintain their healthy development and other crucial processes. As a conclusion, with no vesicles, humans and other pluricellular organisms could not have existed, because the essential cellular chemical processes would have no other method to pass onto another key materials [66].

Since there is increasing support that vesicle trafficking, including the release of EVs, is a highly important process in tumorigenesis, embryogenesis and tissue remodeling, in this paragraph we present an extensive discussion on the EVs convection in haptotaxis with hydrodynamical dissipation (i.e., a novel mechanism for vesicle migration).

#### *2.6.2. Mathematical model*

Vesicles are closed membranes floating in an aqueous solution (see Fig. 1). These membranes act as a barrier that efficiently controls permeability. The vesicles mimic maybe the most primitive and mechanically flexible dividing interfaces between the inside and the outside of a cell. Generally, the fluid enclosed by the membrane is incompressible in order that the vesicle evolves at a constant volume. Moreover, the membrane exchanges no phospholipid molecules with the solution, its area remaining constant as time passes [64]. Helfrich [45] described very well the vesicle's bending energy in its equilibrium state, which is compatible with the constraints above, i.e. constant volume and area. Even if the model is relatively simple it generates various equilibrium profiles, such as, discocytes (resembling red blood cells), stomatocytes, as well as forms presenting higher topologies (such as n-genus torus) that have been also observed experimentally [46]. We identify works studying alignments of vesicle in shear flows [47], fluctuations out of equilibrium [48], lift forces [49, 50], migration of vesicle in the proximity of a substrate [51, 52] or in gravity fields [53] and also vesicle tumbling [54]. One may note several recent experiments dealing with vesicle migration [55-58].

Considering the vesicle migration, we acknowledge that hydrodynamical dissipation in the neighboring fluid as well as inside the vesicle is present, and, in principle, between the two mono-layers which may glide with respect to each other. Furthermore, during motion on the substrate the dynamics of a vesicle may be restricted not only by the hydrodynamical flow but also by bonds breaking and restoring mechanisms that occur on the substrate (see [64]). It is obvious that the slowest mechanism limits the motion. Here we focus on a situation where

well the vesicle's bending energy in its equilibrium state, which is compatible with the constraints above, i.e. constant volume and area. Even if the model is relatively simple it generates various equilibrium profiles, such as, discocytes (resembling red blood cells), stomatocytes, as well as forms presenting higher topologies (such as n-genus torus) that have been also observed experimentally [46]. We identify works studying alignments of vesicle in

One may note several recent experiments dealing with vesicle migration [55-58].

enzymes by a process called phagocytosis. These enzymes break down food cells into smaller

Secretory vesicles are frequently associated with nerve cells in humans or animals. Their membranes sacs contain neurotransmitters. Nervous system through hormonal signals activates these components. Through the process of exocytosis, the secretory vesicle's outer membrane adheres to the nerve terminal and releases neurotransmitters in the area of the nerve endings, named the synaptic cleft. Neurotransmitters transport information from one nerve terminal to the next, across the entire central nervous system, way up to the brain [66].

Vesicles, in their role as cellular mechanism are internally appointed for transport, uptake and storage of numerous imperative bodily functions. Without these tiny bags wrapped in membranes, cells could not make the exchange of materials necessary to maintain their healthy development and other crucial processes. As a conclusion, with no vesicles, humans and other pluricellular organisms could not have existed, because the essential cellular chemical

Since there is increasing support that vesicle trafficking, including the release of EVs, is a highly important process in tumorigenesis, embryogenesis and tissue remodeling, in this paragraph we present an extensive discussion on the EVs convection in haptotaxis with hydrodynamical

Vesicles are closed membranes floating in an aqueous solution (see Fig. 1). These membranes act as a barrier that efficiently controls permeability. The vesicles mimic maybe the most primitive and mechanically flexible dividing interfaces between the inside and the outside of a cell. Generally, the fluid enclosed by the membrane is incompressible in order that the vesicle evolves at a constant volume. Moreover, the membrane exchanges no phospholipid molecules with the solution, its area remaining constant as time passes [64]. Helfrich [45] described very well the vesicle's bending energy in its equilibrium state, which is compatible with the constraints above, i.e. constant volume and area. Even if the model is relatively simple it generates various equilibrium profiles, such as, discocytes (resembling red blood cells), stomatocytes, as well as forms presenting higher topologies (such as n-genus torus) that have been also observed experimentally [46]. We identify works studying alignments of vesicle in shear flows [47], fluctuations out of equilibrium [48], lift forces [49, 50], migration of vesicle in the proximity of a substrate [51, 52] or in gravity fields [53] and also vesicle tumbling [54]. One

Considering the vesicle migration, we acknowledge that hydrodynamical dissipation in the neighboring fluid as well as inside the vesicle is present, and, in principle, between the two mono-layers which may glide with respect to each other. Furthermore, during motion on the substrate the dynamics of a vesicle may be restricted not only by the hydrodynamical flow but also by bonds breaking and restoring mechanisms that occur on the substrate (see [64]). It is obvious that the slowest mechanism limits the motion. Here we focus on a situation where

processes would have no other method to pass onto another key materials [66].

may note several recent experiments dealing with vesicle migration [55-58].

dissipation (i.e., a novel mechanism for vesicle migration).

*2.6.2. Mathematical model*

parts that can be better absorbed by other cells.

328 Selected Topics in Applications of Quantum Mechanics

Fig. 1: Schematic view of a vesicle emphasizing its microscopic structure: a bilayer of phospholipidic molecules. *Ladh* is the adherence length of the membrane **Figure 1.** Schematic view of a vesicle emphasizing its microscopic structure: a bilayer of phospholipidic molecules. *L adh* is the adherence length of the membrane

hydrodynamics are the limiting factors and we give out dissipation associated with bonds on the substrate. Considering the vesicle migration, we acknowledge that hydrodynamical dissipation in the neighboring fluid as well as inside the vesicle is present, and, in principle, between the two

Let us imagine a vesicle that initially adheres on a flat surface. We then consider an adhesion gradient along the substrate. The vesicle then moves in the direction of increasing adhesion energy (see Fig. 2) - it is named haptotaxis (a motion induced by an adhesion gradient). mono-layers which may glide with respect to each other. Furthermore, during motion on the substrate the dynamics of a vesicle may be restricted not only by the hydrodynamical flow but also by bonds breaking and restoring mechanisms that occur on the substrate (see [64]). It is

A highly permeable vesicle can be pulled into a fluid without opposing any resistance (and without modifying the inner area), whereas an impermeable one would be subjected to a drag force [64]. The assumption of local impermeability is legitimate. This entails that the fluid velocity at the membrane is equal that of the membrane itself [59]. obvious that the slowest mechanism limits the motion. Here we focus on a situation where hydrodynamics are the limiting factors and we give out dissipation associated with bonds on the substrate. Let us imagine a vesicle that initially adheres on a flat surface. We then consider an

On a vesicle's scale (*R* ~10*μm*) and for the expected velocities (*V* ~1*μm* /*s*), the dissipative processes fully dominate the dynamics. The energy added instantly dissipates in various degrees of freedom. Local dissipation caused by molecular reorganization, characterized by Leslie's coefficient, is negligible with respect to the hydrodynamics modes [60]. adhesion gradient along the substrate. The vesicle then moves in the direction of increasing adhesion energy (see Fig. 2) - it is named haptotaxis (a motion induced by an adhesion gradient). A highly permeable vesicle can be pulled into a fluid without opposing any resistance

If dissipation is dominated by bulk effects, as shown in [59], we are in the position to write down the basic governing equations for convective vesicles in a geometry depicted in Fig. 3, since we also know and it was proved the velocity field obeys Stokes equations [59]. (and without modifying the inner area), whereas an impermeable one would be subjected to a drag force [64]. The assumption of local impermeability is legitimate. This entails that the fluid velocity at the membrane is equal that of the membrane itself [59].

In an original atmospheric system, the non-even distribution of ascending water droplets is determined by the interplay between solar energy-induced thermal gradients, thermal diffusivity, friction, and gravity. Ultimately, the mathematics of this model shapes the

[60].

Fig. 2: Stationary vesicle profiles are depicted. The vesicle is moving from the left (smaller adhesion) to the right (stronger adhesion); a few discretization points are represented and the arrow allows following one of these at three **Figure 2.** Stationary vesicle profiles are depicted. The vesicle is moving from the left (smaller adhesion) to the right (stronger adhesion); a few discretization points are represented and the arrow allows following one of these at three successive times. One can observe here the rolling and sliding components of the vesicle's motion. characterized by Leslie's coefficient, is negligible with respect to the hydrodynamics modes [60]. If dissipation is dominated by bulk effects, as shown in [59], we are in the position to write down the basic governing equations for convective vesicles in a geometry depicted in Fig.

various degrees of freedom. Local dissipation caused by molecular reorganization,

, the

**Figure 3.** Convective extracellular vesicles (EVs) geometry. A fluid layer of thickness *d* of EVs, adherent on an extracel‐ lular matrix (ECM), is subjected to a gradient of concentration, where *ΔC* =*C*<sup>1</sup> −*C*<sup>0</sup> >0 is the difference of concentra‐ tion between the front and back boundaries of the fluid layer.

umbrella-like or budding appearance of structures like cumulonimbus clouds. This model can better or uniquely describe those types of structural dynamics not explained under fractal, simple/linear and several other types of models.

Acknowledging that similar patterns occur in various biological spaces, we think that the same mathematical determinism can be ascribed. Thus, some histoarchitectural prototypic struc‐ tures, like the capillary sprouting, embryologic organ, or even tumor buds of some types of cancer lesions might be in fact sculpted in that shape because gradients of molecular cues called morphogens can be deployed within the same manner water droplets can organize within nascent clouds.

Assuming that this organization also applies in biological systems, and that the EVs release can be considered among various processes organizing the budding tissue pattern, we think that the Lorenz model can govern their dynamics too. EVs would be particularly interesting as controllers of the tissue shape specification because they can include enzymatically active components (not found in conventional molecular morphogens), and thus might actively interact with the ECM fibers within their migration. Deployment of certain matrix degrading enzymes (MDEs) by EVs can modify this space while diffusing (event not produced by simple morphogens, attractive chemokines or repulsive semaphorins). This activity changes the topography of the ECM and creates spatial gradients directing the migration of subsequent EVs by haptotaxis - a mechanism better described for cell migration.

Let us consider the following thought biological experiment, equivalent to the B\'{enard experiment: a fluid layer of extracellular vesicles adherent on an ECM, in a haptotactic gradient. The fluid layer presents an unstable stratification of the potential density in a field of forces: the dense fluid is placed in front of the less dense one. We assume that in the basic state the layer of fluid of thickness d is subjected to a gradient of concentration

$$
\beta = \frac{C\_1 - C\_0}{d} > 0\tag{79}
$$

*ΔC* =*C*<sup>1</sup> −*C*<sup>0</sup> >0 is the difference of concentration between the front and back boundaries of the fluid layer. The regime with the fluid at rest and a non-perturbed distribution of concentration, belongs to the thermodynamic branch, which is continuously linking the non-equilibrium stationary state (*ΔC* ≠0) with the equilibrium state (*ΔC* =0) (see Fig. 3).

We examine the evolution of a concentration fluctuation *θ* around the non-perturbed concen‐ tration profile *C*<sup>0</sup> (*z*).

Two dissipative processes tend to maintain the fluid at rest:

**•** friction (motion amortization through viscosity);

umbrella-like or budding appearance of structures like cumulonimbus clouds. This model can better or uniquely describe those types of structural dynamics not explained under fractal,

**Figure 3.** Convective extracellular vesicles (EVs) geometry. A fluid layer of thickness *d* of EVs, adherent on an extracel‐ lular matrix (ECM), is subjected to a gradient of concentration, where *ΔC* =*C*<sup>1</sup> −*C*<sup>0</sup> >0 is the difference of concentra‐

Fig. 2: Stationary vesicle profiles are depicted. The vesicle is moving from the left (smaller adhesion) to the right (stronger adhesion); a few discretization points are represented and the arrow allows following one of these at three successive times. One can observe here the rolling and sliding components of the vesicle's motion.

successive times. One can observe here the rolling and sliding components of the vesicle's motion.

**Figure 2.** Stationary vesicle profiles are depicted. The vesicle is moving from the left (smaller adhesion) to the right (stronger adhesion); a few discretization points are represented and the arrow allows following one of these at three

If dissipation is dominated by bulk effects, as shown in [59], we are in the position to write down the basic governing equations for convective vesicles in a geometry depicted in Fig. 3, since we also know and it was proved the velocity field obeys Stokes equations [59].

dissipative processes fully dominate the dynamics. The energy added instantly dissipates in various degrees of freedom. Local dissipation caused by molecular reorganization, characterized by Leslie's coefficient, is negligible with respect to the hydrodynamics modes

Fig. 2: Stationary vesicle profiles are depicted. The vesicle is moving from the left (smaller adhesion) to the right (stronger adhesion); a few discretization points are represented and the arrow allows following one of these at three successive times. One can observe here the rolling and sliding components of the vesicle's motion.

dissipative processes fully dominate the dynamics. The energy added instantly dissipates in various degrees of freedom. Local dissipation caused by molecular reorganization, characterized by Leslie's coefficient, is negligible with respect to the hydrodynamics modes

write down the basic governing equations for convective vesicles in a geometry depicted in Fig.

3, since we also know and it was proved the velocity field obeys Stokes equations [59].

If dissipation is dominated by bulk effects, as shown in [59], we are in the position to

*m* and for the expected velocities ( ~1 / ) *V ms*

, the

*m* and for the expected velocities ( ~1 / ) *V ms*

, the

On a vesicle's scale ( ~ 10 ) *R*

330 Selected Topics in Applications of Quantum Mechanics

simple/linear and several other types of models.

tion between the front and back boundaries of the fluid layer.

On a vesicle's scale ( ~ 10 ) *R*

[60].

[60].

**•** ECM degradation subsequent to MDE's activity allowing vesicle trespassing - which lowers the concentration of the ECM, thus diminishing the forward, or advancing force.

The instability cannot be developed unless the EV is accelerated enough to overcome the effect of these dissipative processes. The gradient of concentration *β* which is the control parameter of this instability has to surpass a critical value *βC*. Over this critical value, an organized structure of convection cells may appear.

For a one component fluid, the mass, momentum and internal energy equations are the expressions (see the fractal - nonfractal transition method [8-22]):

$$\begin{aligned} \frac{\partial \mathbf{\hat{p}}}{\partial t} + \nabla \cdot (\rho \mathbf{v}) &= 0 \\ \frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\overset{\leftrightarrow}{\Pi} + \rho \mathbf{v} \mathbf{v}) &= \rho \mathbf{g} \\ \frac{\partial (\rho \mathbf{c})}{\partial t} + \nabla \cdot (\rho \mathbf{c} \mathbf{v} + \mathbf{j}\_d) &= -\overset{\leftrightarrow}{\Pi} \otimes (\nabla \mathbf{v}) \end{aligned} \tag{80}$$

where *ρ* represents the mass density of the fluid, *v* its speed, *g* acceleration of a field of forces, *ε* the internal energy of the unit volume, and **j***<sup>d</sup>* the flux of ECM degraded by signals received from EVs. Here *Π* <sup>↔</sup> is the stress tensor and ⊗ denotes the product of two tensors,

$$
\stackrel{\leftrightarrow}{A} \otimes \stackrel{\leftrightarrow}{B} = A\_{ij}B\_{ji}
$$

and we use Einstein's summation convention (implicit sum over repeating indices). The stress tensor can be written

$$
\begin{array}{ccc}
\leftrightarrow & \leftrightarrow & \leftrightarrow \\
\Pi = \Pi & + \Pi
\end{array}
\tag{81}
$$

*Π* <sup>↔</sup> *e* is the equilibrium part and depends on the state of the system. *Π* <sup>↔</sup> *<sup>v</sup>* represents the nonequilibrium part and is named, viscous stress tensor. At equilibrium, this part vanishes. For an isotropic medium at rest,

$$\stackrel{\leftrightarrow}{\Pi}^{\epsilon} = \begin{pmatrix} p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p \end{pmatrix} = p \stackrel{\leftrightarrow}{I} \tag{82}$$

where *p* is the hydrostatic pressure. For viscous systems at non-equilibrium, the viscous stress tensor is not null. According to Eq. (81) and Eq. (82), the stress tensor will be, for homogeneous and isotropic viscous systems, at non-equilibrium

$$\stackrel{\leftrightarrow}{\Pi} = p \stackrel{\leftrightarrow}{I} + \stackrel{\leftrightarrow}{\Pi}^v \tag{83}$$

We start with the following assumptions:

**a.** the fluid is Newtonian; as a result the stress tensor is given by Eq. (83), where the viscous stress tensor is [8-22]

Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models http://dx.doi.org/10.5772/59203 333

$$\boldsymbol{\Pi}^{\boldsymbol{\upsilon}}\_{\alpha\boldsymbol{\mu}} = -\eta \left( \frac{\partial \boldsymbol{\upsilon}\_{\alpha}}{\partial \boldsymbol{\chi}\_{\boldsymbol{\beta}}} + \frac{\partial \boldsymbol{\upsilon}\_{\boldsymbol{\beta}}}{\partial \boldsymbol{\chi}\_{\boldsymbol{\alpha}}} - \frac{2}{3} \boldsymbol{\mathsf{S}}\_{\alpha\boldsymbol{\mu}} \left( \boldsymbol{\nabla} \cdot \mathbf{v} \right) \right) - \boldsymbol{\zeta} \boldsymbol{\mathsf{S}}\_{\alpha\boldsymbol{\mu}} \left( \boldsymbol{\nabla} \cdot \mathbf{v} \right),$$

with coefficients *η* and *ζ* independent of velocity, the tangential (shear) and bulk viscosity, respectively;

**b.** ECM degrading by MDEs from EVs is described by the Fourier equation

$$
\mathbf{j}\_d = -\mathbf{\hat{\lambda}} \nabla \mathbf{\hat{C}} \tag{84}
$$

where *λ* is the haptotactic coefficient;

For a one component fluid, the mass, momentum and internal energy equations are the

expressions (see the fractal - nonfractal transition method [8-22]):

¶r +Ñ× r = ¶

*t*

332 Selected Topics in Applications of Quantum Mechanics

*t*

**v**

*t*

from EVs. Here *Π*

tensor can be written

an isotropic medium at rest,

and isotropic viscous systems, at non-equilibrium

We start with the following assumptions:

stress tensor is [8-22]

*Π* <sup>↔</sup> *e* ( ) 0

«

¶ r + Ñ× P+ r = r ¶

( ) ( ) () *<sup>d</sup>*

¶ re + Ñ × re + = -PÄ Ñ ¶

«

(80)

**vj v**

**vv g**

where *ρ* represents the mass density of the fluid, *v* its speed, *g* acceleration of a field of forces, *ε* the internal energy of the unit volume, and **j***<sup>d</sup>* the flux of ECM degraded by signals received

> *A B ABij ji* « « Ä =

and we use Einstein's summation convention (implicit sum over repeating indices). The stress

equilibrium part and is named, viscous stress tensor. At equilibrium, this part vanishes. For

*p pI p*

where *p* is the hydrostatic pressure. For viscous systems at non-equilibrium, the viscous stress tensor is not null. According to Eq. (81) and Eq. (82), the stress tensor will be, for homogeneous

*v*

**a.** the fluid is Newtonian; as a result the stress tensor is given by Eq. (83), where the viscous

« « æ ö ç ÷ P= = ç ÷ ç ÷ è ø

*p I* « ««

P=P +P (81)

P= +P (83)

<sup>↔</sup> *<sup>v</sup>* represents the non-

(82)

«« « *e v*

is the equilibrium part and depends on the state of the system. *Π*

*<sup>e</sup> p*

<sup>↔</sup> is the stress tensor and ⊗ denotes the product of two tensors,

( ) ( )

**v**

**c.** haptotactic energy expansion is linear

$$
\delta \mathfrak{J} \mathfrak{p} = \mathfrak{p} - \mathfrak{p}\_0 = -\mathfrak{p}\_0 \mathfrak{a} \mathfrak{z} = -\mathfrak{p}\_0 \mathfrak{a} \mathfrak{k}\_h \mathfrak{f} \text{(C}-\text{C}\_0) = -\mathfrak{p}\_0 \mathfrak{X} \mathfrak{f} \text{(C}-\text{C}\_0) \tag{85}
$$

where we used the expression of the haptotactic energy

$$\mathfrak{c} = k\_h (\mathcal{C} - \mathcal{C}\_0)$$

*kh* being the haptotactic energy constant. In Eq. (85) *α* is the haptotactic energy expansion constant and *χ* =*αkh* is the haptotactic expansion constant;

**d.** the fluid satisfies a state equation: consequently, its internal energy is (up to a constant factor)

$$
\varepsilon = k\_b \mathbf{C} \tag{86}
$$

where *kb* is the state constant;

**e.** in most liquids, thermal expansion is small. We choose everywhere a constant density, denoted by *ρ*0, except the momentum equation.

With these approximations, the system of Eqs. (80) leads to the Boussinesq type system of equations

$$\begin{aligned} \nabla \cdot \mathbf{v} &= 0\\ \rho\_0 \left[ \frac{\partial \mathbf{v}}{\partial t} + \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} \right] + \nabla p &= \left( \rho\_0 + \delta \rho \right) \mathbf{g} + \eta \nabla^2 \mathbf{v} \\\ \frac{\partial \mathbf{C}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{C} &= \frac{\lambda}{\rho\_0 k\_b} \nabla^2 \mathbf{C} \end{aligned} \tag{87}$$

where *ρ* is the perturbed density

$$
\mathfrak{p} = \mathfrak{p}\_0 + \delta \mathfrak{p} \tag{88}
$$

The first equation (87) represents the incompressibility condition for the fluid.

Convection occurs in the fluid layer when the forward, or advancing force, resulted from energy expansion, surpasses the viscous forces. We may define now a Rayleigh type number identical with the Eqs associated to fractal - nonfractal transition [61]

$$R = \frac{\left| \mathbf{F}\_{\rm acc} \right|}{\left| \mathbf{F}\_{\rm visc} \right|} \approx \frac{\left| \frac{\delta \mathfrak{p} \mathbf{g}}{\rho\_0} \right|}{\left| \frac{\mathfrak{p} \nabla^2 \mathbf{v}}{\rho\_0} \right|} \tag{89}$$

The density perturbation satisfies, according to Eq. (85)

$$\frac{\delta\mathfrak{p}}{\mathfrak{p}\_0} \approx \chi\mathsf{A}\mathsf{C} \tag{90}$$

On the other side, from the internal energy equation Eq. (79), it results

$$w \approx \frac{\lambda}{\rho\_o k\_b} \frac{1}{d} \tag{91}$$

Replacing Eqs. (90) and (91) in Eq. (89), and taking into account Eq. (79), we get a biological Rayleigh number

$$R = \frac{\chi \mathcal{B} \rho\_0 k\_b g}{\nu \lambda} d^4 \tag{92}$$

where *ν* =*η* / *ρ*0 is the cinematic viscosity. For the Bénard convection, the biological Rayleigh number plays the part of a control parameter. The convection occurs for

$$
\mathcal{R} > \mathcal{R}\_{\text{critical}}
$$

Most of the time, *R* is controlled by *β*, the gradient of concentration.

Within a biological context, **g** can be specified by polar/linear topography of semaphorins or/ and chemokines, signals typically creating stable gradients to which EVs can respond.

where *ρ* is the perturbed density

334 Selected Topics in Applications of Quantum Mechanics

Rayleigh number

The first equation (87) represents the incompressibility condition for the fluid.

*asc visc*

0 *<sup>C</sup>* dr » cD

*v*

number plays the part of a control parameter. The convection occurs for

Most of the time, *R* is controlled by *β*, the gradient of concentration.

0

1 *b*

*k d*

Replacing Eqs. (90) and (91) in Eq. (89), and taking into account Eq. (79), we get a biological

where *ν* =*η* / *ρ*0 is the cinematic viscosity. For the Bénard convection, the biological Rayleigh

*R Rcritical* >

On the other side, from the internal energy equation Eq. (79), it results

**F**

<sup>r</sup> = » hÑ

**F v**

identical with the Eqs associated to fractal - nonfractal transition [61]

*R*

The density perturbation satisfies, according to Eq. (85)

Convection occurs in the fluid layer when the forward, or advancing force, resulted from energy expansion, surpasses the viscous forces. We may define now a Rayleigh type number

> 0 2

**g**

dr

0

r

<sup>0</sup> r = r + dr (88)

<sup>r</sup> (90)

<sup>l</sup> » <sup>r</sup> (91)

<sup>0</sup> *<sup>b</sup>* <sup>4</sup> *k g R d* cbr <sup>=</sup> nl (92)

(89)

We choose as reference state the rest stationary state (*vS* =0), for which the last two equations in the system of Eqs. (87) reduce to

$$\begin{cases} \nabla p\_{\mathcal{S}} = -\mathfrak{p}\_{\mathcal{S}} \mathbf{g} \hat{\mathbf{z}} = -\mathfrak{p}\_{0} [1 - \mathfrak{x} (\mathbf{C}\_{\mathcal{S}} - \mathbf{C}\_{0})] \mathbf{g} \hat{\mathbf{z}} \\ \nabla^{2} \mathbf{C}\_{\mathcal{S}} = 0 \end{cases} \tag{93}$$

where *z* ^ is the versor of the vertical direction. We assume pressure and concentration varies only along the vertical direction, due to the geometry of the experiment. For concentration, the boundary conditions read

$$\mathcal{C}(\mathbf{x}, \mathbf{y}, \mathbf{0}) = \mathcal{C}\_{\mathbf{0}} ; \; \mathcal{C}(\mathbf{x}, \mathbf{y}, \mathbf{d}) = \mathcal{C}\_{\mathbf{x}}$$

Integrating the second Eq. (93) with these boundary conditions, it results that, in the stationary reference state, the profile of the concentration in the vertical direction is linear

$$\mathbf{C}\_{\mathcal{S}} = \mathbf{C}\_{0} - \mathfrak{B}\mathbf{z} \tag{94}$$

with *β*, the gradient of concentration. Replacing Eq. (94) in first Eq. (93) and integrating, we get

$$p\_S(\mathbf{z}) = p\_0 - \rho\_0 \mathbf{g}\left(1 + \frac{\chi\beta\mathbf{z}}{2}\right)\mathbf{z} \tag{95}$$

The characteristics of the system in this state are independent of the kinetic coefficients *η* and *λ* which appear in Eqs. (87). We study the stability of the reference state using the small perturbations method. The perturbed state is characterized by

$$\begin{aligned} \mathbf{C} &= \mathbf{C}\_{\mathcal{S}}(\mathbf{z}) + \boldsymbol{\Theta}(\mathbf{r}, t) \\ \boldsymbol{\uprho} &= \boldsymbol{p}\_{\mathcal{S}}(\mathbf{z}) + \boldsymbol{\updelta}\boldsymbol{p}(\mathbf{r}, t) \\ \boldsymbol{p} &= \boldsymbol{p}\_{\mathcal{S}}(\mathbf{z}) + \boldsymbol{\updelta}\boldsymbol{p}(\mathbf{r}, t) \\ \mathbf{v} &= \boldsymbol{\updelta}\mathbf{v}(\mathbf{r}, t) = (\boldsymbol{\upmu}, \boldsymbol{v}, \boldsymbol{w}) \end{aligned} \tag{96}$$

As can be seen from Eqs. (96), the perturbations are functions of coordinate and time. Replacing Eqs. (96) in the evolution equations of the Boussinesq approximation Eqs. (87) and taking into account Eq. (94) and Eq. (95), we get, in the linear approximation, the following equations for the perturbations

$$\begin{aligned} \nabla \cdot \delta \mathbf{v} &= 0 & \mathbf{a} \\ \frac{\partial \delta \mathbf{v}}{\partial t} &= -\frac{1}{\rho\_0} \nabla \delta p + \nu \nabla^2 \delta \mathbf{v} + \mathcal{g} \chi \theta \hat{z} & \mathbf{b} \\ \frac{\partial \Theta}{\partial t} &= \partial x + K \nabla^2 \Theta & \mathbf{c} \end{aligned} \tag{97}$$

where *<sup>K</sup>* <sup>=</sup> *<sup>λ</sup> ρ*0*kb* is a coefficient. We pass to non-dimensional variables in Eqs. (97), using the transformations: **r** ′ = **r** *<sup>d</sup>* ; *<sup>t</sup>* ′ <sup>=</sup> *<sup>t</sup> <sup>d</sup>* <sup>2</sup> / *<sup>K</sup>* ; *<sup>θ</sup>* ′ <sup>=</sup> *<sup>θ</sup>* ( *<sup>ν</sup><sup>K</sup> <sup>g</sup>χ<sup>d</sup>* <sup>3</sup> ) ; *δ***v**′ <sup>=</sup> *<sup>δ</sup>***<sup>v</sup>** *<sup>K</sup>* / *<sup>d</sup>* and *<sup>δ</sup> <sup>p</sup>* ′ <sup>=</sup> *<sup>δ</sup><sup>p</sup>* ( *<sup>ρ</sup>*0*ν<sup>K</sup> d* 2 ) .

Using the standard method from [8-22], it results the biological Lorentz system.

#### *2.6.3. Validity of theoretical model*

Some results are evident:


Implications of Quantum Informational Entropy in Some Fundamental Physical and Biophysical Models http://dx.doi.org/10.5772/59203 337

Fig. 4: Bénard-Rayleigh model patterns representative for biological instances: A) for fingerprint like distribution of skin cells, B) for fish or amphibian scales. **Figure 4.** Bénard-Rayleigh model patterns representative for biological instances: A) for fingerprint like distribution of skin cells, B) for fish or amphibian scales.

#### **Conclusions 3. Conclusions**

2

0 a <sup>1</sup> <sup>ˆ</sup> <sup>b</sup>

**v**

c

<sup>=</sup> *<sup>δ</sup><sup>p</sup>* ( *<sup>ρ</sup>*0*ν<sup>K</sup> d* 2 ) .

is a coefficient. We pass to non-dimensional variables in Eqs. (97), using the

*<sup>K</sup>* / *<sup>d</sup>* and *<sup>δ</sup> <sup>p</sup>* ′

<sup>=</sup> *<sup>δ</sup>***<sup>v</sup>**

**•** in [50], and similar other references ([64] etc.), the EVs behavior, under shear flow close to a substrate, was proved to be quite similar to the one encountered in two dimensional simulations, so we are confident that the 2D assumptions captures the essential features of

**•** different control parameter values for the Lorenz system can create shape distributions similar to the cordonal appearance of fingerprints (see Fig. 4, A), or complex skin tissue tiles

**•** the biological thought experiment equivalent to Bénard's experiment, involving a fluid layer of extracellular vesicles adherent to an extracellular matrix, in a haptotactic gradient can be *checked experimentally* today to a high degree of accuracy. We think that suitable test systems would be the *embryological ones* (i.e., the development of branched vessels in membranes avian eggshell membranes, serous membranes of the peritoneal cavity; or the budding development of lung alveoli, or of fingerprints), and, similar, *inflammatory ones* (i.e., the emergence of neoangionetic vessels driven by inflammatory proximities) - all of which apparently start as point like spots displayed in a comb-like appearance along a rectilinear

**•** we analyze the problem of EVs migration in haptotaxis, though most of the reasoning applies to chemotaxis (migration of cells biased towards a gradient of diffusible MDEs) as well as to a variety of driving forces - all of which include the possibility to specify an active

**•** the resulted system of equations exhibits complex behavior, hard to control, the two occurring convective rolls: either going in one direction, or in the opposite one - means

(97)

0

*w K*

¶q =b + Ñ q ¶

*t*

where *<sup>K</sup>* <sup>=</sup> *<sup>λ</sup>*

transformations: **r**

*ρ*0*kb*

′ = **r** *<sup>d</sup>* ; *<sup>t</sup>* ′ <sup>=</sup> *<sup>t</sup> <sup>d</sup>* <sup>2</sup> / *<sup>K</sup>* ; *<sup>θ</sup>* ′

336 Selected Topics in Applications of Quantum Mechanics

*2.6.3. Validity of theoretical model*

Some results are evident:

the 3D EVs;

or arched origin;

parameter value within the model;

patterning the EVs spreading.

Ñ×d =

**v v**

2

<sup>=</sup> *<sup>θ</sup>* ( *<sup>ν</sup><sup>K</sup> <sup>g</sup>χ<sup>d</sup>* <sup>3</sup> ) ; *δ***v**′

Using the standard method from [8-22], it results the biological Lorentz system.

**•** we build the first Lorenz model for extracellular vesicles migration;

like scale appendages in the amphibian covering (see Fig. 4, B);

*p gz <sup>t</sup>*

¶d = - Ñd + nÑ d + cq ¶ r

i) We establish a relationship between the SL(2R) group and the canonic formalism. It Considering the above, we can write the following conclusions :

Considering the above, we can write the following conclusions:


Gaussians of the same average is characterized by the constant value of the informational energy. In addition, it is equivalent to the ergodic hypothesis. For a constant value of the informational energy, we obtain uncertainty egalitarian relationships and, particularly, for the

complex coordinate, we prove that it has a "hidden symmetry", which is expressed by the homographical transformations group in three parameters. This group (also an achievement of SL(2R)) functions as a synchronization group both in phase and in amplitude, among the oscillators of the same ensemble. The simultaneous invariance related to two different

iv) Assuming that de Broglie's theory is materialized through a periodic field in a

linear harmonic oscillator, we show that the informational energy is quantified;


### **Author details**

Maricel Agop1,2\*, Alina Gavriluț<sup>3</sup> , Călin Buzea<sup>4</sup> , Lăcrămioara Ochiuz<sup>5</sup> , Dan Tesloianu6 , Gabriel Crumpei7 and Cristina Popa8

\*Address all correspondence to: m.agop@yahoo.com

1 Lasers, Atoms and Molecules Physics Laboratory, University of Science and Technology, Lille, France

2 Department of Physics, "Gh. Asachi"' Technical University, Iași, România

3 Department of Mathematics, "Al.I. Cuza" University from Iași, Romania

4 National Institute of Research and Development for Technical Physics, Iași, România

5 Faculty of Pharmacy, Department of Pharmaceutical Technology University of Medicine and Pharmacy "Gr.T. Popa", Iași, România

6 IV Internal Medicine Department, Clinical Emergency Hospital "Sf. Spiridon", University of Medicine and Pharmacy "Gr.T. Popa", Iași, România

7 Psychiatry, Psychotherapy and Counselling Center, Iași, România

8 Dentistry Department, University of Medicine and Pharmacy "Gr.T. Popa", Iași, România

#### **References**

**iii.** We prove that informational energy (in the sense of Onicescu) is a measure of the

**iv.** Assuming that de Broglie's theory is materialized through a periodic field in a

**v.** The synchronization group among the oscillators of the same ensemble admits three

**vi.** Complex measures in the study of certain physical systems dynamics need the use

1 Lasers, Atoms and Molecules Physics Laboratory, University of Science and Technology,

4 National Institute of Research and Development for Technical Physics, Iași, România

, Călin Buzea<sup>4</sup>

2 Department of Physics, "Gh. Asachi"' Technical University, Iași, România

3 Department of Mathematics, "Al.I. Cuza" University from Iași, Romania

differentiable 1-forms and one differentiable 2-form which is absolutely invariant on the group. The existence of a parallel transport in Levi-Civita's sense, in which case the 2-form in Lobacevski's metric form in Poincare representation, implies through a variation principle, equations of Ernst type for the gravitational field of vacuum;

of a space-time endowed with a special topology, namely, the fractal space-time [6,

, Lăcrămioara Ochiuz<sup>5</sup>

, Dan Tesloianu6

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338 Selected Topics in Applications of Quantum Mechanics

7] (see also [71] for details).

and Cristina Popa8

\*Address all correspondence to: m.agop@yahoo.com

**Author details**

Gabriel Crumpei7

Lille, France

Maricel Agop1,2\*, Alina Gavriluț<sup>3</sup>

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ro. Phys. J. B 29, 311-316, 2002.

*tion*, Eur. Phys. J. E 10, 175-189, 2003.

*cle propulsion*, Phys. Rev. E 56, 3776, 1997.

*Shear Flow*, Phys. Rev. Lett. 83, 880-883, 1999.


## **Physical Vacuum is a Special Superfluid Medium**

### V.I. Sbitnev

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59040

#### **1. Introduction**

A dramatic situation in physical understanding of the nature emerged in the late of 19th century. Observed phenomena on micro scales came into contradiction with the general positions of classical physics. It was a time of the origination of new physical ideas explaining these phenomena. Actually, in a very short period, postulates of the new science, quantum mechanics were formulated. The Copenhagen interpretation was first who proposed an ontological basis of quantum mechanics [1]. These positions can be stated in the following points: (a) the description of nature is essentially probabilistic; (b) a quantum system is completely described by a wave function; (c) the system manifests wave–particle duality; (d) it is not possible to measure all variables of the system at the same time; (e) each measurement of the quantum system entails the collapse of the wave function.

Can one imagine a passage of a quantum particle (the heavy fullerene molecule [2], for example) through all slits, in once, at the interference experiment? Following the Copenhagen interpretation, the particle does not exists until it is registered. Instead, the wave function represents it existence within an experimental scene [3].

Another interpretation was proposed by Louis de Broglie [4], which permits to explain such an experiment. In de Broglie's wave mechanics and the double solution theory there are two waves. There is the wave function that is a mathematical construct. It does not physically exist and is used to determine the probabilistic results of experiments. There is also a physical wave guiding the particle from its creation to detection. As the particle moves from a source to a detector, the particle perturbs the wave field and gets a reverse effect from it. As a result, the physical wave guides the particle along some optimal trajectory up to its detection.

A question arises, what is the de Broglie physical wave? Recently, Couder and Fort [5] has executed the experiment with the classical oil droplets bouncing on the oil surface. A remark‐ able observation is that an ensemble of the droplets passing through the barrier having two

gates shows the interference fringes typical for the two slit experiment. Their explanation is that the droplet while moving on the surface induces on this surface the weak Faraday waves. The latter provide the guidance conditions for the droplets. In this perspective, we can draw conclusion that the de Broglie physical wave can be represented by perturbations of the ether when the particle moves through it. In order to describe behavior of such an unusual medium we shall use the Navier-Stokes equation with slightly modified some terms. As the final result we shall get the Schrödinger equation.

In physical science of the New time the assumption for the existence of the ether medium was originally used to explain propagation of light and the long-range interactions. As for the propagation of light, the wave ideas of Huygens and Fresnel require the existence of a continuous intermediate environment between a source and a receiver of the light - the lightbearing ether. It is instructive to compare here the two opposite doctrines about the nature of light belonging to Sir Isaac Newton and Christian Huygens. Newton maintained the theory that the light was made up of tiny particles, corpuscles. They spread through an empty space in accordance with the law of the classical mechanics. Christian Huygens (a contemporary of Newton), believed that the light was made up of waves vibrating up and down perpendicu‐ larly to the direction of its propagation, as waves on a water surface. One can imagine all space populated everywhere densely by Huygens's vibrators. All vibrators are silent until a wave reaches them. As soon as a wave front reaches them, the vibrators begin to radiate waves on the frequency of the incident wave. So, the infinitesimal volume *δV* is populated by infinite amount of the vibrators with frequencies of visible light. These vibrators populate the ether facilitating propagation of the light waves through the space.

In order to come to idea about existence of the intermediate medium (ether) that penetrates overall material world, we begin from the fundamental laws of classical physics. Three Newton's laws first published in Mathematical Principles of Natural Philosophy in 1687 [6] we recognize as basic laws of physics. Namely: (a) the first law postulates existence of inertial reference frames: an object that is at rest will stay at rest unless an external force acts upon it; an object that is in motion will not change its velocity unless an external force acts upon it. The inertia is a property of the bodies to resist to changing their velocity; (b) the second law states: the net force applied to a body with a mass *M* is equal to the rate of change of its linear momentum in an inertial reference frame

$$
\vec{F} = M\vec{a} = M\frac{d\vec{v}}{dt};\tag{1}
$$

(c) the third law states: for every action there is an equal and opposite reaction.

Leonard Euler had generalized the second Newton's laws on motion of deformable bodies [7]. We rewrite this law for such media. Let the deformable body be in volume *ΔV* and has the mass *M*. We divide Eq. (1) by *ΔV* and determine the time-dependent mass density *ρ<sup>M</sup>* =*MΔV* [8]. In this case, we shall understand *F* <sup>→</sup> as force per volume. Then the second law takes a form:

Physical Vacuum is a Special Superfluid Medium http://dx.doi.org/10.5772/59040 347

$$
\vec{F} = \frac{d\rho\_M \vec{v}}{dt} = \rho\_M \frac{d\vec{v}}{dt} + \vec{v} \frac{d\rho\_M}{dt}.\tag{2}
$$

The total derivatives in the right side can be written through partial derivatives:

gates shows the interference fringes typical for the two slit experiment. Their explanation is that the droplet while moving on the surface induces on this surface the weak Faraday waves. The latter provide the guidance conditions for the droplets. In this perspective, we can draw conclusion that the de Broglie physical wave can be represented by perturbations of the ether when the particle moves through it. In order to describe behavior of such an unusual medium we shall use the Navier-Stokes equation with slightly modified some terms. As the final result

In physical science of the New time the assumption for the existence of the ether medium was originally used to explain propagation of light and the long-range interactions. As for the propagation of light, the wave ideas of Huygens and Fresnel require the existence of a continuous intermediate environment between a source and a receiver of the light - the lightbearing ether. It is instructive to compare here the two opposite doctrines about the nature of light belonging to Sir Isaac Newton and Christian Huygens. Newton maintained the theory that the light was made up of tiny particles, corpuscles. They spread through an empty space in accordance with the law of the classical mechanics. Christian Huygens (a contemporary of Newton), believed that the light was made up of waves vibrating up and down perpendicu‐ larly to the direction of its propagation, as waves on a water surface. One can imagine all space populated everywhere densely by Huygens's vibrators. All vibrators are silent until a wave reaches them. As soon as a wave front reaches them, the vibrators begin to radiate waves on the frequency of the incident wave. So, the infinitesimal volume *δV* is populated by infinite amount of the vibrators with frequencies of visible light. These vibrators populate the ether

In order to come to idea about existence of the intermediate medium (ether) that penetrates overall material world, we begin from the fundamental laws of classical physics. Three Newton's laws first published in Mathematical Principles of Natural Philosophy in 1687 [6] we recognize as basic laws of physics. Namely: (a) the first law postulates existence of inertial reference frames: an object that is at rest will stay at rest unless an external force acts upon it; an object that is in motion will not change its velocity unless an external force acts upon it. The inertia is a property of the bodies to resist to changing their velocity; (b) the second law states: the net force applied to a body with a mass *M* is equal to the rate of change of its linear

> = = ; <sup>r</sup> <sup>r</sup> <sup>r</sup> *<sup>d</sup> F Ma M <sup>v</sup>*

Leonard Euler had generalized the second Newton's laws on motion of deformable bodies [7]. We rewrite this law for such media. Let the deformable body be in volume *ΔV* and has the mass *M*. We divide Eq. (1) by *ΔV* and determine the time-dependent mass density *ρ<sup>M</sup>* =*MΔV*

(c) the third law states: for every action there is an equal and opposite reaction.

*dt* (1)

<sup>→</sup> as force per volume. Then the second law takes a form:

we shall get the Schrödinger equation.

346 Selected Topics in Applications of Quantum Mechanics

facilitating propagation of the light waves through the space.

momentum in an inertial reference frame

[8]. In this case, we shall understand *F*

$$\frac{d\,\rho\_M}{dt} = \frac{\partial\,\rho\_M}{\partial t} + \left(\vec{v}\,\nabla\right)\rho\_M. \tag{3}$$

$$\frac{d\vec{v}}{dt} = \frac{\partial \vec{v}}{\partial t} + \left(\vec{v}\nabla\right)\vec{v}.\tag{4}$$

Eq. (3) equated to zero is seen to be the continuity equation. As for Eq. (4) we may rewrite the rightmost term in detail

$$(\vec{v}\nabla)\vec{v} = \nabla\frac{v^2}{2} - \left[\vec{v}\times[\nabla\times\vec{v}]\right].\tag{5}$$

As follows from this formula, the first term, multiplied by the mass, is gradient of the kinetic energy. It represents a force applied to the fluid element for its shifting on the unit of length, *δS*. The second term gives acceleration of the fluid element directed perpendicularly to the velocity *v* <sup>→</sup> . Let the fluid element move along some curve in 3D space. Tangent to the curve in each point points to orientation of the body motion. In turn, the vector *ω* <sup>→</sup> = ∇ ×*v* → is perpen‐ dicular to the plane, within which an arbitrarily small segment of the curve is situated. This vector characterizes a quantitative measure of the vortex motion. It is called *vorticity*. The vector product *v* <sup>→</sup> ×*ω* <sup>→</sup> is perpendicular to the both vectors *v* <sup>→</sup> and *ω* <sup>→</sup> . It shows the acceleration of the fluid element

The term (5) entering in the Navier-Stokes equation [9, 10] is responsible for emergence of vortex structures. The Navier-Stokes equation stems from Eq. (2) if we omit the rightmost term, representing the continuity equation, and specify forces in this equation in detail:

$$
\rho\_M \left( \frac{\partial \vec{v}}{\partial t} + \left( \vec{v} \nabla \right) \vec{v} \right) = \frac{\vec{F}}{\Delta V} + \mu(t) \nabla^2 \vec{v} - \rho\_M \nabla \left( \frac{P}{\rho\_M} \right) \tag{6}
$$

This equation contains two modifications represented in the two last terms from the right side: the dynamic viscosity *μ* depends on time and the rightmost term has a slightly modified view, namely *ρ<sup>M</sup>* ∇(*P* / *ρ<sup>M</sup>* )=∇*P* −*P* ∇ln(*ρ<sup>M</sup>* ). This modification will be important for us when we shall begin to derive the Schrödinger equation. However first we shall examine the Helmholtz vortices with time dependent dynamical viscosity.

#### **2. Vortex dynamics**

The second term from the right in Eq. (6) represents the viscosity of the fluid (*μ* is the dynamic viscosity, its units are N⋅s/m<sup>2</sup> =kg/(m⋅s)). Let us suppose that the fluid is ideal, barotropic, and the mass forces are conservative [10]. At assuming that the external force is conservative, we apply to this equation the operator curl. We get right away the equation for the vorticity:

$$\frac{\partial \vec{\boldsymbol{\alpha}}}{\partial t} + (\vec{\boldsymbol{\alpha}} \cdot \nabla)\vec{\boldsymbol{\upsilon}} = \nu(t)\nabla^2 \vec{\boldsymbol{\alpha}}.\tag{7}$$

Here *ν* (*t*)=*μ*(*t*)/ *ρ<sup>M</sup>* is the kinematic viscosity. Its dimensionality is m2 /s what corresponds to dimensionality of the diffusion coefficient. The rightmost term describes dissipation of the energy stored in the vortex. As a result, the vortex with the lapse of time will disappear.

With omitted the term from the right (i.e., *ν* =0) the Helmholtz theorem reads: (i) if fluid particles form, in any moment of the time, a vortex line, then the same particles support the vortex line both in the past and in the future; (ii) ensemble of the vortex lines traced through a closed contour forms a vortex tube. Intensity of the vortex tube is constant along its length and does not change in time. The vortex tube (a) either goes to infinity by both endings; (b) or these endings lean on walls of bath containing the fluid; (c) or these endings are locked to each on other forming a vortex ring.

Assuming that the fluid is a physical vacuum, which meets the requirements specified earlier, we must say that the viscosity vanishes. In that case, the vorticity *ω* <sup>→</sup> is concentrated in the center of the vortex, i.e., in the point. Mathematical representation of the vorticity is *δ*- function. Such singularity can be a source of possible divergences of computations in further.

We shall not remove the viscosity. Instead of that, we hypothesize that even if there is an arbitrary small viscosity, because of the zero-point oscillations in the vacuum, the vortex does not disappear completely. The vortex can be a long-lived object. The foundation for that hypothesis is the observation (performed by French scientific team [5, 11, 12]) of behavior of the droplets moving on the oil surface, on which the waves Faraday exist. Here an important moment is that the Faraday waves are supported slightly below the super-critical threshold. Due to this trick the droplets can live on the oil surface arbitrary long, before they disappear in the oil. The Faraday waves that are supported near the super-critical threshold may play a role analogous to the zero-point oscillations of the vacuum.

Observe that the bouncing droplet simulates some aspects of quantum mechanics, stimulating theoretical investigations in this area [13-20]. It is interesting to note in this place that Grössing considers a quantum particle as a dissipative phase-locked steady state, where an amount of zero-point energy of the wave-like environment is absorbed by the particle, and then during a characteristic relaxation time is dissipated into the environment again [14].

Here we shall give a simple model of such a picture. Let us look on the vortex tube in its crosssection which is oriented along the axis *z* and its center is placed in the coordinate origin of the plane (*x*, *y*). Eq. (7), written down in the cross-section of the vortex, is as follows

#### Physical Vacuum is a Special Superfluid Medium http://dx.doi.org/10.5772/59040 349

$$\frac{\partial \, \alpha \nu}{\partial t} = \nu \text{g}(t) \left( \frac{\partial^2 \, \alpha}{\partial r^2} + \frac{1}{r} \frac{\partial \, \alpha}{\partial r} \right). \tag{8}$$

We do not write a sign of vector on top of *ω* since *ω* is oriented strictly along the axis *z*. We introduce time-dependent the kinematic viscosity. For the sake of simplicity, let it be looked as

$$\log(t) = \cos(\Omega t + \phi) = \frac{e^{i(\Omega t + \phi)} + e^{-i(\Omega t + \phi)}}{2} \tag{9}$$

where *Ω* is an oscillation frequency and *ϕ* is the uncertain phase.

**2. Vortex dynamics**

viscosity, its units are N⋅s/m<sup>2</sup>

348 Selected Topics in Applications of Quantum Mechanics

on other forming a vortex ring.

The second term from the right in Eq. (6) represents the viscosity of the fluid (*μ* is the dynamic

the mass forces are conservative [10]. At assuming that the external force is conservative, we apply to this equation the operator curl. We get right away the equation for the vorticity:

<sup>2</sup> ( ) () .

 nw

dimensionality of the diffusion coefficient. The rightmost term describes dissipation of the energy stored in the vortex. As a result, the vortex with the lapse of time will disappear.

With omitted the term from the right (i.e., *ν* =0) the Helmholtz theorem reads: (i) if fluid particles form, in any moment of the time, a vortex line, then the same particles support the vortex line both in the past and in the future; (ii) ensemble of the vortex lines traced through a closed contour forms a vortex tube. Intensity of the vortex tube is constant along its length and does not change in time. The vortex tube (a) either goes to infinity by both endings; (b) or these endings lean on walls of bath containing the fluid; (c) or these endings are locked to each

Assuming that the fluid is a physical vacuum, which meets the requirements specified earlier,

of the vortex, i.e., in the point. Mathematical representation of the vorticity is *δ*- function. Such

We shall not remove the viscosity. Instead of that, we hypothesize that even if there is an arbitrary small viscosity, because of the zero-point oscillations in the vacuum, the vortex does not disappear completely. The vortex can be a long-lived object. The foundation for that hypothesis is the observation (performed by French scientific team [5, 11, 12]) of behavior of the droplets moving on the oil surface, on which the waves Faraday exist. Here an important moment is that the Faraday waves are supported slightly below the super-critical threshold. Due to this trick the droplets can live on the oil surface arbitrary long, before they disappear in the oil. The Faraday waves that are supported near the super-critical threshold may play a

Observe that the bouncing droplet simulates some aspects of quantum mechanics, stimulating theoretical investigations in this area [13-20]. It is interesting to note in this place that Grössing considers a quantum particle as a dissipative phase-locked steady state, where an amount of zero-point energy of the wave-like environment is absorbed by the particle, and then during

Here we shall give a simple model of such a picture. Let us look on the vortex tube in its crosssection which is oriented along the axis *z* and its center is placed in the coordinate origin of

w

*t*

Here *ν* (*t*)=*μ*(*t*)/ *ρ<sup>M</sup>* is the kinematic viscosity. Its dimensionality is m2

¶

we must say that the viscosity vanishes. In that case, the vorticity *ω*

role analogous to the zero-point oscillations of the vacuum.

singularity can be a source of possible divergences of computations in further.

a characteristic relaxation time is dissipated into the environment again [14].

the plane (*x*, *y*). Eq. (7), written down in the cross-section of the vortex, is as follows

w

¶ + ×Ñ Ñ

<sup>r</sup> rr r *<sup>t</sup>*

=kg/(m⋅s)). Let us suppose that the fluid is ideal, barotropic, and

*v =* (7)

/s what corresponds to

<sup>→</sup> is concentrated in the center

**Figure 1.** Vorticity *ω*(*r*, *t*) and velocity *v*(*r,t*) as functions of *r* and *t* for *Γ* =1, *ν* =1, *Ω* =*π*, and *n*=16. These parame‐ ters are conditional in order to show clearly oscillations of the vortex in time. The solution does not decay with time.

Solution of the equation (8) in this case is as follows

$$\alpha(r,t) = \frac{\Gamma}{4\pi \left(\nu/\Omega\right) \left(\sin(\Omega t + \phi) + n\right)} \exp\left\{-\frac{r^2}{4\left(\nu/\Omega\right) \left(\sin(\Omega t + \phi) + n\right)}\right\}.\tag{10}$$

Here *Γ* is the integration constant having dimension m2 /s. An extra number *n* > 1. It prevents appearance of singularity in the cases when sin(*Ωt* + *ϕ*) tends to-1. This function at choosing the parameters *Γ* =1, *ν* =1, *Ω* =*π*, and *n*=16 is shown in Fig. 1(a).

The velocity of the fluid matter around the vortex results from the integration of the vorticity function

$$w(r,t) = \frac{1}{r} \Big|\_{0}^{r} o(r',t)r' dr' = \frac{\Gamma}{2\pi r} \left( 1 - \exp\left\{ -\frac{r^2}{4\left(\nu/\Omega\right)\left(\sin\left(\Omega t + \phi\right) + n\right)} \right\} \right). \tag{11}$$

Fig. 1(b) shows behavior of this function at the same input parameters.

In particular, for *n*=0 and *Ω t* <<1 this solution is close to the Lamb–Oseen vortex solution [21]

$$\alpha(r,t) = \frac{\Gamma}{4\pi\nu t} \mathcal{C}^{-r^2/4\nu t},\tag{12}$$

$$w(r,t) = \frac{\Gamma}{2\pi r} \left(1 - e^{-r^2/4\nu t}\right). \tag{13}$$

As seen from here, the Lamb–Oseen solution decays with time since the viscosity *ν* >0.

One can see from the solutions (10) and (11), depending on the distance to the center the functions *ω*(*r*, *t*) and *v*(*r,t*) show typical behavior for the vortices. The both functions do not decay with time, however. Instead of that, they demonstrate pulsations on the frequency *Ω*. Amplitude of the pulsations is the smaller, the larger value of the parameter *n*. At *n* tending to infinity the amplitude of the pulsations tends to zero. At the same time the vortex disappears entirely.

**Figure 2.** Periodic energy exchange between the vortex and vacuum fluctuations

The undamped solution was obtained thanks to assumption, that the kinematic viscosity is a periodic function of time, namely, *ν g*(*t*)=*ν*cos(*Ω t* + *ϕ*). The viscosity in the quantum realm is not a good concept, however. Most likely, it manifests itself through interaction of the quantum object with vacuum fluctuations. According to Eq. (9), there are half-periods when the energy of the vortex is lost at scattering on the vacuum fluctuations, and there are other half-periods when the vacuum returns this energy to the vortex, Fig. 2. On the whole, the viscosity of the fluid medium, within which the vortex tube evolves, in the average remains at zero. It can mean that this medium is superfluid. Such a scenario is not unusual. For example, at transition of helium to the superfluid phase [22] coherent Cooper pairs of electrons arise through the exchange by phonons. This attraction is due to the electron–phonon interaction. The phonons are thermal excitations of a lattice. In that case, they play a role of the background medium.

( )( ) <sup>0</sup> <sup>2</sup> <sup>1</sup> ( , ) ( ', ) ' ' 1 exp . <sup>2</sup> 4 sin( )

In particular, for *n*=0 and *Ω t* <<1 this solution is close to the Lamb–Oseen vortex solution [21]

<sup>2</sup> <sup>4</sup> (,) , <sup>4</sup> G

( ) <sup>2</sup> <sup>4</sup> (,) 1 . <sup>2</sup> G

As seen from here, the Lamb–Oseen solution decays with time since the viscosity *ν* >0.

One can see from the solutions (10) and (11), depending on the distance to the center the functions *ω*(*r*, *t*) and *v*(*r,t*) show typical behavior for the vortices. The both functions do not decay with time, however. Instead of that, they demonstrate pulsations on the frequency *Ω*. Amplitude of the pulsations is the smaller, the larger value of the parameter *n*. At *n* tending to infinity the amplitude of the pulsations tends to zero. At the same time the vortex disappears

The undamped solution was obtained thanks to assumption, that the kinematic viscosity is a periodic function of time, namely, *ν g*(*t*)=*ν*cos(*Ω t* + *ϕ*). The viscosity in the quantum realm is not a good concept, however. Most likely, it manifests itself through interaction of the quantum object with vacuum fluctuations. According to Eq. (9), there are half-periods when the energy of the vortex is lost at scattering on the vacuum fluctuations, and there are other half-periods when the vacuum returns this energy to the vortex, Fig. 2. On the whole, the viscosity of the fluid medium, within which the vortex tube evolves, in the average remains at zero. It can mean that this medium is superfluid. Such a scenario is not unusual. For example, at transition of helium to the superfluid phase [22] coherent Cooper pairs of electrons arise through the

p


pn

æ ö ì ü ï ï = = -- ç ÷ í ý ï ï W W+ + è ø î þ <sup>ò</sup>

*<sup>r</sup> <sup>r</sup> r t r t r dr r r t n*

n

n

*v* (11)

nf


*e* (12)

G

p

Fig. 1(b) shows behavior of this function at the same input parameters.

w

**Figure 2.** Periodic energy exchange between the vortex and vacuum fluctuations

w

350 Selected Topics in Applications of Quantum Mechanics

entirely.

Qualitative view of the vortex tube in its cross-section is shown in Fig. 3. Values of the velocity *v* are shown by grey color ranging from light grey (minimal velocities) to dark grey (maximal ones). A visual image of this picture can be a hurricane (tropical cyclone [23]) shown from the top. In the center of the vortex, a so-called eye of the hurricane (the vortex core) is well viewed. Here it looks as a small light grey disk, where the velocities have small values. In the very center of the disk, in particular, the velocity vanishes. Observe that in the region of the hurricane eye a wind is really very weak, especially near the center. This is in stark contrast to conditions in the region of the eyewall, where the strongest winds exist (in Fig. 3 it looks as a dark grey annular region enclosing the light grey inner area). The eyewall of the vortex tube (a zone where the velocity reaches maximal values) has the nonzero radius.

**Figure 3.** Cross-section of the vortex tube in the plane (*x*, *y*). Values of the velocity *v* are shown in grey ranging from light grey (min *v*) to dark grey (max *v*). Density of the pixels represents magnitude of the vorticity *ω*. Core of the vor‐ tex is well visible in the center.

Let us find the radius of the vortex core. In order to evaluate this radius we equate to zero the first derivative by *r* of equation (11)

$$\frac{\exp\left\{-\frac{r^2}{4\left(\nu/\Omega\right)\left(\sin\left(\Omega t + \phi\right) + n\right)}\right\} \left(2\frac{r^2}{4\left(\nu/\Omega\right)\left(\sin\left(\Omega t + \phi\right) + n\right)} + 1\right) - 1}{2\pi r^2} = 0. \tag{14}$$

The radius is a root of this equation

$$r\_{\vee} = 2\sqrt{a\_0(n + \sin(\Omega \, t + \phi))} \cdot \sqrt{\frac{\nu}{\Omega}}.\tag{15}$$

Here *a*0≈1.2564312 is a root of the equation ln(2*a*<sup>0</sup> + 1)−*a*<sup>0</sup> =0. One can see, that *rv* is an oscillating function. The larger Ω, the more quickly the vortex trembles. As Ω increases, the vortex radius decreases. However it grows with increasing the number *n*. Let us evaluate the radius *rv* at choosing the viscosity ν equal to *ν*¯ =ℏ / 2*m*. Here *ν*¯ is the diffusion coefficient of the Brownian sub-quantum particles wandering in the Nelson's aether [24], see Appendix A. In the case of electron *ν*¯ =ℏ / 2*m*≈ 5.79∙10-5 m2 /s. As for Ω, let it be equal to 2mc2 /ℏ, or approximately 1.6∙10<sup>21</sup> radians per second for electron. Here *c* is the speed of light and *m* is the electron mass. Then we have (ν/Ω)1/2 ≈ 1.93∙10-13 m. This length is seen to be smaller then the Compton wavelength, λC = 2.426∙10-12 m, in about 12 times. So, for choosing *n* ≈ 31 we find from Eq. (15) that the radius of the vortex is about the Compton wavelength. From the above one can see that, on a distance about the Compton wavelength, virtual particles can be involved into a vorticity dancing around the electron core, by polarizing the electron charge. This dancing happens at trembling motion of the electron with the frequency Ω = 2mc2 /ℏ. That oscillating motion has a deep relation to the so-called "Zitterbewengung" [25].

One can give a general solution of Eq. (8) which has the following presentation

$$\begin{aligned} \rho v(r,t) &= \frac{\Gamma}{4\pi \left(\int\_0^t \nu(\tau)d\tau + \sigma^2\right)} \exp\left[-\frac{r^2}{4\left(\int\_0^t \nu(\tau)d\tau + \sigma^2\right)}\right], \\\\ v(r,t) &= \frac{\Gamma}{2\pi r} \left(1 - \exp\left\{-\frac{r^2}{4\left(\int\_0^t \nu(\tau)d\tau + \sigma^2\right)}\right\}\right). \end{aligned} \tag{17}$$

The viscosity function *ν* (*t*) is a quasi-periodic function or even is represented by a color noise. The integral of the viscosity function memorizes integrally character of the viscosity of the medium. Due to this memory effect, the vortex may live a long enough. As for interpretation of these solutions with the quantum-mechanical point of view, we may say that there exists a regular exchange by quanta with the vacuum fluctuations, Fig. 2. The integral accumulates all cases of the exchange with the vacuum. The constant *σ* having dimension of length, prevents appearance of singularities. One can see that even at *ν* ≡0, but *σ* >0, abundance of long-lived vortices can exist in the vacuum. Such vortices are "ghosts" in the superfluid being invisible without interaction.

#### **2.1. Vortex rings and vortex balls**

The radius is a root of this equation

352 Selected Topics in Applications of Quantum Mechanics

electron *ν*¯ =ℏ / 2*m*≈ 5.79∙10-5 m2

w

<sup>v</sup> <sup>0</sup> 2 ( sin( )) .

Here *a*0≈1.2564312 is a root of the equation ln(2*a*<sup>0</sup> + 1)−*a*<sup>0</sup> =0. One can see, that *rv* is an oscillating function. The larger Ω, the more quickly the vortex trembles. As Ω increases, the vortex radius decreases. However it grows with increasing the number *n*. Let us evaluate the radius *rv* at choosing the viscosity ν equal to *ν*¯ =ℏ / 2*m*. Here *ν*¯ is the diffusion coefficient of the Brownian sub-quantum particles wandering in the Nelson's aether [24], see Appendix A. In the case of

/s. As for Ω, let it be equal to 2mc2

radians per second for electron. Here *c* is the speed of light and *m* is the electron mass. Then we have (ν/Ω)1/2 ≈ 1.93∙10-13 m. This length is seen to be smaller then the Compton wavelength, λC = 2.426∙10-12 m, in about 12 times. So, for choosing *n* ≈ 31 we find from Eq. (15) that the radius of the vortex is about the Compton wavelength. From the above one can see that, on a distance about the Compton wavelength, virtual particles can be involved into a vorticity dancing around the electron core, by polarizing the electron charge. This dancing happens at

trembling motion of the electron with the frequency Ω = 2mc2

(,) exp

nt t s

( , ) 1 exp 2

= --

*<sup>r</sup> r t <sup>r</sup> <sup>t</sup>*

G

p G

= -

One can give a general solution of Eq. (8) which has the following presentation

4 4 0 0

*<sup>r</sup> r t t t*

( ) ( )

4 0

í ý æ öæ ö ï ï ç ÷ç ÷ + + è øè ø î þ ò ò

*d d*

a deep relation to the so-called "Zitterbewengung" [25].

p = + W+ ×

n

*r an t* (15)

2

2

*v* (17)

.

ì ü ï ï

,

2 2

2

*d*

( )

í ý æ ö ï ï ç ÷ <sup>+</sup>

nt t s

è ø î þ è ø ò

The viscosity function *ν* (*t*) is a quasi-periodic function or even is represented by a color noise. The integral of the viscosity function memorizes integrally character of the viscosity of the medium. Due to this memory effect, the vortex may live a long enough. As for interpretation of these solutions with the quantum-mechanical point of view, we may say that there exists a regular exchange by quanta with the vacuum fluctuations, Fig. 2. The integral accumulates all cases of the exchange with the vacuum. The constant *σ* having dimension of length, prevents appearance of singularities. One can see that even at *ν* ≡0, but *σ* >0, abundance of long-lived

æ ö ì ü ç ÷ ï ï

nt t s

/ℏ, or approximately 1.6∙10<sup>21</sup>

/ℏ. That oscillating motion has

(16)

fW

> If we roll up the vortex tube in a ring and glue together its opposite ends we obtain a vortex ring. A result of such an operation put into the (*x*, *y*) plane is shown in Fig 4. Position of points on the helicoidal vortex ring in the Cartesian coordinate system is given by

$$\begin{cases} x = (r\_1 + r\_0 \cos(\alpha \underline{\phi}\_2 t + \phi\_2)) \cos(\alpha \underline{\phi} t + \phi\_1), \\ y = (r\_1 + r\_0 \cos(\alpha \underline{\phi}\_2 t + \phi\_2)) \sin(\alpha \underline{\phi} t + \phi\_1), \\ z = r\_0 \sin(\alpha \underline{\phi}\_2 t + \phi\_2). \end{cases} \tag{18}$$

Here *r*0 is the radius of the tube. And *r*1 represents the distance from the center of the tube (pointed in the figure by arrow c) to the center of the torus located in the origin of coordinates (*x,y,z*). A body of the tube, for the sake of visualization is colored in cyan. Eq. (18) parametrized by *t* gives a helicoidal vortex ring shown in this figure. Parameters *ω*1 and *ω*2 are frequencies of rotation along the arrow a about the center of the torus (about the axis *z*) and rotation along the arrow b about the center of the tube (about the axis pointed by arrow c), respectively. Phases *ϕ*1 and *ϕ*<sup>2</sup> have uncertain quantities ranging from 0 to 2*π*. By choosing the phases within this interval with a small increment, we may fill the torus by the helicoidal vortices everywhere densely. The vorticity is maximal along the center of the tube. Whereas the velocity of rotation about this center in the vicinity of it is minimal. However, the velocity grows as a distance from the center increases. After reaching of some maximal value the velocity further begins to decrease.

Let the radius *r*1 in Eq. (18) tends to zero. The helicoidal vortex ring in this case will transform into a vortex ring enveloping a spherical ball. The vortex ring for the case *r* 0=4, *r* 1 ≈ 0, *ω*<sup>2</sup> =3*ω*1, and *ϕ*<sup>2</sup> =*ϕ*<sup>1</sup> =0 drawn by thick curve colored in deep green is shown in Fig. 5(a). Motion of an elementary vortex clot along the vortex ring (along the thick curve colored in deep green) takes place with a velocity

$$\vec{\boldsymbol{\upsilon}}\_{\boldsymbol{\alpha}} = \begin{pmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{\alpha},\boldsymbol{\varepsilon}} - \boldsymbol{r}\_{0}\boldsymbol{\alpha}\_{2}\sin(\alpha\_{2}t + \phi\_{2})\cos(\alpha\_{1}t + \phi\_{1}) - \boldsymbol{r}\_{0}\boldsymbol{\alpha}\_{1}\cos(\alpha\_{2}t + \phi\_{2})\sin(\alpha\_{1}t + \phi\_{1}) \\\\ -\boldsymbol{r}\_{l}\boldsymbol{\alpha}\_{1}\sin(\alpha\_{l}t + \phi\_{1}) \\\\ \boldsymbol{\upsilon}\_{\boldsymbol{\alpha},\boldsymbol{\varepsilon}} - \boldsymbol{r}\_{0}\boldsymbol{\alpha}\_{2}\sin(\alpha\_{2}t + \phi\_{2})\sin(\alpha\_{l}t + \phi\_{1}) + \boldsymbol{r}\_{0}\boldsymbol{\alpha}\_{1}\cos(\alpha\_{2}t + \phi\_{2})\cos(\alpha\_{l}t + \phi\_{1}) \\\\ +\boldsymbol{r}\_{l}\boldsymbol{\alpha}\_{1}\sin(\alpha\_{l}t + \phi\_{1}) \\\\ \boldsymbol{\upsilon}\_{\boldsymbol{\alpha},\boldsymbol{\varepsilon}} = \boldsymbol{r}\_{0}\boldsymbol{\alpha}\_{2}\cos(\alpha\_{2}t + \phi\_{1}) \end{pmatrix} \tag{19}$$

The velocity of the clot at the initial time is *vR*,*<sup>x</sup>* =0, *vR*,*<sup>y</sup>* =*r*0*ω*1, *vR*,*<sup>z</sup>* =*r*0*ω*<sup>2</sup> =3*r*0*ω*<sup>1</sup> (the initial point (*x, y, z*)=(4, 0, 0) is on the top of the ball). We designate this velocity as *v* → <sup>+</sup>. Through the time *t* =*πω*<sup>1</sup> the elementary vortex clot returns to the top position. The velocity in this case is equal

**Figure 4.** Helicoidal vortex ring: *r*0=2, *r*1=3, *ω*<sup>2</sup> =12*ω*1, *ϕ*<sup>2</sup> =*ϕ*<sup>1</sup> =0.

to *vR*,*<sup>x</sup>* =0, *vR*,*<sup>y</sup>* =*r*0*ω*1, *vR*,*<sup>z</sup>* = −*r*0*ω*<sup>2</sup> = −3*r*0*ω*1. We designate this velocity as *v* → <sup>−</sup>. Sum of the two opposite velocities, *v* → <sup>+</sup> and *v* → <sup>−</sup>, gives the velocity *v* → <sup>0</sup> =(0, 2*r*0*ω*1, 0). During *t* =(1 + 3*k*)*π* / 3*ω*1 and *t* =(2 + 3*k*)*π* / 3*ω*<sup>1</sup> (*k*=1, 2, ⋯) the clot travels through the positions 1 and 2 both in the forward and in backward directions, respectively. In the vicinity of these points the velocities *v* <sup>→</sup> + and *v* → − yield the resulting velocity *v* → <sup>0</sup> directed along the circle lying in the plane (*x, y*).

**Figure 5.** (a) Helicoidal vortex ring (colored in deep green) convoluted into the vortex ball. The input parameters of the ball are as follows *r*0=4, *r*1=0.01 ≪ 1, *ω*<sup>2</sup> =3*ω*1, *ϕ*<sup>2</sup> =*ϕ*<sup>1</sup> =0. The radius *r*0 represents a mean radius of the ball, where the velocity *v*0 reaches a maximal value. The ratio *ω*<sup>2</sup> / *ω*<sup>1</sup> =3 was chosen with the aim not to overload the picture by superfluous curves. (b) The vortex ball rotating about axis *z* with the maximal velocity *v*<sup>0</sup> that is reached on the surface of the ball.

The ball can be filled everywhere densely by other rings at adding them with other phases *ϕ*<sup>1</sup> and *ϕ*2 ranging from 0 to 2*π*. The velocity *v* → <sup>0</sup> for any ring will lie on the same circles centered on the axis *z*. We see a dense ball that rolls along the axis *y*, Fig. 5(b). Observe that the ball pulsates on the frequency Ω as it rolls along its path, as it follows from the above computations. Perfect modes describing the rolling ball are spherical harmonics [26].

#### **3.3. Derivation of the Schrödinger equation**

to *vR*,*<sup>x</sup>* =0, *vR*,*<sup>y</sup>* =*r*0*ω*1, *vR*,*<sup>z</sup>* = −*r*0*ω*<sup>2</sup> = −3*r*0*ω*1. We designate this velocity as *v*

<sup>−</sup>, gives the velocity *v*

→

<sup>0</sup> directed along the circle lying in the plane (*x, y*).

*t* =(2 + 3*k*)*π* / 3*ω*<sup>1</sup> (*k*=1, 2, ⋯) the clot travels through the positions 1 and 2 both in the forward

**Figure 5.** (a) Helicoidal vortex ring (colored in deep green) convoluted into the vortex ball. The input parameters of the ball are as follows *r*0=4, *r*1=0.01 ≪ 1, *ω*<sup>2</sup> =3*ω*1, *ϕ*<sup>2</sup> =*ϕ*<sup>1</sup> =0. The radius *r*0 represents a mean radius of the ball, where the velocity *v*0 reaches a maximal value. The ratio *ω*<sup>2</sup> / *ω*<sup>1</sup> =3 was chosen with the aim not to overload the picture by superfluous curves. (b) The vortex ball rotating about axis *z* with the maximal velocity *v*<sup>0</sup> that is reached on the surface

and in backward directions, respectively. In the vicinity of these points the velocities *v*

opposite velocities, *v*

of the ball.

yield the resulting velocity *v*

→ <sup>+</sup> and *v* →

354 Selected Topics in Applications of Quantum Mechanics

→

**Figure 4.** Helicoidal vortex ring: *r*0=2, *r*1=3, *ω*<sup>2</sup> =12*ω*1, *ϕ*<sup>2</sup> =*ϕ*<sup>1</sup> =0.

→

<sup>0</sup> =(0, 2*r*0*ω*1, 0). During *t* =(1 + 3*k*)*π* / 3*ω*1 and

<sup>−</sup>. Sum of the two

<sup>→</sup> + and *v* → − The third term in the right side of Eq. (6) deals with the pressure gradient. One can see, however, it is slightly differ from the pressure gradient presented in the customary Navier-Stokes equation [9, 10]. One can rewrite this term in detail

$$\rho\_M \nabla \left(\frac{P}{\rho\_M}\right) = \nabla P - P \nabla \ln(\rho\_M). \tag{20}$$

The first term, ∇*P*, is the customary pressure gradient represented in the Navier-Stokes equation. Whereas, the second term, *P* ∇ln(*ρ<sup>M</sup>* ), is an extra term describing changing the logarithm of the density along increment of length (the entropy increment) multiplied by *P*. It may mean that change of the pressure is induced by change of the entropy per length, or else by change of the information flow [27, 28] per length. This term has signs typical of the osmotic pressure, mentioned by Nelson [24].

Let us consider in this respect the pressure *P* in more detail. We shall represent the pressure consisting of two parts, *P*1 and *P*2. We begin from the Fick's law [14]. The law says that the diffusion flux, *J*, is proportional to the negative value of the density gradient *J*=−*D*∇*ρ<sup>M</sup>* . Here *D* is the diffusion coefficient *v*¯ =ℏ / 2*m* [24], see Nelson's definition in Appendix A. Since the term *v*¯∇ *J* has dimension of the pressure, we define *P*<sup>1</sup> as the pressure having diffusion nature

$$P\_1 = \overline{\nu} \,\nabla \vec{J} = -\frac{\hbar^2}{4m^2} \nabla^2 \rho\_M. \tag{21}$$

Observe that the kinetic energy of the diffusion flux is (*m* / 2)(*J* / *ρ<sup>M</sup>* )2. It means that there exists one more pressure as the average momentum transfer per unit area per unit time:

$$P\_2 = \frac{\rho\_M}{2} \left(\frac{\vec{J}}{\rho\_M}\right)^2 = \frac{\hbar^2}{8m^2} \frac{(\nabla \rho\_M)^2}{\rho\_M}.\tag{22}$$

Now we can see that sum of the two pressures, *P*1+*P*2, divided by *ρM* gives a term

$$\mathcal{Q}\_M = \frac{\hbar^2}{8m^2} \left(\frac{\nabla \rho\_M}{\rho\_M}\right)^2 - \frac{\hbar^2}{4m^2} \frac{\nabla^2 \rho\_M}{\rho\_M}. \tag{23}$$

One can see that accurate to the divisor *m* this term represents the quantum potential.

To bring the expression (23) to a form of the quantum potential, we need to introduce instead of the mass density *ρM* the probability density *ρ* according to the following presentation

$$
\rho\_M = \frac{M}{\Delta V} = \frac{mN}{\Delta V} = m\rho.\tag{24}
$$

Here the mass *M* is a product of an elementary mass *m* by the number of these masses, *N*, filling the volume *ΔV* . Then the mass density *ρ<sup>M</sup>* is defined as a product of the elementary mass *m* by the density of quasi-particles *ρ* = *N* / *ΔV* . We can imagine the quasi-particle as a long-lived local heterogeneity, which moves with the current velocity *v* <sup>→</sup> and probably has the vorticity. Let us divide the Navier-Stokes equation Eq. (6) by the probability density *ρ*. We obtain

$$m\left(\frac{\partial \vec{v}}{\partial t} + (\vec{v}\nabla)\vec{v}\right) = \frac{\vec{F}}{N} + \nu(t)\nabla^2 m \,\vec{v} - \nabla Q \tag{25}$$

Here *F* <sup>→</sup> / *<sup>N</sup>* is the force per one the quasi-particle. The kinetic viscosity *<sup>ν</sup>* (*t*)=*μ*(*t*)/ *<sup>ρ</sup><sup>M</sup>* is represented through the diffusion coefficient *v*¯ =ℏ / 2*m* [24], *ν*(*t*)=2*v*¯ *g*(*t*)=*ν g*(*t*), *ν* =ℏ / *m*, where *g* (*t)* is the dimensionless time dependent function. The function *Q* here is the real quantum potential

$$\mathcal{Q} = \frac{\hbar^2}{8m} \left(\frac{\nabla \rho}{\rho}\right)^2 - \frac{\hbar^2}{4m} \frac{\nabla^2 \rho}{\rho}. \tag{26}$$

Grössing noticed that the term ∇*Q*, the gradient of the quantum potential, describes a completely thermalized fluctuating force field [13, 14]. Here the fluctuating force is expressed via the gradient of the pressure divided by the density distribution of sub-quantum particles chaotically moving in the environment. Perhaps, they are virtual particle-antiparticle pairs.

Since the pressure provides a basis of the quantum potential, as was shown above, it would be interesting to interpret an osmotic nature of the pressure [24]. The interpretation can be the following (see Appendix A): a semipermeable membrane where the osmotic pressure mani‐ fests itself is an instant, which divides the past and the future (that is, the 3D brane of our being represents the semipermeable membrane in the 4D world). In other words, the thermalized fluctuating force field described by Grössing [13, 14] is asymmetric with respect to the time arrow.

#### **3.1. Transition to the Schrödinger equation**

2 2 2 2 2 2 . 8 4

*M M*

To bring the expression (23) to a form of the quantum potential, we need to introduce instead of the mass density *ρM* the probability density *ρ* according to the following presentation

Here the mass *M* is a product of an elementary mass *m* by the number of these masses, *N*, filling the volume *ΔV* . Then the mass density *ρ<sup>M</sup>* is defined as a product of the elementary mass *m* by the density of quasi-particles *ρ* = *N* / *ΔV* . We can imagine the quasi-particle as a long-lived

n

represented through the diffusion coefficient *v*¯ =ℏ / 2*m* [24], *ν*(*t*)=2*v*¯ *g*(*t*)=*ν g*(*t*), *ν* =ℏ / *m*, where *g* (*t)* is the dimensionless time dependent function. The function *Q* here is the real quantum

> 2 2 2 2 . 8 4 r

Grössing noticed that the term ∇*Q*, the gradient of the quantum potential, describes a completely thermalized fluctuating force field [13, 14]. Here the fluctuating force is expressed via the gradient of the pressure divided by the density distribution of sub-quantum particles chaotically moving in the environment. Perhaps, they are virtual particle-antiparticle pairs.

Since the pressure provides a basis of the quantum potential, as was shown above, it would be interesting to interpret an osmotic nature of the pressure [24]. The interpretation can be the following (see Appendix A): a semipermeable membrane where the osmotic pressure mani‐ fests itself is an instant, which divides the past and the future (that is, the 3D brane of our being represents the semipermeable membrane in the 4D world). In other words, the thermalized fluctuating force field described by Grössing [13, 14] is asymmetric with respect to the time

r

æ ö Ñ Ñ = - ç ÷ è ø

<sup>→</sup> / *<sup>N</sup>* is the force per one the quasi-particle. The kinetic viscosity *<sup>ν</sup>* (*t*)=*μ*(*t*)/ *<sup>ρ</sup><sup>M</sup>* is

 r

 r

 r

*<sup>Q</sup> m m* (23)

*M mN <sup>m</sup> V V* (24)

*v v v* (25)

h h *<sup>Q</sup> m m* (26)

<sup>→</sup> and probably has the vorticity.

 r

 r

r

æ ö Ñ Ñ <sup>=</sup> ç ÷ è ø h h *M M <sup>M</sup>*

One can see that accurate to the divisor *m* this term represents the quantum potential.

== = . D D *<sup>M</sup>*

Let us divide the Navier-Stokes equation Eq. (6) by the probability density *ρ*. We obtain

( ) <sup>2</sup>

ç ÷ + Ñ = + Ñ -Ñ

<sup>r</sup> <sup>r</sup> r r <sup>r</sup> *<sup>m</sup> tm Q t N*

( ) æ ö ¶

*v F*

r

r

356 Selected Topics in Applications of Quantum Mechanics

local heterogeneity, which moves with the current velocity *v*

è ø ¶

Here *F*

potential

arrow.

The current velocity *v* contains two component – irrotational and solenoidal [10] that relate to vortex-free and vortex motions of the medium, respectively. The basis for the latter is the Kelvin-Stokes theorem. Scalar and vector fields underlie of manifestation of the irrotational and solenoidal velocities

$$
\vec{\boldsymbol{\upsilon}} \cdot \vec{\boldsymbol{\upsilon}} = \vec{\boldsymbol{\upsilon}}\_S + \vec{\boldsymbol{\upsilon}}\_R = \frac{1}{m} \nabla S + \vec{\boldsymbol{\upsilon}}\_R. \tag{27}
$$

Here subscripts *S* and *R* hint to scalar and vector (rotational) potentials underlying emergence of these two components of the velocity. These velocities are submitted by the following equations

$$\begin{cases} \left(\nabla \cdot \vec{v}\_{\mathcal{S}}\right) \neq 0, \qquad \left[\nabla \times \vec{v}\_{\mathcal{S}}\right] = 0, \\ \left(\nabla \cdot \vec{v}\_{\mathcal{R}}\right) = 0, \qquad \left[\nabla \times \vec{v}\_{\mathcal{R}}\right] = \vec{\alpha}. \end{cases} \tag{28}$$

The scalar field is represented by the scalar function *S* – action in classical mechanics. Both velocities are perpendicular to each other. We may define the momentum and the kinetic energy

$$\begin{cases} \vec{p} = m\vec{v} = \nabla S + m\vec{v}\_R, \\ m\frac{v^2}{2} = \frac{1}{2m}(\nabla S)^2 + m\frac{v\_R^2}{2}. \end{cases} \tag{29}$$

Now we may rewrite the Navier-Stokes equation (25) in the more detailed form

$$\begin{split} \frac{\partial}{\partial t} \left( \nabla S + m \vec{\boldsymbol{v}}\_{R} \right) + \underbrace{\frac{1}{2m} \nabla \left( \left( \nabla S \right)^{2} + m^{2} \boldsymbol{v}\_{R}^{2} \right) + \left[ \vec{\boldsymbol{\phi}} \times \left( \nabla S + m \vec{\boldsymbol{v}}\_{R} \right) \right]}\_{\left( \vec{a} \right)} = \\ - \nabla U - \nabla Q + \underbrace{\nu(t) \nabla^{2} \left( \nabla S + m \vec{\boldsymbol{v}}\_{R} \right)}\_{\left( \vec{b} \right)}. \end{split} \tag{30}$$

Note that the term embraced by the curly bracket (*a*) stems from (*v* <sup>→</sup> ∇)*v* <sup>→</sup> =∇*v* <sup>2</sup> / 2 + *ω* <sup>→</sup> ×*v* <sup>→</sup> , see Eq. (5). Here we take into account that the external force is conservative, i.e., *F* <sup>→</sup> / *<sup>N</sup>* <sup>=</sup> −∇*<sup>U</sup>* , where *U* is the potential energy relating to the single quasi-particle. The term embraced by the curly bracket (*b*) describes the viscosity of the medium. As was said above the viscosity coefficient in the average is equal to zero.

Let us rewrite Eq. (30) by regrouping the terms

$$\nabla \left( \frac{\partial}{\partial t} S + \frac{1}{2m} \left( \nabla S \right)^2 + \frac{m}{2} \boldsymbol{\upsilon}\_k^2 + \boldsymbol{U} + \boldsymbol{Q} - \nu(\mathbf{t}) \nabla^2 S \right) = -m \frac{\partial}{\partial t} \vec{\boldsymbol{\upsilon}}\_k - \left[ \vec{\boldsymbol{\alpha}} \times \left( \nabla S + m \vec{\boldsymbol{\upsilon}}\_k \right) \right] + \nu(\mathbf{t}) m \nabla^2 \vec{\boldsymbol{\upsilon}}\_k. \tag{31}$$

We assume that fluctuations of the viscosity about zero occur much more frequent, than characteristic time of displacements of the quasi-particles. For that reason, we omit the term *ν*(*t*)∇2*S* by supposing in the first approximation, that the medium is absolutely superfluidthere are no energy sources and sinks. By multiplying this equation from the left by *v* → *<sup>S</sup>* we find that the right part of this equation vanishes since (*v* → *<sup>S</sup>* ⋅*v* → *<sup>R</sup>*)=(*v* → *<sup>S</sup>* ⋅*ω* <sup>→</sup> )=0. The left part vanishes if the expression under the brackets is constant. As a result, we come to the following modified Hamilton-Jacobi equation

$$\frac{\partial}{\partial t}S + \frac{1}{2m}(\nabla S)^2 + \frac{m}{2}v\_R^2 + \
U(\vec{r}) - \frac{\hbar^2}{2m} \left(\frac{\nabla^2 \rho}{2\rho} - \left(\frac{\nabla \rho}{2\rho}\right)^2\right) = C. \tag{32}$$

The modification is due to adding the quantum potential (26). In this equation, *C* is an integration constant. We see that the third term in this equation represents energy of the vortex. On the other hand, we can see that the vortex given by Eq. (7) is replenished by the kinetic energy coming from the scalar field *S*, namely via the term (*ω* <sup>→</sup> ⋅∇)*v* <sup>→</sup> . Solutions of these two equations, Eq. (7) and Eq. (32), describing dynamics of the vortex and scalar fields, depend on each other.

Both the continuity equation

$$\frac{\partial \rho}{\partial t} + (\vec{v} \cdot \nabla)\rho = 0,\tag{33}$$

which stems from Eq. (3), and the quantum Hamilton-Jacobi equation (32) can be extracted from the following Schrödinger equation

$$\mathrm{i}\hbar\frac{\partial\Psi}{\partial t} = \frac{1}{2m}(-\mathrm{i}\hbar\nabla + m\vec{\upsilon}\_R)^2\Psi^\nu + U(\vec{r})\Psi^\nu - C\Psi^\nu. \tag{34}$$

The kinetic momentum operator (−iℏ∇ + *mv* → *<sup>R</sup>*) contains the term *mv* → *<sup>R</sup>* describing a contribution of the vortex motion. This term is analogous to the vector potential multiplied by the ratio of the charge to the light speed, which appears in quantum electrodynamics [29]. Appearance of this term in this equation is conditioned by the Helmholtz theorem.

By substituting into Eq. (34) the wave function *Ψ* represented in a polar form

$$\Psi' = \sqrt{\rho} \cdot \exp\left( \text{ iS/} \hbar \right) \tag{35}$$

and separating on real and imaginary parts we come to Eqs. (32) and (33). So, the Navier-Stokes equation (6) with the slightly expanded the pressure gradient term can be reduced to the Schrödinger equation if we take into consideration also the continuity equation.

( ) ( ) <sup>2</sup> <sup>1</sup> 2 2 ( ) () . 2 2

We assume that fluctuations of the viscosity about zero occur much more frequent, than characteristic time of displacements of the quasi-particles. For that reason, we omit the term *ν*(*t*)∇2*S* by supposing in the first approximation, that the medium is absolutely superfluid-

the expression under the brackets is constant. As a result, we come to the following modified

<sup>2</sup> 2 2 <sup>1</sup> 2 2 ( ) ( ) . 2 2 22 2

The modification is due to adding the quantum potential (26). In this equation, *C* is an integration constant. We see that the third term in this equation represents energy of the vortex. On the other hand, we can see that the vortex given by Eq. (7) is replenished by the kinetic

equations, Eq. (7) and Eq. (32), describing dynamics of the vortex and scalar fields, depend on

( ) 0,

¶ + ×Ñ =

<sup>1</sup> <sup>2</sup> i (i ) () . <sup>2</sup>

→

¶ = - Ñ+ + -

<sup>r</sup> <sup>r</sup> h h *m Ur C <sup>R</sup> t m*

r

r

which stems from Eq. (3), and the quantum Hamilton-Jacobi equation (32) can be extracted

Y

of the vortex motion. This term is analogous to the vector potential multiplied by the ratio of the charge to the light speed, which appears in quantum electrodynamics [29]. Appearance of

 YY

*<sup>R</sup>*) contains the term *mv*

→ *<sup>S</sup>* ⋅*v* → *<sup>R</sup>*)=(*v* → *<sup>S</sup>* ⋅*ω*

r

r

*t m <sup>m</sup> Rv* (32)

 r

 r

<sup>→</sup> ⋅∇)*v*

*v* (33)

*v* (34)

→

= × exp i { *S* h } (35)

*<sup>R</sup> R RR v v S+mv v* (31)

wn

→

<sup>→</sup> )=0. The left part vanishes if

<sup>→</sup> . Solutions of these two

*<sup>R</sup>* describing a contribution

*<sup>S</sup>* we find

n

*t m t 2*

358 Selected Topics in Applications of Quantum Mechanics

that the right part of this equation vanishes since (*v*

energy coming from the scalar field *S*, namely via the term (*ω*

r

*t*

¶

this term in this equation is conditioned by the Helmholtz theorem.

Y

By substituting into Eq. (34) the wave function *Ψ* represented in a polar form

 r

Hamilton-Jacobi equation

Both the continuity equation

from the following Schrödinger equation

The kinetic momentum operator (−iℏ∇ + *mv*

Y

¶

each other.

æ ö ¶ ¶ Ñ + Ñ + + + - Ñ =- -é ´ Ñ ù+ Ñ ç ÷ ë û è ø ¶ ¶ *<sup>m</sup>* rr r r *S S UQ t S m t m*

there are no energy sources and sinks. By multiplying this equation from the left by *v*

æ ö ¶ Ñ Ñæ ö + Ñ+ + - - = ç ÷ ç ÷ ¶ ç ÷ è ø è ø *<sup>m</sup>* <sup>r</sup> <sup>h</sup> *S S Ur <sup>C</sup>*

**Figure 6.** Probability density distribution from scattering the fullerene molecules on the grating containing 9 slits: de Broglie wavelength is 5 pm and the distance between slits is 250 nm.

There are confirmations that the Schrödinger equation is deduced from the Feynman path integral [30, 31]. Therefore, for searching solutions of the Schrödinger equation we may apply the path integral. The solution of the Schrödinger equation (34) with the potential that simulates a grating with *N* slits has the following view [32]

$$\left|\Psi^{\nu}(\boldsymbol{y},\boldsymbol{z})\right> = \frac{1}{N\sqrt{1 + \mathbf{i}\frac{\mathcal{L}\boldsymbol{y}}{2\pi\sigma b^{2}}}} \sum\_{n=0}^{N-1} \exp\left\{ -\frac{\left(z - \left(n - \frac{N-1}{2}\right)d\right)^{2}}{2b^{2}\left(1 + \mathbf{i}\frac{\mathcal{L}\boldsymbol{y}}{2\pi\sigma b^{2}}\right)} \right\}.\tag{36}$$

Here *λ* is the de Broglie wavelength, *d* is the distance between slits, *b* is the slit width, and **i**=√-1. In this calculation we have used *λ* =5 pm, *b* =5⋅10<sup>3</sup> *λ* , and *d* =5⋅10<sup>4</sup> *λ* =10*b* =250 nm . By choosing *N*=9 slits, for example, we find the interference pattern shown in Fig. 6 as the density distri‐ bution function [32]. This function is a scalar product of the wave function |*Ψ*(*y*, *z*) , namely:

$$p(\mathbf{y}, \mathbf{z}) = \left\langle \Psi(\mathbf{y}, \mathbf{z}) \middle| \Psi(\mathbf{y}, \mathbf{z}) \right\rangle. \tag{37}$$

A useful unit of length at observation of the interference patterns is the Talbot length:

$$
\mu\_{\rm T} = 2\frac{d^2}{\lambda}.\tag{38}
$$

This length bears name of Henry Fox Talbot who discovered in 1836 [33] a beautiful interfer‐ ence pattern, named further the Talbot carpet [34, 35].

The particles, incident on the slit grating, come from a distant coherent source. The de Broglie wavelength of the particle, *λ* =*h* / *p* (*h* is the Planck constant, and *p* is the particle momentum) is a main characteristics binding the corpuscular Newtonian physics with the wave Huygens' physics. It is that we call now the wave-particle dualism. The de Broglie pilot wave being represented by the complex-valued wave function |*Ψ* fills all ambient space, except of opaque objects, which determine the boundary conditions. The particle passes from the source to a place of detection along the optimal trajectory, Bohmian trajectory [27]. The equation describing motion of the Bohmian particle can be found, for example, in [36]. There is the unique trajectory for each the particle, the vortex ball in our case. However, an attempt to measure exact position of the ball along the trajectory together with its velocity fails. Namely, there is no way to measure simultaneously the complemen‐ tary parameters, such as coordinate and velocity, what follows from the uncertainly principle [37].

Fig. 7 shows in lilac color Bohmian trajectories divergent from the slit grating. The probability density distribution is shown here in grey color ranging from white for *p=*0 to light grey for max *p*. Bundle of the Bohmian trajectories imitates a fluid flow through the obstacle, containing slits, relatively well. One can see that characteristic streamlets are formed in the flow, along which particles move. Such a vision of hydrodynamical behaviors of quantum systems is typical for many scientists since the formation of the quantum mechanics up to our days [38-42]. Principal moment is that the Schrödinger equation describes the expiration of the superfluid medium, which depend on the boundary conditions and other devices perturbing it (as, for example, the slit gratings, collimators and others). The vortex balls move along optimal directions of the flows – along the Bohmian paths.

**Figure 7.** Interference pattern of the coherent flow of the fullerene molecules with the de Broglie wavelength *λ* =5 pm within a zone *y* ≤6*yT* from the grating containing 9 slits. Lilac curves against the grey background represent the Boh‐ mian trajectories.

#### **4. Physical vacuum as a superfluid medium**

Here *λ* is the de Broglie wavelength, *d* is the distance between slits, *b* is the slit width, and **i**=√-1.

*N*=9 slits, for example, we find the interference pattern shown in Fig. 6 as the density distri‐ bution function [32]. This function is a scalar product of the wave function |*Ψ*(*y*, *z*) , namely:

> Y

2 2 . l= *d*

This length bears name of Henry Fox Talbot who discovered in 1836 [33] a beautiful interfer‐

The particles, incident on the slit grating, come from a distant coherent source. The de Broglie wavelength of the particle, *λ* =*h* / *p* (*h* is the Planck constant, and *p* is the particle momentum) is a main characteristics binding the corpuscular Newtonian physics with the wave Huygens' physics. It is that we call now the wave-particle dualism. The de Broglie pilot wave being represented by the complex-valued wave function |*Ψ* fills all ambient space, except of opaque objects, which determine the boundary conditions. The particle passes from the source to a place of detection along the optimal trajectory, Bohmian trajectory [27]. The equation describing motion of the Bohmian particle can be found, for example, in [36]. There is the unique trajectory for each the particle, the vortex ball in our case. However, an attempt to measure exact position of the ball along the trajectory together with its velocity fails. Namely, there is no way to measure simultaneously the complemen‐ tary parameters, such as coordinate and velocity, what follows from the uncertainly

Fig. 7 shows in lilac color Bohmian trajectories divergent from the slit grating. The probability density distribution is shown here in grey color ranging from white for *p=*0 to light grey for max *p*. Bundle of the Bohmian trajectories imitates a fluid flow through the obstacle, containing slits, relatively well. One can see that characteristic streamlets are formed in the flow, along which particles move. Such a vision of hydrodynamical behaviors of quantum systems is typical for many scientists since the formation of the quantum mechanics up to our days [38-42]. Principal moment is that the Schrödinger equation describes the expiration of the superfluid medium, which depend on the boundary conditions and other devices perturbing it (as, for example, the slit gratings, collimators and others). The vortex balls move along

*λ* , and *d* =5⋅10<sup>4</sup>

( *yz yz* ,) (,) . (37)

*y* (38)

*λ* =10*b* =250 nm . By choosing

In this calculation we have used *λ* =5 pm, *b* =5⋅10<sup>3</sup>

360 Selected Topics in Applications of Quantum Mechanics

ence pattern, named further the Talbot carpet [34, 35].

optimal directions of the flows – along the Bohmian paths.

principle [37].

*pyz* (,) =

Y

T

A useful unit of length at observation of the interference patterns is the Talbot length:

The Schrödinger equation (34) describes a flow of the peculiar fluid that is the physical vacuum. The vacuum contains pairs of particle-antiparticles. The pair, in itself, is the Bose particle that stays at a temperature close to zero. In aggregate, the pairs make up Bose-Einstein condensate. It means that the vacuum represents a superfluid medium [43]. A 'fluidic' nature of the space itself is exhibited through this medium. Another name of such an 'ideal fluid' is the ether [29].

The physical vacuum is a strongly correlated system with dominating collective effects [44] and the viscosity equal to zero. Nearest analogue of such a medium is the superfluid helium [22], which will serve us as an example for further consideration of this medium. The vacuum is defined as a state with the lowest possible energy. We shall consider a simple vacuum consisting of electron-positron pairs. The pairs fluctuate within the first Bohr orbit having energy about 13.6⋅<sup>2</sup> eV≈<sup>27</sup> eV. Bohr radius of this orbit is *<sup>r</sup>* <sup>1</sup> <sup>≈</sup>5.29⋅10<sup>−</sup>11 m. These fluctuations occur about the center of their masses. The total mass of the pair, *m*p, is equal to doubled mass of the electron, *m*. The charge of the pair is zero. The total spin of the pair is equal to 0. The angular momentum, *L*, is nonzero, however. For the first Bohr orbit *L* =ℏ. The velocity of rotating about this orbit is *<sup>L</sup>* / (*r*1*m*)≈2.192⋅<sup>10</sup> <sup>6</sup> m/s. It means that there exist an elementary vortex. Ensemble of such vortices forms a vortex line.

**Figure 8.** The dispersion relation *ε* vs. *p*. The dotted curve shows the non-relativistic square dispersion relation *ε* ∼ *p* <sup>2</sup> . The hump on the curve is a contribution of the roton component *pR <sup>f</sup>* (*<sup>p</sup>* <sup>−</sup> *pR*), *pR* / ℏ≈1.89⋅<sup>10</sup> 10 m-1, and *σ* =0.5*pR*.

We may evaluate the dispersion relation between the energy, *ε*(*p*)=ℏ*ω*, and wave number, *p* =ℏ*k*, as it done in [45]. As follows from the Schrödinger equation (34) we have:

$$
\omega(p) = \frac{1}{2m\_p} \left( p + p\_R f(p - p\_R) \right)^2. \tag{39}
$$

Here *pR* = *L* /*r* <sup>1</sup> =*mpvR* is the momentum of the rotation. The function *f*(*p*-*p* R) is a form-factor relating to the electron-positron pairs rotating about the center of their mass *mp*. The formfactor describes dispersion of the momentum *p* around *p*<sup>R</sup> conditioned by fluctuations about the ground state with the lowest energy. The form-factor is similar to the Gaussian curve

$$\int f(p - p\_R) = \exp\left\{-\frac{\left(p - p\_R\right)^2}{2\sigma^2}\right\}.\tag{40}$$

Here *σ* is the variance of this form-factor. It is smaller or close to *p* R. The dispersion relation (39) is shown in Fig. 8. The hump on the curve is due to the contribution of the rotating electronpositron pair about the center of their masses. These rotating objects are named rotons [45].

Rotons are ubiquitous in vacuum because of a huge availability of pairs of particle-antiparticle. The movement of the roton in the free space is described by the Schrödinger equation

$$\|h\frac{\partial \Psi^{\nu}}{\partial t} = \frac{1}{2m\_p}(-\mathrm{i}\hbar \nabla + m\_p \vec{v}\_R)^2 \Psi^{\nu} - C\Psi. \tag{41}$$

The constant *C* determines an uncertain phase shift of the wave function, and most possible this phase relates to the chemical potential of a boson (the electron-positron pair) [45]. We shall not take into account contribution of this term in the dispersion diagram because of its smallness. As follows from the above consideration of the Navier-Stokes equation, Eq. (41) can be reduced to the Euler equation

$$\frac{\partial \vec{\boldsymbol{\upsilon}}\_{R}}{\partial t} + [\vec{\boldsymbol{\alpha}} \times \vec{\boldsymbol{\upsilon}}\_{R}] = -\frac{\nabla P}{m\_{p}\rho},\tag{42}$$

that describes a flow of the inviscid incompressible fluid under the pressure field *P*. One can see from here that the Coriolis force appears as a restoring force, forcing the displaced fluid particles to move in circles. The Coriolis force is the generating force of waves called inertial waves [45]. The Euler equation admits a stationary solution for uniform swirling flow under the pressure gradient along *z*.

**Figure 8.** The dispersion relation *ε* vs. *p*. The dotted curve shows the non-relativistic square dispersion relation

We may evaluate the dispersion relation between the energy, *ε*(*p*)=ℏ*ω*, and wave number,

Here *pR* = *L* /*r* <sup>1</sup> =*mpvR* is the momentum of the rotation. The function *f*(*p*-*p* R) is a form-factor relating to the electron-positron pairs rotating about the center of their mass *mp*. The formfactor describes dispersion of the momentum *p* around *p*<sup>R</sup> conditioned by fluctuations about the ground state with the lowest energy. The form-factor is similar to the Gaussian curve

*p* =ℏ*k*, as it done in [45]. As follows from the Schrödinger equation (34) we have:

 = +- *R R p*

e

362 Selected Topics in Applications of Quantum Mechanics

( ) <sup>1</sup> <sup>2</sup> ( ) ( ). <sup>2</sup>

( ) ( ) exp . 2

ì ü ï ï - -= -í ý

Here *σ* is the variance of this form-factor. It is smaller or close to *p* R. The dispersion relation (39) is shown in Fig. 8. The hump on the curve is due to the contribution of the rotating electronpositron pair about the center of their masses. These rotating objects are named rotons [45].

Rotons are ubiquitous in vacuum because of a huge availability of pairs of particle-antiparticle. The movement of the roton in the free space is described by the Schrödinger equation

¶ = - Ñ+ -

<sup>1</sup> <sup>2</sup> i (i ) <sup>2</sup>

<sup>r</sup> h h *<sup>p</sup> <sup>R</sup> p*

Y

*t m*

¶

*<sup>R</sup> <sup>R</sup>*

. The hump on the curve is a contribution of the roton component *pR <sup>f</sup>* (*<sup>p</sup>* <sup>−</sup> *pR*), *pR* / ℏ≈1.89⋅<sup>10</sup> 10 m-1,

*p p pfp p <sup>m</sup>* (39)

*p p fp p* (40)

2 2

Y

*m C*

 Y.

*v* (41)

s

ï ï î þ

*ε* ∼ *p* <sup>2</sup>

and *σ* =0.5*pR*.

**Figure 9.** The formation of twisted vortex state [46]. The vortices have their propagating ends bent to the side wall of the rotating cylinder. As they expand upwards into the vortex-free state, the ends of the vortex lines rotate around the cylinder axis. The twist is nonuniform because boundary conditions allow it to unwind at the bottom solid wall. The

figure gives a snapshot (at time *t* =25*Ω* <sup>−</sup><sup>1</sup> , where *Ω* is the angular velocity.) of a numerical simulation of 23 vortices initially generated near the bottom end (*t*=0). Courtesy kindly by Erkki Thuneberg.

Formation of the swirling flow, the twisted vortex state, has been studied in the superfluid 3 He-B [46]. These observations give us a possibility to suppose the existence of such phenomena in the physical vacuum. The twisted vortex states observed in the superfluid 3 He-B are closely related to the inertial waves in rotating classical fluids. The superfluid initially is at rest [46]. The vortices are nucleated at a bottom disk platform rotating with the angular velocity *Ω* about axis *z*. As the platform rotates they propagate upward by creating the twisted vortex state spontaneously, Fig. 9. The Coriolis forces take part in this twisting. The twisted vortices grow upward along the cylinder axis [47].

Analogous experiment with nucleating vortices can be realized when the lower disk *A* rotates in the vacuum, Fig. 10. In this case, the vortices are viewed as the dancing electron-positron pairs on the first Bohr orbit. As the vortices grow upward the spontaneous twisted vortex states arise. The latter by reaching upper fixed disk *B* can capture it into rotation.

**Figure 10.** Rotation of the superfluid fluid is not uniform but takes place via a lattice of quantized vortices, whose cores (colored in yellow) are parallel to the axis of rotation [46, 47]. Green arrows are the vorticity ω. Small black arrows indicate the circulation of the velocity *v*R around the cores. The vacuum is supported between two non-ferromagnetic disks, *A* and *B*, fixed on center shafts, **1** and **2**, of electric motors [48], see Fig. 11. Radiuses of the both disks are *R*=82.5 mm and distance between them can vary from 1 to 3 mm and more. The vortex bundle rotates rigidly with the disk *A*. As soon as the vortex bundle reaches the top disk *B* it begins rotation as well.

Pr. V. Samohvalov has shown through the experiment [48], that the vortex bundle induced by rotating the bottom non-ferromagnetic disk *A* leads to rotation of the upper fixed initially nonferromagnetic disk *B*, Fig. 11. Both disks at room temperature have been placed in the container with technical vacuum at 0.02 Torr, The utmost number of the vortices that may be placed on the square of the disk *A* is *N*max =(2*π<sup>R</sup>* <sup>2</sup> ) / (2*πr*<sup>1</sup> 2 )=2∙1018, where *R*=82.5 mm is the radius of the disk and *<sup>r</sup>* <sup>1</sup> <sup>≈</sup>5.29⋅10<sup>−</sup>11 m is the radius of the first Bohr orbit. Really, the number of the vortices situated on the square, *N*, is considerably smaller. It can be evaluated by multiplying *N*max by a factor *δ*. This factor is equal to the ratio of the geometric mean of the velocities *vR* <sup>=</sup><sup>ℏ</sup> / (*r*<sup>1</sup> *<sup>m</sup>*)≈2.192⋅<sup>10</sup> <sup>6</sup> m/s and *VD* =*R Ω* to their arithmetic mean. Here *Ω* is an angular rate of the disk A. So, we have

Formation of the swirling flow, the twisted vortex state, has been studied in the superfluid

He-B [46]. These observations give us a possibility to suppose the existence of such phenomena

related to the inertial waves in rotating classical fluids. The superfluid initially is at rest [46]. The vortices are nucleated at a bottom disk platform rotating with the angular velocity *Ω* about axis *z*. As the platform rotates they propagate upward by creating the twisted vortex state spontaneously, Fig. 9. The Coriolis forces take part in this twisting. The twisted vortices grow

Analogous experiment with nucleating vortices can be realized when the lower disk *A* rotates in the vacuum, Fig. 10. In this case, the vortices are viewed as the dancing electron-positron pairs on the first Bohr orbit. As the vortices grow upward the spontaneous twisted vortex states

**Figure 10.** Rotation of the superfluid fluid is not uniform but takes place via a lattice of quantized vortices, whose cores (colored in yellow) are parallel to the axis of rotation [46, 47]. Green arrows are the vorticity ω. Small black arrows indicate the circulation of the velocity *v*R around the cores. The vacuum is supported between two non-ferromagnetic disks, *A* and *B*, fixed on center shafts, **1** and **2**, of electric motors [48], see Fig. 11. Radiuses of the both disks are *R*=82.5 mm and distance between them can vary from 1 to 3 mm and more. The vortex bundle rotates rigidly with the disk *A*.

Pr. V. Samohvalov has shown through the experiment [48], that the vortex bundle induced by rotating the bottom non-ferromagnetic disk *A* leads to rotation of the upper fixed initially non-

As soon as the vortex bundle reaches the top disk *B* it begins rotation as well.

He-B are closely

in the physical vacuum. The twisted vortex states observed in the superfluid 3

arise. The latter by reaching upper fixed disk *B* can capture it into rotation.

3

upward along the cylinder axis [47].

364 Selected Topics in Applications of Quantum Mechanics

$$N = N\_{\text{max}} \frac{\sqrt{\upsilon\_R \cdot V\_D}}{\upsilon\_R + V\_D} = N\_{\text{max}} \sqrt{\frac{V\_D}{\upsilon\_R}} \approx 6 \cdot 10^{15} \tag{43}$$

at the angular rate *Ω*=160 1/s [48] the disk velocity *V*D=13.2 m/s. Now we can evaluate the kinetic energy of the vortex bundle induced by the rotating disk *A*. This kinetic energy is *E* = *N* ⋅*mpvR* <sup>2</sup> / 2≈ 0.026 J. This energy is sufficient for transfer of the moment of force to the disk *B*. Measured in the experiment [48] the torque is about 0.01 N∙m. So, the disk *B* can be captured by the twisted vortex.

**Figure 11.** Basic diagram (a) and general view of the device (b) for researching mass dynamics effects [48]: 1 and 2 are shafts with mounted on them electric motors; 3 and 4 are steel plates with mounted on them electromagnetic brakes; 5 and 6 (the disks A and B, see Fig. 10) are disks rigidly fixed on flanges of the rotors of the electric motors. Courtesy kindly by Vladimir Samokhvalov.

The formation of the growing twisted vortices can be confirmed with attraction of mod‐ ern methods of interference of light rays passing through the gap between the disks. Light traveling along two paths through the space between the disks undergoes a phase shift manifested in the interference pattern [29] as it was shown in the famous experiment of Aharonov and Bohm [49].

#### **5. Conclusion**

The Schrödinger equation is deduced from two equations, the continuity equation and the Navier-Stokes equation. At that, the latter contains slightly modified the gradient pressure term, namely, ∇*P* →*ρ<sup>M</sup>* ∇(*P* / *ρ<sup>M</sup>* )=∇*P* + *P* ∇ln(*ρ<sup>M</sup>* ). The extra term *P* ∇ln(*ρ<sup>M</sup>* ) describes change of the pressure induced by change of the entropy ln(*ρ<sup>M</sup>* ) per length. In this case, the modified gradient pressure term can be reduced to the quantum potential through using the Fick's law. In the law we replace also the diffusion coefficient by the factor ℏ / 2*m*, where ℏ is the reduced Planck constant and *m* is mass of the particle.

We have shown that a vortex arising in a fluid can exist infinitely long if the viscosity undergoes periodic oscillations between positive and negative values. At that, the viscosity, in average on time, stays equal to zero. It can mean that the fluid is superfluid. In our case, the superfluid consists of pairs of particle-antiparticle representing the Bose-Einstein condensate.

As for the quantum reality, such a periodic regime can be interpreted as exchange of the energy quanta of the vortex with the vacuum through the zero-point vacuum fluctuations. In reality, these fluctuations are random, covering a wide range of frequencies from zero to infinity. Based on this observation we have assumed that the fluctuations of the vacuum ground state can support long-lived existence of vortex quantum objects. The core of such a vortex has nonzero radius inside of which the velocity tends to zero. In the center of the vortex, the velocity vanishes. The velocity reaches maximal values on boundary of the core, and then it decreases to zero as the distance to the vortex goes to infinity.

The experimental observations of the Couder's team [5, 11, 12] can have far-reaching ontolog‐ ical perspectives in regard of studying our universe. Really, we can imagine that our world is represented by myriad of baryonic and lepton "droplets" bouncing on a super-surface of some unknown dark matter. A layer that divides these "droplets", i.e., particles, and the dark matter is the superfluid vacuum medium. This medium, called also the ether [24], is populated by the particles of matter ("droplets"), which exist in it and move through it [29, 50, 51]. The particle traveling through this medium perturbs virtual particle-antiparticle pairs, which, in turn, create both constructive and destructive interference at the forefront of the particle [30]. Thus, the virtual pairs interfering each other provide an optimal, Bohmian, path for the particle.

Assume next, that the baryonic matter is similar, say, on "hydrophobic" fluid, whereas the dark matter, say, is similar to "hydrophilic" fluid. Then the baryonic matter will diverge each from other on cosmological scale owing to repulsive properties of the dark matter, like soap spots diverge on the water surface. Observe that this phenomenon exhibits itself through existence of the short-range repulsive gravitational force that maintains the incompatibility between the dark matter and the baryonic matter [52, 53]. At that, the dark matter stays invisible. One can imagine that the zero-point vacuum fluctuations are nothing as weak ripples on a surface of the dark matter.

#### **Appendix A: Nelson's derivation of the Schrödinger equation**

**5. Conclusion**

366 Selected Topics in Applications of Quantum Mechanics

The Schrödinger equation is deduced from two equations, the continuity equation and the Navier-Stokes equation. At that, the latter contains slightly modified the gradient pressure term, namely, ∇*P* →*ρ<sup>M</sup>* ∇(*P* / *ρ<sup>M</sup>* )=∇*P* + *P* ∇ln(*ρ<sup>M</sup>* ). The extra term *P* ∇ln(*ρ<sup>M</sup>* ) describes change of the pressure induced by change of the entropy ln(*ρ<sup>M</sup>* ) per length. In this case, the modified gradient pressure term can be reduced to the quantum potential through using the Fick's law. In the law we replace also the diffusion coefficient by the factor ℏ / 2*m*, where ℏ is

We have shown that a vortex arising in a fluid can exist infinitely long if the viscosity undergoes periodic oscillations between positive and negative values. At that, the viscosity, in average on time, stays equal to zero. It can mean that the fluid is superfluid. In our case, the superfluid

As for the quantum reality, such a periodic regime can be interpreted as exchange of the energy quanta of the vortex with the vacuum through the zero-point vacuum fluctuations. In reality, these fluctuations are random, covering a wide range of frequencies from zero to infinity. Based on this observation we have assumed that the fluctuations of the vacuum ground state can support long-lived existence of vortex quantum objects. The core of such a vortex has nonzero radius inside of which the velocity tends to zero. In the center of the vortex, the velocity vanishes. The velocity reaches maximal values on boundary of the core, and then it decreases

The experimental observations of the Couder's team [5, 11, 12] can have far-reaching ontolog‐ ical perspectives in regard of studying our universe. Really, we can imagine that our world is represented by myriad of baryonic and lepton "droplets" bouncing on a super-surface of some unknown dark matter. A layer that divides these "droplets", i.e., particles, and the dark matter is the superfluid vacuum medium. This medium, called also the ether [24], is populated by the particles of matter ("droplets"), which exist in it and move through it [29, 50, 51]. The particle traveling through this medium perturbs virtual particle-antiparticle pairs, which, in turn, create both constructive and destructive interference at the forefront of the particle [30]. Thus, the virtual pairs interfering each other provide an optimal, Bohmian, path for the particle.

Assume next, that the baryonic matter is similar, say, on "hydrophobic" fluid, whereas the dark matter, say, is similar to "hydrophilic" fluid. Then the baryonic matter will diverge each from other on cosmological scale owing to repulsive properties of the dark matter, like soap spots diverge on the water surface. Observe that this phenomenon exhibits itself through existence of the short-range repulsive gravitational force that maintains the incompatibility between the dark matter and the baryonic matter [52, 53]. At that, the dark matter stays invisible. One can imagine that the zero-point vacuum fluctuations are nothing as weak ripples

consists of pairs of particle-antiparticle representing the Bose-Einstein condensate.

the reduced Planck constant and *m* is mass of the particle.

to zero as the distance to the vortex goes to infinity.

on a surface of the dark matter.

Nelson proclaim that the medium through which a particle moves contains myriad subparticles that accomplish Brownian motions by colliding with each other chaotically. The Brownian motions is described by the Wiener process with the diffusion coefficient

$$
\overline{\nu} = \frac{\hbar}{2m}.\tag{44}
$$

Here *m* is mass of the particle and ℏ=*h* / 2*π* is the reduced Planck constant. Here we use *ν* with the upper bar in order to avoid confusion with the kinematic viscosity adopted in hydrody‐ namics. As seen this motion has a quantum nature [24] in contrast to the macroscopic Brownian motions where the diffusion coefficient has a view *v*¯ =*kT* / *mβ* <sup>−</sup><sup>1</sup> ; here *k* is Boltzmann constant, *T* is a temperature, and *β* <sup>−</sup><sup>1</sup> is the relaxation time.

Two equations are main in the article [24]. The position **x** (*t)* of the Brownian particle, being subjected either by external forces or by currents in the medium, can be written by two equivalent equations:

$$d\mathbf{x}(t) = \mathbf{b}(\mathbf{x}(t), t)dt \,\, +d\mathbf{w}(t),\tag{45}$$

$$d\mathbf{x}(t) = \mathbf{b}\_{\bullet}(\mathbf{x}(t), t)dt + d\mathbf{w}\_{\bullet}(t) \tag{46}$$

Here *w*(*t*) and *w*\* (*t*) are the Wiener processes, both have equivalent properties. Variables *b* and *b*\* are vector-valued forward and backward functions on space-time, respectively. In fact, they are the mean forward and mean backward measured quantities

$$\mathbf{b}(\mathbf{x}(t),t) = \lim\_{\Delta t \to 0\_{+}} E\_{t} \frac{\mathbf{x}(t+\Delta t) - \mathbf{x}(t)}{\Delta t},\tag{47}$$

$$\mathbf{b}\_{\bullet}(\mathbf{x}(t),t) = \lim\_{\Delta t \to 0\_{+}} E\_{t} \frac{\mathbf{x}(t) - \mathbf{x}(t-\Delta t)}{\Delta t}. \tag{48}$$

Here *Et* denotes the conditional expectation (average) given the state of the system at time *t*, and 0+means that *Δt* tends to 0 through positive values. Thus *b*(*x*(*t*), *t*) and *b*\* (*x*(*t*), *t*) are again stochastic variables [54, 55]. It is instructive to compare calculus (46) and (47) with classical calculations of infinitesimal small increments

$$w(t) = \lim\_{\Delta t \to 0\_+} \frac{\mathbf{x}(t + \Delta t) - \mathbf{x}(t)}{\Delta t} = \lim\_{\Delta t \to 0\_+} \frac{\mathbf{x}(t) - \mathbf{x}(t - \Delta t)}{\Delta t}. \tag{49}$$

One can see that these calculations are symmetrical with respect to the time arrow, whereas the calculations (46) and (47) are not, in general (see below).

It should be noted that *b*(*x*(*t*), *t*) and *b*\* (*x*(*t*), *t*) are not real velocities. The real current velocity of the particle is calculated as

$$
\vec{\psi}(t) = \frac{1}{2} \left( \mathbf{b}(\mathbf{x}(t), t) + \mathbf{b}\_\*(\mathbf{x}(t), t) \right). \tag{50}
$$

There is a one more velocity, which is represented via difference of *b*(*x*(*t*), *t*) and *b*\* (*x*(*t*), *t*) :

$$
\vec{u}(t) = \frac{1}{2} \left( \mathbf{b}(\mathbf{x}(t), t) - \mathbf{b}\_\bullet(\mathbf{x}(t), t) \right). \tag{51}
$$

According to Einstein's theory of Brownian motion, *u* <sup>→</sup> (*t*) is the velocity acquired by a Brownian particle, in equilibrium with respect to an external force, to balance the osmotic force [19]. For this reason, this velocity is named the osmotic velocity. It can be expressed in the following form

$$
\vec{u}(t) = \overline{\nu} \,\nabla \left( \ln(\rho(t)) \right) = \frac{\hbar}{m} \frac{\nabla R(t)}{R(t)},\tag{52}
$$

where *ρ*(*t*) is the probability density of **x**(*t*) and *R*(*t*)=*ρ*(*t*) 1/2 is the probability density amplitude. The current velocity, in turn, is expressed through gradient of a scalar field *S* called the action

$$
\vec{v}(t) = -\frac{\hbar}{m}\nabla S(t). \tag{53}
$$

The both equations, (44) and (45), introduced above are important for derivation of the Schrödinger equation. The derivation of the equation is provided by the use of the wave function presented in the polar form

$$\Psi' = R \exp\left(i S/\hbar\right),\tag{54}$$

by replacing the velocities *v* <sup>→</sup> (*t*) and *u* <sup>→</sup> (*t*) in the initial equations. It should be noted that Nelson departs from two equations describing directed the forward and backward Brownian motions which are written down for real-valued functions. In order to come to the Schrödinger equation he has used a complex-valued wave function exp{*R+***i***S*} instead of the generally accepted *R*exp{i*S* / ℏ}. Obviously, this discrepancy are eliminated by replacing exp{*R*}→*R* .

Observe that the wave function represented in the polar form (53) is used for getting equations underlying the Bohmian mechanics [27]. These two equations are the continuity equation and the Hamilton-Jacobi equation containing an extra term known as the Bohmian quantum potential. The quantum potential has the following view:

$$\mathcal{Q} = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R} = -\frac{\hbar}{2} \frac{\left(\nabla R \vec{u}\right)}{R} = -\frac{m}{2} u^2 - \frac{\hbar}{2} (\nabla \vec{u}).\tag{55}$$

One can see that the quantum potential depends only on the osmotic velocity, which is expressed through difference of the forward and backward averaged quantities (46) and (47). These forward and backward quantities can be interpreted as uncompensated flows through a "semipermeable membrane" which represents an instant dividing the past and the future. Following to Licata and Fiscaletti, who have shown that the quantum potential has relation to the Bell length indicating a non-local correlation [28], one can add that the non-local correlation exists also between the past and the future. E. Nelson as one can see has considered a particle motion through the ether populated by sub-particles experiencing accidental collisions with each other. The Brownian motions of the sub-particles submits to the Wiener process with the diffusion coefficient ν proportional to the Plank constant as shown in Eq. (43). The ether behaves itself as a free-friction fluid.

#### **Acknowledgements**

0 0 ( ) () () ( ) ( ) = lim lim . D ® + + D ® +D - - -D <sup>=</sup> *t t* D D *xt t xt xt xt t*

= + <sup>r</sup>

There is a one more velocity, which is represented via difference of *b*(*x*(*t*), *t*) and *b*\*

*t t*

One can see that these calculations are symmetrical with respect to the time arrow, whereas

( ) \* <sup>1</sup> ( ) ( ( ), ) ( ( ), ) . <sup>2</sup>

( ) \* <sup>1</sup> ( ) ( ( ), ) ( ( ), ) . <sup>2</sup> <sup>=</sup> - <sup>r</sup>

particle, in equilibrium with respect to an external force, to balance the osmotic force [19]. For this reason, this velocity is named the osmotic velocity. It can be expressed in the following

( ) ( ) ( ) ln( ( )) , ( )

The current velocity, in turn, is expressed through gradient of a scalar field *S* called the action

The both equations, (44) and (45), introduced above are important for derivation of the Schrödinger equation. The derivation of the equation is provided by the use of the wave

departs from two equations describing directed the forward and backward Brownian motions

 r

<sup>Ñ</sup> =Ñ = <sup>r</sup> <sup>h</sup> *R t*

n

Y

<sup>→</sup> (*t*) and *u*

*u t t*

*v* (49)

(*x*(*t*), *t*) are not real velocities. The real current velocity

(*x*(*t*), *t*) :

*v t tt tt* **bx b x** (50)

*ut t t t t* **bx b x** (51)

1/2

( ) = ( ). - Ñ r h *t St <sup>m</sup> <sup>v</sup>* (53)

= *R S* exp i , { h } (54)

<sup>→</sup> (*t*) in the initial equations. It should be noted that Nelson

<sup>→</sup> (*t*) is the velocity acquired by a Brownian

*m Rt* (52)

is the probability density amplitude.

*t*

368 Selected Topics in Applications of Quantum Mechanics

It should be noted that *b*(*x*(*t*), *t*) and *b*\*

of the particle is calculated as

form

the calculations (46) and (47) are not, in general (see below).

According to Einstein's theory of Brownian motion, *u*

where *ρ*(*t*) is the probability density of **x**(*t*) and *R*(*t*)=*ρ*(*t*)

function presented in the polar form

by replacing the velocities *v*

The author thanks Mike Cavedon for useful and valuable remarks and offers. The author thanks also Miss Pipa (quantum portal administrator) for preparing a program drawing Fig. 7.

#### **Author details**

V.I. Sbitnev1,2

1 St. Petersburg B.P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Leningrad district, Russia

2 Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

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### **Husimi Distribution and the Fisher Information Husimi Distribution and the Fisher Information**

Sergio Curilef and Flavia Pennini Sergio Curilef1 and Flavia Pennini1, 2

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59126

#### 1. Introduction

10.5772/59126

In this chapter we review some aspects of the concept of the Fisher information measure in phase space for two specific systems: the Landau Diamagnetism and the Rigid rotator. The indispensable tool in this proposal is a quasi probability called Husimi distribution [1], which is frequently employed to characterize the quantum and classical behavior [2] of systems. Also, it possesses interesting applications in several areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory and Nanotechnology [3–10]. Its main properties are: 1) it is definite positive in all phase space, 2) it possesses no correct marginal properties, 3) it permits to calculate the expectation values of observables in quantum mechanics similarly to the classical case [11], and 4) it is a special type of probability that simultaneously approximate location of position and momentum in phase space. It is important to note that this quasi probability is constructed by definition as the expectation value of the density operator in a basis of coherent states [12]. Details about the formulation of coherent states and the obtaining of Husimi distribution from these can be found on our chapter that it can be read in Ref. [13]. The main propose of this chapter is to present to the reader interesting problems in physics, such as, the harmonics oscillator [5], the Landau diamagnetism model [8, 14] and, the rigid rotator [7, 15], analyzed from a point of view of the information measures. In particular, we will put emphasis in the Fisher Information measure and its construction starting from a well-defined set of coherent states.

In our previous contribution published in Ref. [13] we research about a special semi-classical measure, the Wehrl entropy, as an important application of the Husimi distribution. In the present study we analyze some consequences of obtaining the Husimi distribution; for instance, the Fisher information for fundamental problems in physics for which the coherent states formulation is well defined.

In physics, great attention has been paid to the Landau diamagnetism which consists in a particle charged in a uniform magnetic field. For our purpose we will use a complete description of the Husimi Distributionin three dimensions in order to study such system, so as it was shown in our previous contributions (see Ref. [13], where we have discussed some limiting cases as high and low temperatures. From the present analysis, when three dimensions are considered, naturally arises a lower temperature

©2012 Author(s), licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

bound, whereby it is not possible to work in all finite temperatures. Such discussion is explained with details in Ref. [13].

The other system, that we take into account here, is the linear rigid rotator and its corresponding 3*D* anisotropic version. We analyze phase space delocalization and obtain the concomitant semiclassical Fisher information measure constructed by using Husimi Distributionconstructed from suitable basis of coherent states.

In order to facilitate the understanding of this chapter to the reader, we give the following organization: in section 2 we begin introducing the concepts and methodology that will employ in the rest of the chapters. In section 3 we focus our attention on the Husimi distribution and the Fisher measure for the Landau diamagnetism. In section 4 we study the delocalization into phase space, within a semiclassical context by recourse to the Husimi distribution, for both cases of rigid rotators: linear and 3*D*−anisotropic instances. Finally, some conclusions and open problems are commented in section 5.

#### 2. Previous concepts

This section provides reference material that we consider relevant to conveniently understand the development of this chapter. These are *i)* the Husimi distribution, *ii)* Wehrl entropy, and *iii)* Fisher information measure. In all cases, we refer to the model of the harmonic oscillator in a thermal state.

#### 2.1. Husimi distribution and Wehrl entropy

From the standard statistical mechanics, the thermal density matrix can be represented by

$$
\mathfrak{d} = Z^{-1} e^{-\mathfrak{H}},
\tag{1}
$$

where β = 1/*kBT* the inverse temperature *T*, and *kB* the Boltzmann constant [16], *H*ˆ is the Hamiltonian of the system and *Z* = Tr(*e*−β*H*<sup>ˆ</sup> ) is the partition function.

In the current strategy, the expectation value of the density operator in a basis of coherent states is related to the Husimi distribution as [1]

$$
\mu(z) = \langle z|\hat{\mathfrak{p}}|z\rangle,\tag{2}
$$

where the set {|*z*} denotes the eigenstates of the annihilation operator *a*ˆ, i.e., *a*ˆ|*z* = *z*|*z* defined for all *z* ∈ **C** [12] and they are the coherent states for the system. Therefore, the normalization of the distribution is given by

$$
\int \frac{\mathbf{d}^2 \mathbf{z}}{\pi} \mu(\mathbf{z}) = 1,\tag{3}
$$

where the integration is over the complex plane *z* and the element of integration is an area proportional to phase space element given by d2*z* = d*x*d*p*/2¯*h*.

The set {*En*} stands for the spectrum of an arbitrary Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> , where *<sup>n</sup>* is a positive integer. With these elements the Husimi distribution takes the form

$$
\mu(z) = \frac{1}{Z} \sum\_{n} e^{-\beta E\_n} |\langle z|n\rangle|^2,\tag{4}
$$

where the set {|*n*|} represents energy eigenstates with eigenvalues *En* [4, 5].

A direct application that is additionally a useful measure of localization in phase-space [17, 18] is the Wehrl entropy, which is suitably defined as

$$W = -\int \frac{d^2 z}{\pi} \mu(z) \ln \mu(z). \tag{5}$$

As a consequence of the uncertainty principle, Lieb [4] proved the inequality *W* ≥ 1 which was previously conjectured by Wehrl [17].

For the Hamiltonian *H*ˆ = *h*¯ω[*a*ˆ †*a*ˆ+1/2] of the harmonic oscillator, it is obtained a basis {|*n*} and the spectrum *En* = *h*¯ω(*n* + 1/2), with *n* = 0, 1,... from the complete orthonormal set of eigenstates and eigenvectors, respectively. The algebra allows us to define the following elementary properties:

1. A set of Glauber coherent states is given by [19]

$$|z\rangle = e^{-|z|^2/2} \sum\_{n=0}^{\infty} \frac{z^n}{\sqrt{n!}} |n\rangle. \tag{6}$$

2. The normalization,

2 ime knjige

details in Ref. [13].

coherent states.

2. Previous concepts

of the system and *Z* = Tr(*e*−β*H*<sup>ˆ</sup>

distribution is given by

related to the Husimi distribution as [1]

to phase space element given by d2*z* = d*x*d*p*/2¯*h*.

these elements the Husimi distribution takes the form

2.1. Husimi distribution and Wehrl entropy

bound, whereby it is not possible to work in all finite temperatures. Such discussion is explained with

The other system, that we take into account here, is the linear rigid rotator and its corresponding 3*D* anisotropic version. We analyze phase space delocalization and obtain the concomitant semiclassical Fisher information measure constructed by using Husimi Distributionconstructed from suitable basis of

In order to facilitate the understanding of this chapter to the reader, we give the following organization: in section 2 we begin introducing the concepts and methodology that will employ in the rest of the chapters. In section 3 we focus our attention on the Husimi distribution and the Fisher measure for the Landau diamagnetism. In section 4 we study the delocalization into phase space, within a semiclassical context by recourse to the Husimi distribution, for both cases of rigid rotators: linear and 3*D*−anisotropic instances. Finally, some conclusions and open problems are commented in section 5.

This section provides reference material that we consider relevant to conveniently understand the development of this chapter. These are *i)* the Husimi distribution, *ii)* Wehrl entropy, and *iii)* Fisher information measure. In all cases, we refer to the model of the harmonic oscillator in a thermal state.

ρˆ = *Z*−1*e*−β*H*<sup>ˆ</sup>

where β = 1/*kBT* the inverse temperature *T*, and *kB* the Boltzmann constant [16], *H*ˆ is the Hamiltonian

In the current strategy, the expectation value of the density operator in a basis of coherent states is

where the set {|*z*} denotes the eigenstates of the annihilation operator *a*ˆ, i.e., *a*ˆ|*z* = *z*|*z* defined for all *z* ∈ **C** [12] and they are the coherent states for the system. Therefore, the normalization of the

where the integration is over the complex plane *z* and the element of integration is an area proportional

The set {*En*} stands for the spectrum of an arbitrary Hamiltonian *<sup>H</sup>*<sup>ˆ</sup> , where *<sup>n</sup>* is a positive integer. With

d2*z*

, (1)

*µ*(*z*) = *z*|ρˆ|*z*, (2)

<sup>π</sup> *<sup>µ</sup>*(*z*) = 1, (3)

From the standard statistical mechanics, the thermal density matrix can be represented by

) is the partition function.

$$
\langle n|n'\rangle = \mathfrak{G}\_{n,n'} \tag{7}
$$

where δ*n*,*n* is the Kronecker delta function.

3. The completeness property is contained in the relation

$$\sum\_{n=0}^{\infty} |n'\rangle\langle n| = \hat{1},\tag{8}$$

where 1 represents the identity operator in the defined space of eigenvectors. ˆ

Now, a suitable application of the present theoretical characterization [4] comes from certain calculations of the harmonic oscillator as the Husimi distribution

$$
\mu\_{HO}(z) = \left(1 - e^{-\beta \hbar \alpha}\right) e^{-\left(1 - e^{-\beta \hbar \alpha}\right) |z|^2},\tag{9}
$$

and the Wehrl entropy

$$W\_{HO} = 1 - \ln(1 - e^{-\beta \hbar \alpha}),\tag{10}$$

which are respectively known and useful analytical expressions [4].

#### 2.2. Fisher information measure

A pertinent quantifier of information, which possess innumerable applications in several fields of Physics, is the Fisher information measure [20]. The last years have witnessed a great deal of activity revolving around physical applications of Fisher information measure [20, 21] providing tools to yield most of the canonical Lagrangians of theoretical physics [20, 21] related properly to the Boltzmann entropy [22, 23]. The Fisher information connected with translations of an observable *x* with the consistent probability density ρ(*x*) is given by [24]

$$\mathcal{F} = \int \mathrm{d}x \,\mathfrak{p}(x) \left[ \frac{\partial \ln \mathfrak{p}(x)}{\partial x} \right]^2,\tag{11}$$

and the Cramer–Rao inequality is given by [24]

$$
\Delta \mathbf{x} \ge \mathcal{F}^{-1} \tag{12}
$$

where ∆*x* is the variance for the stochastic variable *x* which is of the form [24]

$$
\Delta \mathbf{x}^2 = \langle \mathbf{x}^2 \rangle - \langle \mathbf{x} \rangle^2 = \int \mathbf{dx} \mathbf{\rho}(\mathbf{x}) \mathbf{x}^2 - \left( \int \mathbf{dx} \mathbf{\rho}(\mathbf{x}) \mathbf{x} \right)^2. \tag{13}
$$

In particular, it is interesting to study its representation appealing to the semiclassical approach (see, for example, Ref. [25] and references therein), whose main tool is a distribution function in phase space in the basis of coherent states. Specially, in this proposal, we pay attention to a particular distribution, the well-known *Q*−function or Husimi distribution.

An original, compact expression in phase space is advanced for the "semiclassical" Fisher information measure, that can be easily derived from the Wehrl-methodology described in Refs. [5] and [6]. The appearance for this measures reads

$$\mathcal{F} = \frac{1}{4} \int \frac{\mathrm{d}^2 z}{\pi} \mu(z) \left\{ \frac{\partial \ln \mu(z)}{\partial |z|} \right\}^2,\tag{14}$$

which will be used in the following sections.

Inserting the *µ*−expression for the harmonic oscillator into Eq. (14) we find its anlytical form

$$
\mathcal{G}\_{HO} = 1 - e^{-\beta \hbar \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}},
\tag{15}
$$

leading to the following limits:

$$\begin{cases} \text{for } T \to 0 \text{ one has } \mathcal{F}\_{HO} = 1\\ \text{for } T \to \infty \text{ one has } \mathcal{F}\_{HO} = 0, \end{cases} \tag{16}$$

as it should be expected.

#### 3. Landau Diamagnetism: Charged particle in a uniform magnetic field

4 ime knjige

2.2. Fisher information measure

consistent probability density ρ(*x*) is given by [24]

and the Cramer–Rao inequality is given by [24]

well-known *Q*−function or Husimi distribution.

which will be used in the following sections.

appearance for this measures reads

leading to the following limits:

as it should be expected.

*F* = 

where ∆*x* is the variance for the stochastic variable *x* which is of the form [24]

*<sup>F</sup>* <sup>=</sup> <sup>1</sup> 4  d2*z* <sup>π</sup> *<sup>µ</sup>*(*z*)

Inserting the *µ*−expression for the harmonic oscillator into Eq. (14) we find its anlytical form

for *T* → 0 one has *FHO* = 1

<sup>2</sup> = 

<sup>d</sup>*x*ρ(*x*) *<sup>x</sup>*<sup>2</sup> <sup>−</sup>

In particular, it is interesting to study its representation appealing to the semiclassical approach (see, for example, Ref. [25] and references therein), whose main tool is a distribution function in phase space in the basis of coherent states. Specially, in this proposal, we pay attention to a particular distribution, the

An original, compact expression in phase space is advanced for the "semiclassical" Fisher information measure, that can be easily derived from the Wehrl-methodology described in Refs. [5] and [6]. The

> ∂ln*µ*(*z*) ∂|*z*|

2

*FHO* <sup>=</sup> <sup>1</sup>−*e*−β*h*¯ω, (15)

for *T* → ∞ one has *FHO* = 0, (16)

<sup>∆</sup>*x*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*2−*<sup>x</sup>*

A pertinent quantifier of information, which possess innumerable applications in several fields of Physics, is the Fisher information measure [20]. The last years have witnessed a great deal of activity revolving around physical applications of Fisher information measure [20, 21] providing tools to yield most of the canonical Lagrangians of theoretical physics [20, 21] related properly to the Boltzmann entropy [22, 23]. The Fisher information connected with translations of an observable *x* with the

> d*x*ρ(*x*) ∂lnρ(*x*) ∂*x*

2

, (11)

. (13)

, (14)

<sup>∆</sup>*<sup>x</sup>* <sup>≥</sup> *F* <sup>−</sup><sup>1</sup> (12)

2

d*x*ρ(*x*) *x*

Diamagnetism is a problem firstly appointed by Landau who showed the discreteness of energy levels for a charged particle in a magnetic field [26]. By the observation of the diverse scenarios in the framework provided by the Landau diamagnetism we can study some relevant physical properties [27– 29] as the role of the size of systems or the influence of boundaries, also the thermodynamic limit or quasi-stationary states. The primary motivation even today for several specialists is to find a useful measure to characterize theoretically every practical consequence of the system and its behavior.

In the past, Feldman and Kahn calculated the proper partition function for this system by appealing to the concept of Glauber coherent states from a set of basis states [30]. This formulation uses classical concepts as electron orbits, even though it contains all quantum effects [30]. This approach was previously used to obtain measures as the Wehrl entropy [17, 18] and Fisher information [31] with the purpose of studying the thermodynamics of the free spinless charged particle in a uniform magnetic field [32], this is the Landau diamagnetism problem. As observed, in such contribution the formulation is not completely consistent because it was necessary to normalize the Husimi distribution in order to arrive to reliable expressions for semiclassical measures [9, 32, 33].

Certainly, because the relevant effects seem to come only from the transverse motion, several efforts are made to describe this problem in two dimensions [9, 28, 29, 32–35]. Furthermore, the discovery of the quantum Hall effect has aroused much interest in understanding the behavior of electrons moving in a plane perpendicular to the magnetic field [35]. The confinement is possible at the *interface* typically between a semiconductor and an insulator, where a quantum well that traps the particles is formed, allowing their motion just in the direction of the interface plane at low energies, forbidding the motion in any other directions.

Conversely, we discuss here this problem in three dimensions, the most complete formulation. However, if the length of the cylindrical geometry of the system is large enough the results are close to those in two dimensions. Despite this latter, it is suggested that the formulation in two dimensions is not sufficient to explain the whole problem. As suggested before, electronic devices are based in interfaces. As a consequence of this line of reasoning, a natural lower temperature bound is theoretically imposed, that appears from the analysis in three dimensions.

#### 3.1. The model of one charged particle in a magnetic field

We introduce the present application giving the essential ingredients of the well-known Landau model for diamagnetism: a spinless charged particle in a magnetic field *B*. Consider the kinetic momentum

$$
\overrightarrow{\pi} = \overrightarrow{p} + \frac{q}{c}\overrightarrow{\mathcal{A}}\,\tag{17}
$$

where *mq* is the mass of a particle of charge *q*, the vector −→*p* is the linear momentum subject to the action of −→*A* , the vector potential.

If we follow the presentation of Feldman et al. [30]), the Hamiltonian reads [30]

$$H = \frac{\overrightarrow{\pi} \cdot \overrightarrow{\pi}}{2m\_q},\tag{18}$$

and the magnetic field is −→*<sup>B</sup>* = −→ ∇ × −→*A* . The vector potential is chosen in the symmetric gauge as −→*<sup>A</sup>* = (−*By*/2,*Bx*/2, 0), which corresponds to a uniform magnetic field along the *<sup>z</sup>*−direction.

By using the formulation of the step-ladder operators [30], one needs to define the step operators as follows [30]

$$\hbar\_{\pm} = \not p\_{\times} \pm i \not p\_{\times} \pm \frac{i\hbar}{2\ell\_{\text{B}}^2} (\hbar \pm i\emptyset),\tag{19}$$

where the length

$$\ell\_{\mathbb{B}} = (\hbar c/q\mathbb{B})^{1/2} \tag{20}$$

is the classical radius of the ground-state Landau orbit [30]. Motion along the *z*−axis is free [30]. For the transverse motion, the Hamiltonian specializes to [30]

$$
\hat{H}\_l = \frac{\hbar\_+\hbar\_-}{2m\_q} + \frac{1}{2}\hbar\Omega\hat{1},
\tag{21}
$$

where an important quantity characterizes the problem, namely,

$$
\Omega = qB/m\_q c,\tag{22}
$$

the cyclotron frequency [36]. The set of eigenstates {|*N*,*m*} is characterized by two quantum numbers: *N* related to the energy and *m* wuth the *z*− projection of the angular momentum. They are consequentely eigenstates of both *H*ˆ*<sup>t</sup>* , the Hamilronian, and *L*ˆ*z*, the angular momentum operator [30], thus

$$
\langle \hat{H}\_l | N, m \rangle = \left( N + \frac{1}{2} \right) \hbar \Omega \left| N, m \right\rangle = E\_N | N, m \rangle \tag{23}
$$

and

$$
\hat{L}\_{\mathcal{L}}|N,m\rangle = m\hbar|N,m\rangle. \tag{24}
$$

The eigenvalues of *<sup>L</sup>*ˆ*<sup>z</sup>* are not bounded by below, because *<sup>m</sup>* takes the values <sup>−</sup>∞,...,−1, 0, 1,...,*<sup>N</sup>* [30]. This fact agrees with the energies (*N* +1/2)*h*¯Ω that are infinitely degenerate [36]. As seen below, for estimation purposes, the physical relevance of phase-space localization is diminished by this fact. In addition, *Lz* is not an independent constant of the motion [36].

There exists a analogous formulation of an charged particle in a magnetic field by Kowalski that takes into account the geometry of a circle [33] (and for a comparison with the Feldman formulation see Ref. [9]), but at this point, we choose the Feldman formulation to work because the measure is easily defined and the normalization condition and other semiclassical measures are well described.

#### 3.2. Husimi distribution and Wehrl entropy

We will start our present endeavor defining the Hamiltonian *H*ˆ = *H*ˆ*<sup>t</sup>* + *H*ˆ*<sup>l</sup>* where *H*ˆ*<sup>t</sup>* = *h*¯Ω(*N*ˆ + 1/2) to describe the transverse motion, being Ω the cyclotron frequency as defined by the Eq. (22) and *N*ˆ the number operator; the Hamiltonian *H*ˆ*<sup>l</sup>* = *p*ˆ 2 *<sup>z</sup>* /2*mq* to represent the longitudinal one-dimensional free motion, for a particle of mass *mq* and charge *q* in a magnetic field *B*. A possible way to define the Husimi function η is given by

$$
\mathfrak{n}(\mathbf{x}, p\_{\mathbf{x}}; \mathbf{y}, p\_{\mathbf{y}}; p\_{\mathbf{z}}) = \langle \mathfrak{a}, \mathfrak{k}, k\_{\mathbf{z}} | \mathfrak{j} | \mathfrak{a}, \mathfrak{k}, k\_{\mathbf{z}} \rangle,\tag{25}
$$

where ρˆ is the thermal density operator and the set {|α,ξ,*kz*} stands for the coherent states for the description in three dimensions. By the direct product |α,ξ,*kz*≡|α,ξ <sup>|</sup>*kz*, the set {|α,ξ} corresponds to the coherent states of the transverse motion and {|*kz*} to the longitudinal motion. Therefore, the thermal density operator is given by

$$\clubsuit = \frac{1}{Z} \mathbf{e}^{-\\$(\hat{H}\_l + \hat{H}\_l)},\tag{26}$$

where β = 1/*kBT*, *T* is the temperature and *kB* the Boltzmann constant. In addition, *Z* is the partition function for motion in three dimensions of the particle. Now, if *Z* can be separated by using *Zt* (the contribution for the transverse motion) and *Zl* (the contribution for the one-dimensional free motion), then the partition function could be written as *Z* = *ZlZt* . Thus, the Husimi function [1] is expressed as

$$\mathfrak{m} = \frac{\mathbf{e}^{-\beta p\_c^2 / 2m\_q}}{Z\_l Z\_l} \sum\_{n,m} \mathbf{e}^{-\beta \hbar \Omega(n+1/2)} |\langle n,m|\alpha, \xi \rangle|^2. \tag{27}$$

where

6 ime knjige

follows [30]

where the length

and

and the magnetic field is −→*<sup>B</sup>* = −→

∇ ×

the transverse motion, the Hamiltonian specializes to [30]

where an important quantity characterizes the problem, namely,

*<sup>H</sup>*ˆ*<sup>t</sup>* <sup>|</sup>*N*,*m* <sup>=</sup>

addition, *Lz* is not an independent constant of the motion [36].

−→*<sup>A</sup>* = (−*By*/2,*Bx*/2, 0), which corresponds to a uniform magnetic field along the *<sup>z</sup>*−direction.

πˆ<sup>±</sup> = *p*ˆ*<sup>x</sup>* ±*ip*ˆ*<sup>y</sup>* ±

By using the formulation of the step-ladder operators [30], one needs to define the step operators as

is the classical radius of the ground-state Landau orbit [30]. Motion along the *z*−axis is free [30]. For

the cyclotron frequency [36]. The set of eigenstates {|*N*,*m*} is characterized by two quantum numbers: *N* related to the energy and *m* wuth the *z*− projection of the angular momentum. They are consequentely

The eigenvalues of *<sup>L</sup>*ˆ*<sup>z</sup>* are not bounded by below, because *<sup>m</sup>* takes the values <sup>−</sup>∞,...,−1, 0, 1,...,*<sup>N</sup>* [30]. This fact agrees with the energies (*N* +1/2)*h*¯Ω that are infinitely degenerate [36]. As seen below, for estimation purposes, the physical relevance of phase-space localization is diminished by this fact. In

eigenstates of both *H*ˆ*<sup>t</sup>* , the Hamilronian, and *L*ˆ*z*, the angular momentum operator [30], thus

 *N* + 1 2 

*<sup>H</sup>*ˆ*<sup>t</sup>* <sup>=</sup> <sup>π</sup>ˆ+πˆ<sup>−</sup> 2*mq* + 1 2 *h*¯Ω1,

*ih*¯ 2<sup>2</sup> B

−→*A* . The vector potential is chosen in the symmetric gauge as

<sup>B</sup> = (*hc*¯ /*qB*)1/<sup>2</sup> (20)

Ω = *qB*/*mqc*, (22)

*<sup>h</sup>*¯Ω|*N*,*m* <sup>=</sup> *EN*|*N*,*m* (23)

*<sup>L</sup>*ˆ*z*|*N*,*m* <sup>=</sup> *mh*¯|*N*,*m*. (24)

(*x*ˆ±*iy*ˆ), (19)

ˆ (21)

$$Z\_l = (\mathcal{L}/h)(2\pi m\_q k\_B T)^{1/2} \qquad \qquad \text{and} \tag{28}$$

$$Z\_{\mathfrak{l}} = \mathfrak{A}m\_q \Omega / (4\pi\mathfrak{H}\sinh(\mathfrak{B}\mathfrak{H}\mathfrak{D}/2)),\tag{29}$$

being *L* the length of the cylinder, *A* = π*R*<sup>2</sup> the area for cylindrical geometry [30]. In addition, the matrix element <sup>|</sup>*n*,*m*|α,ξ|<sup>2</sup> describes the probability of finding the particle in the coherent state <sup>|</sup>α,<sup>ξ</sup>. Its expression was defined previously [37].

The distribution η is written as:

$$
\mathfrak{h} = \mathfrak{n}\_{\varGamma}(p\_{\mathbb{Z}}) \mathfrak{n}\_{\varGamma}(\mathfrak{x}, p\_{\mathbb{X}}; \mathfrak{y}, p\_{\mathbb{Y}}), \tag{30}
$$

where η has been separated as a function of two distributions, namely, η*<sup>l</sup>* = η*l*(*pz*) and η*<sup>t</sup>* = η*t*(*x*, *px*;*y*, *py*). The explicit form of the Hamiltonian *H*ˆ*<sup>l</sup>* makes to miss the dependence on the variable *z*. Therefore, summing in Eq. (27) we solve

$$\mathfrak{m}\_l = \frac{\mathbf{e}^{-\mathfrak{P}p\_\varepsilon^2/2m\_q}}{\mathbf{Z}\_l},\tag{31}$$

$$\mathfrak{m}\_{\mathsf{I}} = \frac{2\pi\hbar}{\mathcal{A}m\_{\mathsf{q}}\Omega} \left(1 - \mathbf{e}^{-\|\mathsf{H}\Omega\|}\right) \mathbf{e}^{-\left(1 - \mathbf{e}^{-\|\mathsf{H}\Omega\|}\right) |\mathsf{a}|^{2} / 2\ell\_{\mathsf{B}}^{2}},\tag{32}$$

where the length <sup>B</sup> is defined by the Eq. (20). From expressions (31) and (32), we emphasize again that η*l*(*pz*) describes the free motion of the particle in the magnetic field direction and η*t*(*x*, *px*;*y*, *py*) the Landau levels due to the circular motion in a transverse plane to the magnetic field, similar to the harmonic oscillator of Eq. (9) since |*z*| <sup>2</sup> → |α<sup>|</sup> <sup>2</sup>/2<sup>2</sup> B. Consequently Eqs. (30), (31) and (32) together contain the complete description of the system. We noticed both distributions are naturally normalized in a standard form, i.e.,

$$\int \frac{\mathrm{d}z \mathrm{d}p\_z}{h} \mathfrak{n}\_l(p\_z) = 1,\tag{33}$$

and

$$\int \frac{d^2 \alpha d^2 \xi}{4\pi^2 \ell\_\mathbb{B}^4} \eta\_\mathbb{I}(\chi, p\_x; \mathfrak{y}, p\_\mathfrak{y}) = 1. \tag{34}$$

In consequence, both Eqs. (31) and (32), under conditions (33) and (34), allow us to get a close form for the Wehrl entropy. Furthermore, using one of the most basic property of the entropy, the additivity, we can state *W*total = *Wl* +*Wt* . Hence,

$$W\_l = -\int \frac{\mathrm{d}z \mathrm{d}p\_\sharp}{h} \eta\_l(p\_\sharp) \ln \eta\_l(p\_\sharp),\tag{35}$$

$$W\_l = -\int \frac{\mathrm{d}^2 \mathrm{ad}^2 \xi}{4\pi^2 \ell\_\mathrm{B}^4} \mathfrak{n}\_{\mathrm{I}}(\mathbf{x}, p\_\mathbf{x}; \mathbf{y}, p\_\mathbf{y}) \ln \mathfrak{n}\_{\mathrm{I}}(\mathbf{x}, p\_\mathbf{x}; \mathbf{y}, p\_\mathbf{y}), \tag{36}$$

again, the subindexes *t* and *l* represent respectively the transverse and longitudinal motions.

As a consequence of solving the integrals (35) and (36) we can identify the two entropies, they are

$$W\_l = \frac{1}{2} + \ln\left(\frac{\mathcal{L}}{\lambda}\right),\tag{37}$$

$$W\_l = 1 - \ln\left(1 - \mathbf{e}^{-\beta\hbar\Omega}\right) + \ln\left(\mathbf{g}\right),\tag{38}$$

where λ = *h*/(2π*mqk*B*T* )1/<sup>2</sup> represents the mean thermal length of the particle and *g* = *A*/2π<sup>2</sup> <sup>B</sup> the degeneracy of a Landau level [38].

#### 3.3. Semiclassical behavior

8 ime knjige

*z*. Therefore, summing in Eq. (27) we solve

harmonic oscillator of Eq. (9) since |*z*|

we can state *W*total = *Wl* +*Wt* . Hence,

degeneracy of a Landau level [38].

*Wl* = −

*Wt* = −

in a standard form, i.e.,

and

<sup>η</sup>*<sup>l</sup>* <sup>=</sup> <sup>e</sup>−β*p*<sup>2</sup>

<sup>η</sup>*<sup>t</sup>* <sup>=</sup> <sup>2</sup>π*h*¯

*<sup>z</sup>* /2*mq Zl*

<sup>2</sup> → |α<sup>|</sup>

 d2αd2ξ 4π<sup>2</sup><sup>4</sup> B

d*z*d*pz*

again, the subindexes *t* and *l* represent respectively the transverse and longitudinal motions.

<sup>2</sup> <sup>+</sup>ln

As a consequence of solving the integrals (35) and (36) we can identify the two entropies, they are

*L* λ 

where λ = *h*/(2π*mqk*B*T* )1/<sup>2</sup> represents the mean thermal length of the particle and *g* = *A*/2π<sup>2</sup>

<sup>1</sup>−e−β*h*¯<sup>Ω</sup>

 d2 αd<sup>2</sup> ξ 4π<sup>2</sup><sup>4</sup> B

*Wl* <sup>=</sup> <sup>1</sup>

*Wt* = 1−ln

d*z*d*pz*

where η has been separated as a function of two distributions, namely, η*<sup>l</sup>* = η*l*(*pz*) and η*<sup>t</sup>* = η*t*(*x*, *px*;*y*, *py*). The explicit form of the Hamiltonian *H*ˆ*<sup>l</sup>* makes to miss the dependence on the variable

*<sup>A</sup>mq*<sup>Ω</sup> (1−e−β*h*¯Ω) <sup>e</sup>−(1−e−β*h*¯<sup>Ω</sup>)|α<sup>|</sup>

where the length <sup>B</sup> is defined by the Eq. (20). From expressions (31) and (32), we emphasize again that η*l*(*pz*) describes the free motion of the particle in the magnetic field direction and η*t*(*x*, *px*;*y*, *py*) the Landau levels due to the circular motion in a transverse plane to the magnetic field, similar to the

<sup>2</sup>/2<sup>2</sup>

contain the complete description of the system. We noticed both distributions are naturally normalized

In consequence, both Eqs. (31) and (32), under conditions (33) and (34), allow us to get a close form for the Wehrl entropy. Furthermore, using one of the most basic property of the entropy, the additivity,

, (31)

B. Consequently Eqs. (30), (31) and (32) together

*<sup>h</sup>* <sup>η</sup>*l*(*pz*) = 1, (33)

η*t*(*x*, *px*;*y*, *py*) = 1. (34)

*<sup>h</sup>* <sup>η</sup>*l*(*pz*)lnη*l*(*pz*), (35)

η*t*(*x*,*px*;*y*,*py*)lnη*t*(*x*,*px*;*y*,*py*), (36)

, (37)

+ln(*g*), (38)

<sup>B</sup> the

<sup>B</sup> , (32)

<sup>2</sup>/2<sup>2</sup>

In fact, the classical entropy for a free particle in one dimension and Eq. (37) are coincident. Furthermore, the Eq. (38) is the Wehrl entropy for the transverse motion and possesses a form close to the harmonic oscillator entropy given by the Eq. (10), with the exception of a term associated with the degeneracy. Some properties of entropies that can be directly derived from Eqs. (37) and (38) are:

1. As commented before, *Wl* and the classical entropy for the free motion in one dimension coincide between them. Furthermore, this part of the entropy has to be nonnegative at all temperatures, this is *Wl* ≥ 0. This condition imposes a minimum to the temperature, given by

$$T\_0 = \frac{h^2}{2\pi m\_q e k\_B L^2},\tag{39}$$

where *e* = 2.718281828. Due to this basic property of *Wl*, the system is forced to take high values of temperature, being *T* > *T*0, where the behavior of the system is classical. Equivalently, it is possible to assure that, if *T*/*T*<sup>0</sup> ≥ 1, the length of a thermal wave λ lower than the average of the spacing among particles and quantum considerations are not relevant [39]. In addition, *T*<sup>0</sup> does not depend on external or internal physical parameters related to the system, as the transverse area, external magnetic field, charge of the particle, etc, practically depends only on the size of the system. If the system is large enough, the minimum temperature is low. However, modern electronic systems possess junctions where *L* can be considered almost zero. Thus, minimum temperature required to make applicable the present description is enough high [40].

2. The Wehrl entropy that is associated with transverse motion satisfies *Wt* ≥ 1 + ln(*g*) for all temperatures of the system, which is very nearly the Lieb condition in one dimension [41] with an additional term given by the logarithm of *g*, the degeneracy. The transverse motion is approximately bi-dimensional, but the Landau approach reduces the quantum motion of the particle in a magnetic field to a degenerate spectrum in one dimension essentially recovering the physics of the missing dimension. Therefore, the discussion about the behavior of the Wehrl entropy in light of the Lieb condition does not increase any applicability of the present treatment because the latter is always satisfied. The main problem that appears from the emphasis on the transverse motion is the restricted vision that is obtained of the behavior of the system. [9, 30, 32, 33], which represents the main difference with other contributions that discuss this topic. The combination of reasoning including both motions has sense when the imposition over the temperature is satisfied. For values of the temperature lower than *T*0, the behavior is essentially anomalous, thus this proposal is not applicable.

Additionally, the total entropy is expressed simply as follows

$$W\_{\text{total}} = \frac{3}{2} - \ln(1 - e^{-\beta \hbar \Omega}) + \ln\left(g\right) + \ln\left(\frac{\mathcal{L}}{\lambda}\right). \tag{40}$$

Now, we can discuss some approximate and limiting cases.

In first order of approximation, for *kBT <sup>h</sup>*¯Ω, we have ln(*g*/(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−β*h*¯Ω)) <sup>≈</sup> ln(*AT*/*T*0*L*<sup>2</sup>). If we write the thermal wave length in terms of the temperature *T*0, as λ = *L*(*eT*0/*T* )1/<sup>2</sup> and considering that *V* = *AL*, the entropy (40) is rephrased as follows

$$W\_{\text{total}}^{(1)} = \frac{3}{2} + \ln\left(\frac{\mathcal{V}}{\lambda^3}\right). \tag{41}$$

This is a particular expression for the entropy of a free particle in three dimensions related to the motion of a charged particle into a region of the magnetic field making mention of some geometrical properties of the system.

In second order of approximation, considering the special condition *A* <sup>∼</sup> *L*2, Wehrl entropy is expressed as follows

$$\mathbf{W}\_{\text{total}}^{(2)} \approx \mathbf{W}\_{\text{total}}^{(1)} + \frac{T\_0}{T} \mathbf{g}. \tag{42}$$

As explained before, the Wehrl entropy takes values that are permitted by the Lieb condition, namely, *W* ≥ 1. According to Eq. (42) the slope decreases as temperature increases. This fact also illustrates why the disorder increases as the magnetic field increases too.

The lower bound of temperature is related to values of *T* greater than *T*0, because this approach does not consider any temperature less than *T*0. In addition to this, the behavior of the total Wehrl entropy is reduced to the logarithm of the magnetic field. In order to see what occurs in the limiting case of the lowest temperature, according to Eq. (39), we take systems with *L* → ∞; thus the transverse entropy of Eq. (38) is rewritten as follows

$$W\_l^{T \to 0^+} = 1 + \ln\left(g\right). \tag{43}$$

As aforementioned, the Wehrl entropy is similar to the entropy of the harmonic oscillator and the lowest temperature comes being greater than the bound temperature, thus *W* ≥ 1 [41] as it was conjectured by Wehrl and shown by Lieb. From this condition, it must arrive to the following inequality for the magnetic field

$$\mathbf{g} \ge \mathbf{1},\tag{44}$$

where *g* = *qAB*/*hc* also accounts for the ratio between the flux of the magnetic field *AB* and the quantum of the magnetic flux given by *hc*/*<sup>q</sup>* <sup>=</sup> 4.14×10−7[*gauss cm*2] [14]. Then the inequality (44) adopts the form

$$B \ge \frac{1}{\mathcal{A}} \frac{hc}{q} = B\_0. \tag{45}$$

Moreover, the magnetic field *B*<sup>0</sup> = *hc*/*Aq* becomes to take a bound limiting value representing a minimum value for the external magnetic field. If *A* → ∞, we can study what occurs to the system when the magnetic field close to zero.

Now, we add two comments about the quantum description of particles in magnetic field close to limiting values of temperatures and magnetic fields, respectively:

1. The quantum Hall effect is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields and emerges from the Landau quantization [42, 43] which corresponds to a quantum version of the Hall effect [35]. The degeneracy is given by [14]

$$
\boldsymbol{\phi} = \mathbf{v} \boldsymbol{\phi}\_0,\tag{46}
$$

where φ<sup>0</sup> = *hc*/*q* is the minimum quantity (or quantum) of the magnetic flux. The factor ν takes integer values as ν = 1, 2, 3, . . . and it is related to the "*filling factor*" and simply with the conductivity quantization as σ = ν*q*2/*h*. The subsequent discovery of the fractional quantum Hall effect [34] expand the values ν to rational fractions as ν = 1/3, 1/5, 5/2, 12/5, . . . . Thus the fractional quantum Hall effect relies on other phenomena associated with interactions. In any case, the degeneracy is ν greater than 1 due to the inequality (44), as before, the transverse entropy always satisfies the Lieb bound for all temperatures and large enough systems, obtaining an infinite family of Wehrl entropies

$$W\_l = 1 - \ln(1 - e^{\beta \hbar \Omega}) + \ln \mathbf{v}.\tag{47}$$

The limiting value of ν provides a good descriptor for the integer quantum Hall effect. Conversely, for the fractional values of ν less than 1 are left out the present approach.

2. The Haas-van Alphen effect is other phenomenon that we can discuss. It is observed at low enough values of temperatures, describing oscillations in the magnetization, because the particles tend to occupy the lowest energy states. In the present description it is manifest for finite values of *A* and *B* lower than *B*0. Whereas if the value of the magnetic field decreases a less number of particles can be in the lowest state due to degeneracy is directly proportional to *B* [38]. Then, the transverse Wehrl entropy *Wt* is well defined for values of the magnetic field over *<sup>B</sup>*0, this is *<sup>B</sup>*/*B*<sup>0</sup> <sup>≥</sup> 1 and/or *<sup>g</sup>* <sup>→</sup> <sup>1</sup>+.

#### 3.4. Fisher Information Measure versus degeneracy

10 ime knjige

of the system.

Eq. (38) is rewritten as follows

when the magnetic field close to zero.

limiting values of temperatures and magnetic fields, respectively:

magnetic field

adopts the form

as follows

*W*(1) total <sup>=</sup> <sup>3</sup>

*W*(2) total <sup>≈</sup> *<sup>W</sup>*(1)

*WT*→0<sup>+</sup>

*B* ≥ 1 *A hc*

why the disorder increases as the magnetic field increases too.

<sup>2</sup> <sup>+</sup>ln

This is a particular expression for the entropy of a free particle in three dimensions related to the motion of a charged particle into a region of the magnetic field making mention of some geometrical properties

In second order of approximation, considering the special condition *A* <sup>∼</sup> *L*2, Wehrl entropy is expressed

total + *T*0

As explained before, the Wehrl entropy takes values that are permitted by the Lieb condition, namely, *W* ≥ 1. According to Eq. (42) the slope decreases as temperature increases. This fact also illustrates

The lower bound of temperature is related to values of *T* greater than *T*0, because this approach does not consider any temperature less than *T*0. In addition to this, the behavior of the total Wehrl entropy is reduced to the logarithm of the magnetic field. In order to see what occurs in the limiting case of the lowest temperature, according to Eq. (39), we take systems with *L* → ∞; thus the transverse entropy of

As aforementioned, the Wehrl entropy is similar to the entropy of the harmonic oscillator and the lowest temperature comes being greater than the bound temperature, thus *W* ≥ 1 [41] as it was conjectured by Wehrl and shown by Lieb. From this condition, it must arrive to the following inequality for the

where *g* = *qAB*/*hc* also accounts for the ratio between the flux of the magnetic field *AB* and the quantum of the magnetic flux given by *hc*/*<sup>q</sup>* <sup>=</sup> 4.14×10−7[*gauss cm*2] [14]. Then the inequality (44)

Moreover, the magnetic field *B*<sup>0</sup> = *hc*/*Aq* becomes to take a bound limiting value representing a minimum value for the external magnetic field. If *A* → ∞, we can study what occurs to the system

Now, we add two comments about the quantum description of particles in magnetic field close to

 *V* λ3 

. (41)

*<sup>T</sup> <sup>g</sup>*. (42)

*<sup>t</sup>* = 1+ln(*g*). (43)

*g* ≥ 1, (44)

*<sup>q</sup>* <sup>=</sup> *<sup>B</sup>*0. (45)

In the present subsection we propose a compact expression for the transverse Fisher information measure, taking into account a special way formerly developed in Ref. [6], which is given by

$$\mathcal{F}\_l = \int \frac{\mathrm{d}^2 \mathrm{ad}^2 \xi}{4\pi^2 \ell\_\mathrm{H}^4} \,\eta\_l(\alpha) \left(\frac{\partial \ln \eta\_l(\alpha)}{\partial \alpha}\right)^2. \tag{48}$$

After introducing the known expression for η*t* , we arrive to

$$\mathcal{F}\_l = \frac{2}{\ell\_\text{H}^2} (1 - \mathbf{e}^{-\text{βt}\Omega}). \tag{49}$$

Fisher measure *F<sup>t</sup>* has space dimension (*L*)−<sup>2</sup> and quantifies the ability for estimating the parameter α [44]. This parameter corresponds to the radio of a circular orbit of coherent states. By combining Eqs. (49) and (46) with the definition of <sup>H</sup> we obtain

$$\mathcal{F}\_l = \frac{4\pi\mathbf{v}}{A} (1 - \mathbf{e}^{-\beta\hbar\Omega}),\tag{50}$$

which represents the linear dependence of the measure *F<sup>t</sup>* with the magnetic field through the constant 2 <sup>H</sup> at low temperature.

The inverse exponential dependence on the temperature, of the Fisher information, is clear from Eq. (50). Further, the initial value directly depends on the factor ν.

Now, to complete the description of the movement, we consider the Fisher information measure for the longitudinal motion, this is

$$\mathcal{F}\_l = \int \frac{dzdp\_z}{h} \, \eta\_l(p\_\varepsilon) \left(\frac{\partial \ln \eta\_l(p\_\varepsilon)}{\partial p\_\varepsilon}\right)^2,\tag{51}$$

where *pz* is the variable that we contain in the present discussion, which was previously ignored [32], making a great difference when the results are compared. The function η*<sup>l</sup>* is included into the above equation to get

$$
\mathcal{F}\_l = \frac{\mathfrak{B}}{m}.\tag{52}
$$

As seen before, the Fisher measure in one dimension coincides with the classical one for the free particle [45]. As expected, the total Fisher measure is constructed multiplying Eqs. (50) and (52).

#### 3.5. Additional appointments and consequences

The Wehrl entropy, which we obtain here, depends on multiple parameters, for instance, the degeneracy *g*, and the ratio between the cylinder and thermal lengths, *i.e.*, *L* and λ. The combination of these parameters can effectively give some interesting results. Therefore, in a especial perspective we can see that the harmonic oscillator is behaved as a particular case of the charged particle in a magnetic field. Thus, we can consider, for example, the following relation among parameters:

$$g = \frac{\lambda}{\mathcal{L}} \exp\left(\frac{1}{2}\right),\tag{53}$$

which leads the Wehrl entropy from the Landau diamagnetism to the one-dimensional harmonic oscillator (15). This is a nontrivial approach because the nature of problems are radically different. For instance, the harmonic oscillator, that we use here, is a one-dimensional system, but the Landau diamagnetism is three-dimensional. Consequently, phase spaces are not coincident and measures are not the same.

Besides, we know *g* ≥ 1. But, if we consider the minimum value *g* = 1, we can obtain the bound value of the temperature *T*0, given by Eq. (39), above this value, the present approach is valid. Afterward, we obtain a relationship between both lengths involved into the problem, this is a bound value for the length of the cylinder, *L* ≥ λ/*e*. Thus, for values where this condition is violated, this approach is not valid.

The comparison between the Fisher information measures, for both cited problems, is also possible. Hence, in the same previous line we can propose a comparison of the Fisher information measures, considering measures dimensionally compatible. Originally, the classical Fisher information (11) accounts the localization of the corresponding probability density ρ(*x*), which is approached by Cramer-Rao inequality (12), where ∆*x* is the variance for the stochastic variable *x*. However, the variation of the definition (11) takes into account the localization, not in the variable *x* or any other coordinate, but the localization into phase space. This is well defined for the transverse motion. Moreover, the longitudinal motion is classical, not quantized, and any coherent state formulation is proposed. The quantum counterpart can be defined as a problem of continuous spectrum [46], and a suitable formulation of coherent states is still unknown; for the time, this continues being an open problem. Thus, the classical formulation is used and we have decided to advance evaluating the classical distribution for the longitudinal motion.

In addition, with the purpose of describing the complete motion, we consider now the Fisher information measure for the movement, this is

$$\mathcal{F}' = \frac{\lambda^2}{\ell\_{\text{H}}^2} (1 - \mathbf{e}^{-\beta \hbar \Omega}). \tag{54}$$

where *F* is defined as *F* = *h*<sup>2</sup>*ItIl*/4π in order to compare the trend of this Fisher measure with corresponding one of the harmonic oscillator. These cases are comparable with the harmonic oscillator only if λ<sup>2</sup> = <sup>2</sup> <sup>H</sup> and are depicted with red-solid-line in Fig. 1.

#### 4. Description of the molecular rotation: Rigid rotator

There are few physical systems whose spectrum is analytically known, aside from the previous one we have the anisotropic rigid rotator, which is a system of a single particle that can rotate in several ways. Thermodynamic properties can be analytically described [47]. It is expected that this treatment can characterize important features of molecular systems [48] to apply such concepts to several aspects related to materials [49].

#### 4.1. Linear rigid rotator

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2

<sup>H</sup> at low temperature.

longitudinal motion, this is

equation to get

not the same.

valid.

which represents the linear dependence of the measure *F<sup>t</sup>* with the magnetic field through the constant

The inverse exponential dependence on the temperature, of the Fisher information, is clear from Eq.

Now, to complete the description of the movement, we consider the Fisher information measure for the

where *pz* is the variable that we contain in the present discussion, which was previously ignored [32], making a great difference when the results are compared. The function η*<sup>l</sup>* is included into the above

*<sup>F</sup><sup>l</sup>* <sup>=</sup> <sup>β</sup>

As seen before, the Fisher measure in one dimension coincides with the classical one for the free particle [45]. As expected, the total Fisher measure is constructed multiplying Eqs. (50) and (52).

The Wehrl entropy, which we obtain here, depends on multiple parameters, for instance, the degeneracy *g*, and the ratio between the cylinder and thermal lengths, *i.e.*, *L* and λ. The combination of these parameters can effectively give some interesting results. Therefore, in a especial perspective we can see that the harmonic oscillator is behaved as a particular case of the charged particle in a magnetic field.

which leads the Wehrl entropy from the Landau diamagnetism to the one-dimensional harmonic oscillator (15). This is a nontrivial approach because the nature of problems are radically different. For instance, the harmonic oscillator, that we use here, is a one-dimensional system, but the Landau diamagnetism is three-dimensional. Consequently, phase spaces are not coincident and measures are

Besides, we know *g* ≥ 1. But, if we consider the minimum value *g* = 1, we can obtain the bound value of the temperature *T*0, given by Eq. (39), above this value, the present approach is valid. Afterward, we obtain a relationship between both lengths involved into the problem, this is a bound value for the length of the cylinder, *L* ≥ λ/*e*. Thus, for values where this condition is violated, this approach is not

The comparison between the Fisher information measures, for both cited problems, is also possible. Hence, in the same previous line we can propose a comparison of the Fisher information measures, considering measures dimensionally compatible. Originally, the classical Fisher information (11) accounts the localization of the corresponding probability density ρ(*x*), which is approached by

1 2 

∂lnη*l*(*pz*) ∂*pz*

2

, (51)

*<sup>m</sup>*. (52)

, (53)

*<sup>h</sup>* <sup>η</sup>*l*(*pz*)

(50). Further, the initial value directly depends on the factor ν.

*F<sup>l</sup>* =

3.5. Additional appointments and consequences

*dzd pz*

Thus, we can consider, for example, the following relation among parameters:

*<sup>g</sup>* <sup>=</sup> <sup>λ</sup> *<sup>L</sup>* exp We begin exploring the linear rigid rotator based on the excellent discussion made in Ref. [50] about the coherent states for angular momenta. The Hamiltonian of this simple system is [16]

$$
\hat{H} = \frac{\mathcal{L}^2}{2I\_{\text{xy}}},
\tag{55}
$$

where the operator *L*ˆ <sup>2</sup> is associated with the angular momentum and the parameter *Ixy* is the corresponding inertia momentum. The set {|*IK*} is the set of eigenstates of the Hamiltonian, where we can verify the following relations

$$\begin{aligned} \hat{L}^2|IK\rangle &= I(I+1)\hbar^2|IK\rangle\\ \hat{L}\_z|IK\rangle &= K\hbar|IK\rangle,\end{aligned} \tag{56}$$

with *I* = 0, 1, 2..., for −*I* ≤ *K* ≤ *I*. Additionally, the energy spectrum is given by eigenstates of the operator *H*

$$\mathfrak{e}\_I = \frac{I(I+1)\hbar^2}{2I\_{\rm xy}}.\tag{57}$$

A suitable construction of coherent states is found in Ref. [51, 52] for the lineal rigid rotator, using Schwinger oscillator model of angular momentum, in the fashion

$$|IK\rangle = \frac{(\mathring{a}\_+^\dagger)^{I+K} (\mathring{a}\_-^\dagger)^{I-K}}{\sqrt{(I+K)!(I-K)!}} |0\rangle,\tag{58}$$

where *a*ˆ+, *a*ˆ<sup>−</sup> are the corresponding creation and annihilation operators, respectively, and they show the following basic properties

1. The vacuum state

.

$$|0\rangle \equiv |0,0\rangle$$

2. Orthogonality is satisfied by

$$
\langle \stackrel{\circ}{I} \stackrel{\circ}{K} | I K \rangle = \delta\_{I',I} \delta\_{K',K},
$$

3. The completeness property is contained in the relation

$$\sum\_{I=0}^{\infty} \sum\_{K=-I}^{I} |IK\rangle\langle IK| = \hat{1}.$$

Due to we are interested in two degrees of freedom, the resulting coherent states come from the tensor product of |*z*<sup>1</sup> and |*z*<sup>2</sup> [50, 53], where

$$|z\_1 z\_2\rangle = |z\_1\rangle \otimes |z\_2\rangle,\tag{59}$$

and

$$
\langle \hat{a}\_{+} | z\_{1} z\_{2} \rangle = z\_{1} | z\_{1} z\_{2} \rangle,\tag{60}
$$

$$
\hat{a}\_{-}|z\_{1}z\_{2}\rangle = z\_{2}|z\_{1}z\_{2}\rangle. \tag{61}
$$

Therefore, |*z*1*z*<sup>2</sup> is the coherent state written [50] as

$$\langle z\_1 z\_2 \rangle = e^{-\frac{|z|^2}{2}} e^{z\_1 \hat{d}\_+^\dagger} e^{z\_2 \hat{d}\_-^\dagger} |0\rangle,\tag{62}$$

with

$$|z\_1\rangle = e^{-\frac{|z\_1|^2}{2}} e^{z\_1 \mathcal{d}\_+^\uparrow} |0\rangle,\tag{63}$$

$$|z\_2\rangle = e^{-\frac{|z\_2|^2}{2}} e^{\mathbb{C}z\hat{d}\_-^\dagger}|0\rangle. \tag{64}$$

We need to introduce the suitable notation

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the following basic properties

2. Orthogonality is satisfied by

product of |*z*<sup>1</sup> and |*z*<sup>2</sup> [50, 53], where

Therefore, |*z*1*z*<sup>2</sup> is the coherent state written [50] as

1. The vacuum state

.

and

with

A suitable construction of coherent states is found in Ref. [51, 52] for the lineal rigid rotator, using

(*<sup>I</sup>* +*K*)!(*<sup>I</sup>* <sup>−</sup>*K*)!



Due to we are interested in two degrees of freedom, the resulting coherent states come from the tensor


ˆ


*a*ˆ+|*z*1*z*<sup>2</sup> = *z*1|*z*1*z*<sup>2</sup>, (60)

*a*ˆ−|*z*1*z*<sup>2</sup> = *z*2|*z*1*z*<sup>2</sup>. (61)

<sup>−</sup> |0, (62)

<sup>+</sup> |0, (63)

<sup>−</sup> |0. (64)

where *a*ˆ+, *a*ˆ<sup>−</sup> are the corresponding creation and annihilation operators, respectively, and they show


Schwinger oscillator model of angular momentum, in the fashion

3. The completeness property is contained in the relation

<sup>|</sup>*IK* <sup>=</sup> (*a*<sup>ˆ</sup>

*I K*

∞ ∑ *I*=0

<sup>|</sup>*z*1*z*<sup>2</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*<sup>|</sup>

<sup>|</sup>*z*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*1<sup>|</sup>

<sup>|</sup>*z*<sup>2</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>|</sup>*z*2<sup>|</sup>

2 <sup>2</sup> *ez*1*a*<sup>ˆ</sup> † <sup>+</sup> *ez*2*a*<sup>ˆ</sup> †

> 2 <sup>2</sup> *ez*1*a*<sup>ˆ</sup> †

> 2 <sup>2</sup> *ez*2*a*<sup>ˆ</sup> †

*I* ∑ *K*=−*I*

† +)*<sup>I</sup>*+*K*(*a*<sup>ˆ</sup> † −)*I*−*<sup>K</sup>*

$$|z|^2 = |z\_1|^2 + |z\_2|^2. \tag{65}$$

Using Eqs. (58) and (62) we easily calculate |*z*1*z*<sup>2</sup> and, after a bit of algebra, find

$$|z\_1 z\_2\rangle = e^{-\frac{|z|^2}{2}} \sum\_{n\_+, n\_-} \frac{z\_1^{n\_+}}{\sqrt{n\_+!}} \frac{z\_2^{n\_-}}{\sqrt{n\_-!}} |IK\rangle,\tag{66}$$

where *n*<sup>+</sup> = *I* + *K* and *n*<sup>−</sup> = *I* − *K*. Thus, the probability of obtaining the state |*IK* in the coherent state |*z*1*z*<sup>2</sup> is of the form

$$|\langle IK|z\_1 z\_2\rangle|^2 = e^{-|z|^2} \frac{|z\_1|^{2n\_+}}{n\_+!} \frac{|z\_2|^{2n\_-}}{n\_-!}.\tag{67}$$

The present coherent states satisfy resolution of unity

$$
\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} |z\_1 z\_2\rangle \langle z\_1 z\_2| = 1. \tag{68}
$$

Furthermore, *z*<sup>1</sup> and *z*<sup>2</sup> are continuous variables.

The procedure developed by Anderson *et al.* [4] is easily followed and used to assess the Husimi distribution [1]. In our approach this is defined, from Eq. (4), as

$$
\mu(z\_1, z\_2) = \langle z\_1, z\_2 | \mathfrak{d} | z\_1, z\_2 \rangle,\tag{69}
$$

where the density operator is

$$\Phi = Z\_{2D}^{-1} \exp\left(-\beta \hat{\mathcal{H}}\right). \tag{70}$$

The concomitant rotational partition function *Z*2*<sup>D</sup>* is given in Ref. [16]

$$Z\_{2D} = \sum\_{I=0}^{\infty} (2I+1) \, e^{-I(I+1)\frac{\Theta}{T}},\tag{71}$$

with <sup>Θ</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>/(2*IxykB*). We emphasize that in the present context the performing of the sum Tr <sup>≡</sup> ∑∞ *<sup>I</sup>*=<sup>0</sup> <sup>∑</sup>*<sup>I</sup> <sup>K</sup>*=−*<sup>I</sup>* corresponds to the "operation trace" . Using now the completeness property into Eq. (69) and theEq. (67), we fobtain the Husimi distribution in the form

$$\mu(z\_1, z\_2) = e^{-|z|^2} \frac{\sum\_{I=0}^{\infty} \frac{|z|^d}{(2I)!} e^{-I(I+1)\frac{\Theta}{\overline{I}}}}{\sum\_{I=0}^{\infty} (2I+1) e^{-I(I+1)\frac{\Theta}{\overline{I}}}}. \tag{72}$$

It is easy to show that this distribution is normalized to unity

$$\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \mu(z\_1, z\_2) = 1,\tag{73}$$

where *z*<sup>1</sup> and *z*<sup>2</sup> are given by Eqs. (60), (61), and (65). We must employ the binomial expression (|*z*1| <sup>2</sup> <sup>+</sup> <sup>|</sup>*z*2<sup>|</sup> <sup>2</sup>)4*<sup>I</sup>* and then integrate over the whole complex plane in two dimensions to verify the normalization condition. The differential element of area in the *z*1(*z*2) plane is d<sup>2</sup>*z*<sup>1</sup> = d*x*d*px*/2¯*h* (d2*z*<sup>2</sup> = d*y*d*py*/2¯*h*) [19]. Moreover, we have the phase-space relationships

$$|z\_1|^2 = \frac{1}{4} \left( \frac{\chi^2}{\sigma\_\chi^2} + \frac{p\_\chi^2}{\sigma\_{p\_x}^2} \right),\tag{74}$$

$$|z\_2|^2 = \frac{1}{4} \left( \frac{\text{y}^2}{\sigma\_\text{y}^2} + \frac{p\_\text{y}^2}{\sigma\_{p\_\gamma}^2} \right),\tag{75}$$

where <sup>σ</sup>*<sup>x</sup>* <sup>≡</sup> <sup>σ</sup>*<sup>y</sup>* <sup>=</sup> <sup>√</sup>*h*¯/2*m*<sup>ω</sup> and <sup>σ</sup>*px* <sup>≡</sup> <sup>σ</sup>*py* <sup>=</sup> <sup>√</sup>*m*ω*h*¯/2.

The profile of the Husimi function is similar to that of a Gaussian distribution.

As before, a semiclassical measure of localization is the Wehrl entropy [17], and the Fisher [5] as well. For the present model in two dimensions, the Wehrl entropy reads

$$\mathcal{W} = -\int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \mu(z\_1, z\_2) \ln \mu(z\_1, z\_2), \tag{76}$$

where *µ*(*z*1,*z*2) is given by Eq. (72).

#### *4.1.1. Fisher information measure*

The Fisher measure [5, 20, 21] regards as a semiclassical counterpart of Wehrl entropy [5]. Now, extending the ideas developed in Ref. [5] for the case of the harmonic oscillator in one dimension to the present case in two dimensions, we can define the shift invariant Fisher measure in the fashion

$$\mathcal{F}\_{2D} = \frac{1}{4} \int \frac{\mathrm{d}^2 z\_1}{\pi} \frac{\mathrm{d}^2 z\_2}{\pi} \mu(z\_1, z\_2) \left(\frac{\partial \ln \mu(z\_1, z\_2)}{\partial |z|}\right)^2. \tag{77}$$

From Eq. (72) it is easy to prove that

$$\mathfrak{n}(z\_1, z\_2) = \frac{1}{2} \frac{\partial \ln \mu(z\_1, z\_2)}{\partial |z|} = \frac{\sum\_{I=0}^{\infty} \left[ \frac{|\bar{z}|^{4I-1}}{(2I-1)!} - \frac{|\bar{z}|^{4I+1}}{(2I)!} \right] e^{-I(I+1)\Theta/T}}{\sum\_{I=0}^{\infty} \frac{|\bar{z}|^{4I}}{(2I)!} e^{-I(I+1)\Theta/T}}. \tag{78}$$

Therefore, the corresponding Fisher measure acquires the simpler appearance

$$\mathcal{F}\_{2D} = \int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \,\mu(z\_1, z\_2) \,\mathfrak{n}(z\_1, z\_2)^2,\tag{79}$$

i.e.,

16 ime knjige

(|*z*1|

<sup>2</sup> <sup>+</sup> <sup>|</sup>*z*2<sup>|</sup>

It is easy to show that this distribution is normalized to unity

where <sup>σ</sup>*<sup>x</sup>* <sup>≡</sup> <sup>σ</sup>*<sup>y</sup>* <sup>=</sup> <sup>√</sup>*h*¯/2*m*<sup>ω</sup> and <sup>σ</sup>*px* <sup>≡</sup> <sup>σ</sup>*py* <sup>=</sup> <sup>√</sup>*m*ω*h*¯/2.

where *µ*(*z*1,*z*2) is given by Eq. (72).

*4.1.1. Fisher information measure*

From Eq. (72) it is easy to prove that

<sup>η</sup>(*z*1,*z*2) = <sup>1</sup>

For the present model in two dimensions, the Wehrl entropy reads

*W* = −

*<sup>F</sup>*2*<sup>D</sup>* <sup>=</sup> <sup>1</sup> 4

2

 d2*z*<sup>1</sup> π

∂ln*µ*(*z*1,*z*2)

<sup>∂</sup>|*z*<sup>|</sup> <sup>=</sup> <sup>∑</sup><sup>∞</sup>

d2*z*<sup>2</sup>

 d2*z*<sup>1</sup> π

(d2*z*<sup>2</sup> = d*y*d*py*/2¯*h*) [19]. Moreover, we have the phase-space relationships



The profile of the Husimi function is similar to that of a Gaussian distribution.

 d2*z*<sup>1</sup> π

d2*z*<sup>2</sup>

where *z*<sup>1</sup> and *z*<sup>2</sup> are given by Eqs. (60), (61), and (65). We must employ the binomial expression

normalization condition. The differential element of area in the *z*1(*z*2) plane is d<sup>2</sup>*z*<sup>1</sup> = d*x*d*px*/2¯*h*

As before, a semiclassical measure of localization is the Wehrl entropy [17], and the Fisher [5] as well.

The Fisher measure [5, 20, 21] regards as a semiclassical counterpart of Wehrl entropy [5]. Now, extending the ideas developed in Ref. [5] for the case of the harmonic oscillator in one dimension to the present case in two dimensions, we can define the shift invariant Fisher measure in the fashion

<sup>π</sup> *<sup>µ</sup>*(*z*1,*z*2)

*I*=0 |*z*| 4*I*−1 (2*I*−1)! <sup>−</sup> <sup>|</sup>*z*<sup>|</sup>

> ∑∞ *I*=0 |*z*| 4*I*

∂ln*µ*(*z*1,*z*2) ∂|*z*|

> 4*I*+1 (2*I*)!

d2*z*<sup>2</sup>

<sup>2</sup>)4*<sup>I</sup>* and then integrate over the whole complex plane in two dimensions to verify the

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *x* σ2 *px*

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *y* σ2 *py* <sup>π</sup> *<sup>µ</sup>*(*z*1,*z*2) = 1, (73)

, (74)

, (75)

<sup>π</sup> *<sup>µ</sup>*(*z*1,*z*2) ln*µ*(*z*1,*z*2), (76)

2

*e*−*I*(*I*+1)Θ/*<sup>T</sup>*

(2*I*)! *<sup>e</sup>*−*I*(*I*+1)Θ/*<sup>T</sup>* . (78)

. (77)

$$\mathcal{F}\_{2D} \equiv \langle \mathfrak{n}(z\_1, z\_2)^2 \rangle,\tag{80}$$

where with the notation

$$
\langle \mathcal{G} \rangle = \int \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \mu(z) \, \mathcal{G}, \tag{81}
$$

we refer to the *semi-classical expectation value* of *G*. In Fig. 1 we plot the Fisher information and the Wehrl entropy as a function of the temperature (black-dashed-line), which we compare with the same measures for the transverse Landau diamagnetism (blue-solid-line). At low temperatures, the Fisher information measure describes the inverse-delocalization and takes its maximum value when the Wehrl entropy is minimum. This behavior is reversed for high temperatures. Every curve can be compared with the respective counterpart shown for the harmonic oscillator in one dimension (red-solid-line).

Figure 1. Trends of Fisher Information and Wehrl entropy for the rotator (black-dashed-line) in two dimensions is compared with the transverse Landau diamagnetism (blue-solid-line), the horizontal axis is the normalized temperature τ = *kT* (2*Ixy*)/*h*¯ <sup>2</sup> and τ = *kT*/*h*¯Ω, respectively. Additionally, we show a case where the Landau diamagnetism dimensionally coincides with the one-dimensional harmonic oscillator (red-solid-line). The Wehrl entropy starts in *W*=1. If the normalized temperature increases, the Fisher information decreases while Wehrl entropy increases.

#### 4.2. Rigid rotator in three dimensions

In the present section we consider a more general problem, the model of the rigid rotator in three dimensions, whose Hamiltonian writes [54]

$$
\hat{H} = \frac{\hat{L}\_\chi^2}{2I\_\chi} + \frac{\hat{L}\_\chi^2}{2I\_\chi} + \frac{\hat{L}\_\chi^2}{2I\_\chi},\tag{82}
$$

where the parameters *Ix*, *Iy*, and *Iz* are the inertia momenta. The set {|*IMK*} corresponds to a complete set of eigenvectors of the operator *H*ˆ . The following relations are additionally applied

$$\begin{aligned} \langle \hat{L}^2 |IMK\rangle &= I(I+1)\hbar^2 |IMK\rangle\\ \langle \hat{L}\_{\mathbb{Z}} |IMK\rangle &= K\hbar |IMK\rangle\\ \langle \hat{J}\_{\mathbb{Z}} |IMK\rangle &= M\hbar |IMK\rangle, \end{aligned} \tag{83}$$

with −*I* ≤ *K* ≤ *I* and −*I* ≤ *M* ≤ *I*, where *I* = 0,...,∞,. The elements of set {|*IMK*} satisfy orthogonality and completeness property [54]

$$
\langle \stackrel{\circ}{I} \stackrel{\circ}{M} \stackrel{\circ}{K} |IMK\rangle = \delta\_{I',I} \delta\_{M',M} \delta\_{K',K} \tag{84}
$$

$$\sum\_{I=0}^{\infty} \sum\_{M=-I}^{I} \sum\_{K=-I}^{I} |IMK\rangle\langle IMK| = \hat{1}.\tag{85}$$

If we take *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup> *<sup>x</sup>* + *L*ˆ <sup>2</sup> *<sup>y</sup>* + *L*ˆ <sup>2</sup> *<sup>z</sup>* and assume axial symmetry, i.e., *Ixy* ≡ *Ix* = *Iy*, we can recast the Hamiltonian as

$$\hat{H} = \frac{1}{2I\_{\text{xy}}} \left[ \mathcal{L}^2 + \left( \frac{I\_{\text{xy}}}{I\_{\text{z}}} - 1 \right) \mathcal{L}\_{\text{z}}^2 \right],\tag{86}$$

where the operator *L*ˆ*<sup>z</sup>* represents the projection on the rotation axis *z* of the *L*ˆ 2, which is the angular momentum operator. The concomitant spectrum of energy becomes

$$\varepsilon\_{I,K} = \frac{\hbar^2}{2I\_{\rm xy}} \left[ I(I+1) + \left( \frac{I\_{\rm xy}}{I\_{\rm \varepsilon}} - 1 \right) K^2 \right],\tag{87}$$

where the number *I* is integer and non-negative and it stands for the eigenvalue of the operator *L*ˆ 2, the angular momentum. The range of the other quantum number −*I* ≤ *m* ≤ *I* represents the projections on the intrinsic rotation axis of the rotator. Every state has a degeneracy (2*I* +1). The inertia momenta are quantified by the parameters *Ix* = *Iy* ≡ *Ixy* and *Iz*. The ratio *Ixy*/*Iz* characterizes different "geometrical" issues. For instance, some typical values of *Ixy*/*Iz* are 1, 1/2 and ∞, which correspond to the spherical, the extremely oblate and prolate cases, respectively.

#### *4.2.1. Construction of coherent states*

Again, we cite the work of Morales *et al.* where they construct a suitable set of coherent states for the rigid rotator in Ref. [54] and kindly discuss their mathematical foundations. First, they start introducing the auxiliary quantity

$$X\_{I,M,K} = \sqrt{I!(I+M)!(I-M)!(I+K)!(I-K)!} \tag{88}$$

to obtain [54]

18 ime knjige

where the parameters *Ix*, *Iy*, and *Iz* are the inertia momenta. The set {|*IMK*} corresponds to a complete

*<sup>L</sup>*<sup>ˆ</sup> <sup>2</sup>|*IMK* <sup>=</sup> *<sup>I</sup>*(*<sup>I</sup>* <sup>+</sup>1)*h*¯ <sup>2</sup>|*IMK*

with −*I* ≤ *K* ≤ *I* and −*I* ≤ *M* ≤ *I*, where *I* = 0,...,∞,. The elements of set {|*IMK*} satisfy

 ,*I*δ*M* ,*M*δ*K*

*Ixy Iz* −1 *L*ˆ 2 *z* 

> *Ixy Iz* −1 *K*2

where the operator *L*ˆ*<sup>z</sup>* represents the projection on the rotation axis *z* of the *L*ˆ 2, which is the angular

*I*(*I* +1) +

where the number *I* is integer and non-negative and it stands for the eigenvalue of the operator *L*ˆ 2, the angular momentum. The range of the other quantum number −*I* ≤ *m* ≤ *I* represents the projections on the intrinsic rotation axis of the rotator. Every state has a degeneracy (2*I* +1). The inertia momenta are quantified by the parameters *Ix* = *Iy* ≡ *Ixy* and *Iz*. The ratio *Ixy*/*Iz* characterizes different "geometrical" issues. For instance, some typical values of *Ixy*/*Iz* are 1, 1/2 and ∞, which correspond to the spherical,

Again, we cite the work of Morales *et al.* where they construct a suitable set of coherent states for the rigid rotator in Ref. [54] and kindly discuss their mathematical foundations. First, they start introducing


*<sup>z</sup>* and assume axial symmetry, i.e., *Ixy* ≡ *Ix* = *Iy*, we can recast the

*I*!(*I* + *M*)!(*I* − *M*)!(*I* +*K*)!(*I* −*K*)! (88)

*<sup>z</sup>*|*IMK* = *Mh*¯|*IMK*,


*I* ∑ *K*=−*I*

*<sup>L</sup>*ˆ*z*|*IMK* <sup>=</sup> *Kh*¯|*IMK* (83)

,*<sup>K</sup>* (84)

ˆ (85)

, (86)

, (87)

set of eigenvectors of the operator *H*ˆ . The following relations are additionally applied

*J*ˆ

*I M K*

∞ ∑ *I*=0

*I* ∑ *M*=−*I*

*<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>1</sup> 2*Ixy L*ˆ <sup>2</sup> +

momentum operator. The concomitant spectrum of energy becomes

<sup>ε</sup>*I*,*<sup>K</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*Ixy* 

orthogonality and completeness property [54]

*<sup>x</sup>* + *L*ˆ <sup>2</sup>

*<sup>y</sup>* + *L*ˆ <sup>2</sup>

the extremely oblate and prolate cases, respectively.

*XI*,*M*,*<sup>K</sup>* =

*4.2.1. Construction of coherent states*

the auxiliary quantity

If we take *L*ˆ <sup>2</sup> = *L*ˆ <sup>2</sup>

Hamiltonian as

$$<\langle z\_{1}z\_{2}z\_{3}\rangle = e^{-\frac{|a|^{2}}{2}} \sum\_{IMK} \frac{[(2I)!]^{2} z\_{1}^{(I+M)} z\_{2}^{I} z\_{3}^{(I+K)}}{X\_{I,M,K}} |IMK\rangle,\tag{89}$$

where Morales *et al.* introduced the following supplementary variable

$$\left|u\right|^2 = \left|z\_2\right|^2 (1+\left|z\_1\right|^2)^2 (1+\left|z\_3\right|^2)^2. \tag{90}$$

These coherent states comply at least two requirements: continuity of labeling and resolution of unity. In relation to this latter property, we add

$$
\int d\Gamma |z\_1 z\_2 z\_3\rangle \langle z\_1 z\_2 z\_3| = 1,\tag{91}
$$

where the measure of integration dΓ is given by [54]

$$\mathrm{d}\Gamma = \mathrm{d}\tau \left\{ 4[(1+|z\_1|^2)(1+|z\_3|^2)]^4 |z\_2|^4 - 8[(1+|z\_1|^2)(1+|z\_3|^2)]^2 |z\_2|^2 + 1 \right\} \tag{92}$$

with

$$\mathbf{d}\mathbf{\bar{\sigma}} = \frac{\mathbf{d}^2 z\_1}{\pi} \frac{\mathbf{d}^2 z\_2}{\pi} \frac{\mathbf{d}^2 z\_3}{\pi}. \tag{93}$$

In accordance with this requirement on coherent states, we can assert that the present formulation satisfy the weaker version, because the measure is non-positive definite [54].

#### *4.2.2. Husimi function, Wehrl entropy*

In order to get a valid expression for the Husimi distribution and the Wehrl entropy, a proper formulation of coherent states is essential. Using now Eq. (89) we find

$$\left| \langle IMK \vert z1\ z2z3 \rangle \right|^2 = \frac{e^{-|\mu|^2}}{X\_{I,M,K}^2} \left[ (2I)! \right]^2 \left| z1 \right|^{2(I+M)} \left| z2 \right|^{2I} \left| z3 \right|^{2(I+K)}.\tag{94}$$

Therefore, the rotational partition function is given by

$$Z\_{\mathfrak{3}D} = \sum\_{I=0}^{\infty} \sum\_{K=-I}^{I} \sum\_{M=-I}^{I} e^{-\beta \varepsilon\_{I,K}},\tag{95}$$

i.e.,

$$Z\_{3D} = \sum\_{I=0}^{\infty} \left(2I + 1\right) e^{-I\left(I + 1\right)\frac{\Theta}{T}} \sum\_{K=-I}^{I} e^{-\left(\frac{l\_W}{T\_\natural} - 1\right)K^2 \frac{\Theta}{T}}.\tag{96}$$

We see that *Z*2*<sup>D</sup>* is recovered from *Z*3*<sup>D</sup>* for the limiting case defined as the extremely prolate. The Husimi distribution yields

$$\mu(z\_1, z\_2, z\_3) = \frac{e^{-|\boldsymbol{\mu}|^2}}{Z\_{\mathfrak{M}}} \sum\_{I=0}^{\infty} \frac{(2I)!}{I!} |\boldsymbol{\nu}|^{2I} e^{-I(I+1)\frac{\mathfrak{G}}{\mathfrak{T}}} \times \operatorname{g}(I),\tag{97}$$

where

$$\log(I) = \sum\_{K=-I}^{I} \frac{|\mathbb{Z}\_{\mathfrak{I}}|^{2(I+K)}}{(I+K)!(I-K)!} e^{-\left(\frac{l\upsilon}{l\_{\mathbb{C}}}-1\right)K^{2}\frac{\varTheta}{T}},\tag{98}$$

with

$$|\nu|^2 = (1 + |z\_1|^2)^2 |z\_2|^2,\tag{99}$$

$$|\mu|^2 = |\nu|^2 (1 + |z\_3|^2)^2. \tag{100}$$

Other relevant property that it is easily verified for *µ*(*z*1,*z*2,*z*3) is normalization in the fashion

$$\int \mathrm{d}\Gamma \,\mu(z\_1, z\_2, z\_3) = 1. \tag{101}$$

Now, we obtain the Wehrl entropy in the form

$$\mathcal{W} = \int \mathrm{d}\Gamma \,\mu(z\_1, z\_2, z\_3) \, \ln \mu(z\_1, z\_2, z\_3) . \tag{102}$$

The spherical rotator, that corresponds to another special case, we explicitly obtain

$$\mu(z\_1, z\_2, z\_3) = e^{-|\mu|^2} \frac{\sum\_{I=0}^{\infty} \frac{|u|^{2I}}{I!} e^{-I(I+1)\frac{\Theta}{\overline{\Phi}}}}{\sum\_{I=0}^{\infty} (2I+1)^2 e^{-I(I+1)\frac{\Theta}{\overline{\Phi}}}}. \tag{103}$$

Having the Husimi functions the Wehrl entropy is straightforwardly computed.

In order to emphasize some special cases associated to possible applications we consider several possibilities.


#### *4.2.3. Fisher information measure*

20 ime knjige

i.e.,

where

with

possibilities.

*Ixy*/*Iz* → ∞ (*e.g. CO*2, *C*2*H*2).

Husimi distribution yields

*Z*3*<sup>D</sup>* =

∞ ∑ *I*=0

*<sup>µ</sup>*(*z*1,*z*2,*z*3) = *<sup>e</sup>*−|*u*<sup>|</sup>

*g*(*I*) =

Now, we obtain the Wehrl entropy in the form

(2*<sup>I</sup>* <sup>+</sup>1) *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

2

∞ ∑ *I*=0


<sup>2</sup> = (1+|*z*1<sup>|</sup>

Other relevant property that it is easily verified for *µ*(*z*1,*z*2,*z*3) is normalization in the fashion

*Z*3*<sup>D</sup>*

*I* ∑ *K*=−*I*



The spherical rotator, that corresponds to another special case, we explicitly obtain

<sup>2</sup> ∑<sup>∞</sup> *I*=0 |*u*| 2*I <sup>I</sup>*! *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

∑∞

In order to emphasize some special cases associated to possible applications we consider several

4. The extremely prolate rotator is equivalent to the linear case (all diatomic molecules, *Iz* = 0, this is

*W* = 

*µ*(*z*1,*z*2,*z*3) = *e*−|*u*<sup>|</sup>

Having the Husimi functions the Wehrl entropy is straightforwardly computed.

2. The oblate rotator *Ixy* = *Ix* = *Iy* < *Iz*, being 1/2 ≤ *Ixy*/*Iz* < 1 (*e.g. C*6*H*6).

3. The prolate rotator *Ixy* = *Ix* = *Iy* > *Iz*, thus *Ixy*/*Iz* > 1 (*e.g. PCl*5).

1. The spherical rotator *Ixy* = *Ix* = *Iy* = *Iz*, which corresponds to *Ixy*/*Iz* = 1 (*e.g. CH*4).

*T I* ∑ *K*=−*I e* − *Ixy Iz* −1 *K*<sup>2</sup> <sup>Θ</sup>

We see that *Z*2*<sup>D</sup>* is recovered from *Z*3*<sup>D</sup>* for the limiting case defined as the extremely prolate. The

(2*I*)! *<sup>I</sup>*! <sup>|</sup>*v*<sup>|</sup>

<sup>2</sup>*<sup>I</sup> <sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

*e* − *Ixy Iz* −1 *K*<sup>2</sup> <sup>Θ</sup>

<sup>2</sup>)2|*z*2<sup>|</sup>

<sup>2</sup>(1+|*z*3<sup>|</sup>

*<sup>T</sup>* . (96)

*<sup>T</sup>* ×*g*(*I*), (97)

*<sup>T</sup>* , (98)

. (103)

2, (99)

<sup>2</sup>)2. (100)

dΓ*µ*(*z*1,*z*2,*z*3) = 1. (101)

dΓ*µ*(*z*1,*z*2,*z*3) ln*µ*(*z*1,*z*2,*z*3). (102)

*T*

*T*

*<sup>I</sup>*=<sup>0</sup> (2*<sup>I</sup>* <sup>+</sup>1)<sup>2</sup> *<sup>e</sup>*−*I*(*I*+1) <sup>Θ</sup>

In this circumstance we define the shift invariant Fisher measure in 3*D*−dimensions as

$$\mathcal{F}\_{\mathfrak{3}D} = \frac{1}{4} \int \mathrm{d}\Gamma \,\mu(z\_1, z\_2, z\_3) \left( \frac{\partial \ln \mu(z\_1, z\_2, z\_3)}{\partial |u|} \right)^2 \,. \tag{104}$$

Thus, from Eq. (97) we get

$$\Phi(z\_1, z\_2) = \frac{1}{2} \frac{\partial \ln \mu(z\_1, z\_2)}{\partial |u|} = \frac{\sum\_{I=0}^{\infty} \left[ \frac{|u|^{2l-1}}{(I-1)!} - \frac{|u|^{2l+1}}{(I)!} \right] e^{-I(I+1)\Theta/T}}{\sum\_{I=0}^{\infty} \frac{|u|^{2l}}{(I)!} e^{-I(I+1)\Theta/T}},\tag{105}$$

and, the corresponding Fisher measure can be expressed as

$$\mathcal{F}\_{\mathfrak{3}D} = \int \mathrm{d}\Gamma \,\mu(z\_1, z\_2) \, \mathfrak{g}(z\_1, z\_2, z\_3)^2 = \langle \mathfrak{g}(z\_1, z\_2, z\_3)^2 \rangle. \tag{106}$$

#### 5. Final remarks

In this chapter, we have described some elements to motivate possible and future applications in condensed matter and information theory. Our fundamental discussion is devoted to two interesting systems, those are: the Landau diamagnetism and the rigid rotator in three dimensions. We choose these systems because the quantum mechanics is analytically solved. Specifically, the spectrum and a suitable formulation of coherent states are known without approximations.

In general, quantum distributions as the Husimi distribution, have long been seen as powerful tools for studying the quantum-classical correspondence and semi-classical aspects of quantum mechanics. Then, a crucial starting point in the present strategy, to evaluate some theoretical measures, is to get the Husimi distribution. This is made evoking a convenient set of coherent states in every system. As introduced by Gazeau and Klauder in the context of the harmonic oscillator, we use the same formal perspective of general requirements for formulations of coherent states that we use in the current contribution. Additionally, we have included some mathematical and practical details of the the present formalisms in order to make it instructive in courses of quantum mechanics (for graduates) and easy to apply to specific calculations of theoretical measures.

The present derivation of Husimi distributions is based on the evaluation of the mean value of the density operator in the basis of a single-particle coherent state. Then, after defining the Husimi distribution we are ready to make a possible semiclassical description evaluating (i) the semiclassical Wehrl entropy and (ii) the phase-space location via measures as Fisher information.

Furthermore, we evaluate the probability of observing a quantum state in a coherent state, by projecting the quantum states over the coherent states, as a function of a variable related to the coherent states. We see that the localization of probability and correspondingly the Husimi distribution in the phase space decreases as temperature increases.

As known, while the coherent states are independent-particle states, the Husimi function takes into account collective and environmental effects being necessary many wave packets of independent-particle states to represent them. Furthermore, the thermodynamics of particles in systems does not depend on any coherent states formulation.

Finally, we remark, all results presented here were kindly obtained in an analytical fashion. We show some instances where the Landau diamagnetism is equivalent to the harmonic oscillator and, in the other example, where the linear rigid rotator is reobtained as a particular instance of the formulation in three dimensions. Some indications given in the present work lead to the conclusion that Fisher measure is a better indicator of the delocalization than Wehrl entropy.

We acknowledge partial financial support by FONDECYT 1110827 and CONICYT PSD065.

#### Author details

Sergio Curilef1 and Flavia Pennini1,2

1 Departamento de Física, Universidad Católica del Norte, Antofagasta, Chile

2 Instituto de Física La Plata–CCT-CONICET, Fac. de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, Argentina

#### References


DOI: 10.5772/53846. Available from: http://www.intechopen.com/books/ advances-in-quantum-mechanics/the-husimi-distribution-development-and-applications


22 ime knjige

Author details

References

1990).

439.

Sergio Curilef1 and Flavia Pennini1,2

de La Plata, La Plata, Argentina

independent-particle states to represent them. Furthermore, the thermodynamics of particles in systems

Finally, we remark, all results presented here were kindly obtained in an analytical fashion. We show some instances where the Landau diamagnetism is equivalent to the harmonic oscillator and, in the other example, where the linear rigid rotator is reobtained as a particular instance of the formulation in three dimensions. Some indications given in the present work lead to the conclusion that Fisher measure is a

2 Instituto de Física La Plata–CCT-CONICET, Fac. de Ciencias Exactas, Universidad Nacional

[3] M.C. Gutzwiller, *Chaos in Classical and Quantum Mechanics*, (Springer-Verlag, New York,

[6] F. Olivares, F. Pennini, G.L. Ferri, A. Plastino, Brazilian Journal of Physics 39, 2A (2009).

[7] S. Curilef, F. Pennini, A. Plastino and G.L. Ferri, *J. Phys. A: Math. Theor.* 40 (2007) 5127.

[9] D. Herrera, A.M. Valencia, F. Pennini and S. Curilef, *European Journal of Physics* 29 (2008)

[10] M. Janssen, *Fluctuations and Localization in Mesoscopic Electron Systems*, World Scientific

[13] S. Curilef and F. Pennini (2013). The Husimi Distribution: Development and Applications, Advances in Quantum Mechanics, Prof. Paul Bracken (Ed.), ISBN: 978-953-51-1089-7, InTech,

[11] W.P. Scheleich, *Quantum Optics in phase space*, (Wiley VCH-Verlag, Berlin, 2001).

[12] J.P. Gazeau and J.R. Klauder, *Journal of Physics A: Math Gen.* 32 (1999) 123.

[8] S. Curilef, F. Pennini and A. Plastino, *Physical Review* B 71 (2005) 024420.

[1] K. Husimi *Proceedings of the Physico-Mathematical Society of Japan* 22 (1940) 264.

We acknowledge partial financial support by FONDECYT 1110827 and CONICYT PSD065.

1 Departamento de Física, Universidad Católica del Norte, Antofagasta, Chile

[2] K. Takahashi and N. Saitô, *Physical Review Letters* 55 (1985) 645.

[4] A. Anderson and J.J. Halliwell, *Physical Review* D 48 (1993) 2753.

[5] F. Pennini and A. Plastino, *Physical Review* E 69 (2004) 057101.

Lecture Notes in Physics Vol. 64., 2001

does not depend on any coherent states formulation.

better indicator of the delocalization than Wehrl entropy.

