**The Theoretical Ramifications of the Computational Unified Field Theory**

Jonathan Bentwich

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53804

### **1. Introduction**

Four previous articles (Bentwich, 2012: a-d) have postulated the existence of a novel 'Compu‐ tational Unified Field Theory' (CUFT) which is a candidate 'Theory of Everything' (Brumfiel, 2006; Ellis, 1986; Greene, 2003) – i.e., has the potential of unifying between quantum (Born, 1954) and relativistic models of physical reality (and moreover also possesses the potential of opening new 'vistas' for scientific examination connected with its discovery of a new hypo‐ thetical 'Universal Computational Principle which carries out an extremely rapid computation, c2 /h of a series of Universal Simultaneous Computational Frames, 'USCF's, which give rise to all 'apparent' secondary computational 'physical' properties of 'space', 'time', 'energy' and 'mass'); Indeed, the primary focus of the current manuscript is precisely to explore the potential theoretical ramifications of this novel CUFT – based on the recognition that the (singular) Universal Computational Principle ('י ('solely produces all (apparent) secondary computa‐ tional 'physical' properties of 'space, 'time', 'energy' and 'mass', and hence comprises the sole "reality" (which produces all exhaustive hypothetical inductive and deductive phenomenon through a higher-ordered 'a-causal' computational framework; this may subsequently bear significant theoretical ramifications for all (key) 'material-causal' scientific paradigms as well as point at the discovery of a (new) *'Universal Consciousness Principle Computational Program'*, as well as outline potential resolutions of major Physical 'enigma's;

Hence, the current manuscript traces the potential theoretical ramifications of:

**a.** An 'a-causal' computational framework of the (CUFT's) singular Universal Consciousness Principle's ('י ('responsible for the (higher-ordered) computation of all exhaustive hypo‐ thetical (e.g., empirically knowable) inductive or deductive 'x-y' pairs series – which leads to the discovery of a-*causal 'Universal Consciousness Principle Computational Program'*.

© 2013 Bentwich; licensee InTech. This is an open access article 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. © 2013 Bentwich; licensee InTech. This is a paper 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.

**b.** An exploration of the CUFT's Universal Consciousness Principle's ('י ('and Duality Principle's (Bentwich, 2003c, 2004, 2006) reformalization of all (apparent inductive or deductive) major SROCS computational paradigms (e.g., including: Darwin's 'Natural Selection Principle' (Darwin, 1859) and associated Genetic Encoding hypothesis, Neuro‐ science's Psychophysical Problem of human Consciousness and all inductive and deduc‐ tive Gödel-like SROCS paradigms).

inductive or deductive ('x-y') phenomenon, we must reformulate our scientific understanding in such a way which will allow us to present any such 'x-y' relationship/s as being computed by the singular Universal Consciousness Principle (e.g., as the computation of an exhaustivehypothetical "co-occurring" 'x-y' pairs' series); In that respect, this (novel) 'Universal Con‐ sciousness Principle's' scientific framework shifts Science from its current basic (Cartesian) assumption wherein all natural phenomena can be described as 'material-causal' ('x→y') relationships (e.g., comprising the apparent SROCS computational structure contradicted by the computational Duality Principle) – to an 'a-causal' singular Universal Consciousness Principle which computes the simultaneous "co-occurrences" of any inductive or deductive 'xy' pairs series comprising the various 'pixels' of the USCF's frames (e.g., produced by this

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Finally, it should be noted that a key principle underlying this shift from the current 'materialcausal' (Cartesian) scientific framework towards the CUFT's (proven) higher-ordered singular Universal Consciousness Principle's ('י' ('a-causal' theoretical framework is the acceptance of the impossibility of the existence of any such 'material-causal' ('x-y') relationship/s – i.e., due to the impossibility of any 'physical' entity, attribute (or property) being transferred across any (two subsequent) 'USCF's frames: Thus, apart from the (previously shown) conceptual computational proof of the 'Duality Principle' wherein due to the inevitable 'logical inconsis‐ tency' and 'computational indeterminacy' arising from the SROCS computational structure (which is contradicted by empirical evidence indicating the capacity of these key scientific SROCS paradigms to compute the "existence" or "non-existence" of any particular 'y' element or value) – which points at the existence of the higher-ordered (singular) 'Universal Compu‐ tational/Consciousness Principle that computes the "simultaneous co-occurrences" of any (exhaustive-hypothetical) 'x-y' pairs' series; it is suggested that the inclusion of this computa‐ tional Duality Principle as one of the (seven) theoretical postulates of the CUFT (e.g., specifi‐ cally alongside the CUFT's 'Computational Invariance' and 'Universal Consciousness' postulates) unequivocally asserts that there cannot (in principle) exist any 'material-causal' effect/s (or relationship/s) being transferred across any (two subsequent) USCF's frames! This is because the CUFT's very definition of all four 'physical' properties of 'space', 'time', 'energy' and 'mass' – as secondary computational by-products of the (singular) Universal Computa‐ tional Consciousness' computation of (an extremely rapid series of) 'Universal Simultaneous Computational Frames' (USCF's); and moreover the CUFT's 'Computational Invariance' postulate indication that due to the 'computational variance' of these four (secondary compu‐ tational) 'physical' properties (e.g., as existing only "during" the appearance of the USCF frames but 'non-existence' "in-between" any two such subsequent frames, see Bentwich, 2012:c-d) as opposed to the 'computational invariance' of the 'Universal Consciousness Principle' ('י(', we need to regard only this singular (computationally invariant) 'Universal Consciousness Principle' as "real" whereas all four (secondary computationally variant) 'physical' properties must be regarded as merely 'phenomenal' (i.e., as being comprised in reality only from the singular Universal Consciousness Principle); Therefore, the CUFT's 'Universal Consciousness Principle' advocated that none of these four (secondary computationally variant) 'physical' properties (e.g., of 'space', 'time', 'energy' or 'mass') "really" exists – but rather that there is only this one singular Universal Consciousness Principle which exists (solely) "in-between" any

Universal Consciousness Principle).

**c.** Theoretical Ramifications of the Universal Consciousness Principle.

### **2. A singular 'a-causal' universal consciousness principle computation of all inductive and deductive 'x-y' relationships**

We thus begin with an exploration of three potential theoretical ramifications of the CUFT's description of the operation of the (singular) Universal Consciousness Principle ('י ('which has been shown to compute an extremely rapid series of Universal Simultaneous Computational Frames (USCF's);

The Universal Computational/Consciousness Principle was (previously) shown to encapsulate a singular higher-ordered 'D2' computation of an 'a-causal' computation of the "simultaneous co-occurrences" of all exhaustive hypothetical inductive or deductive (e.g., empirically knowable) 'x-y' pairs series; Therefore, the acceptance of the CUFT's description of the Universal Consciousness Principle necessarily implies that throughout the various (inductive or deductive) disciplines of Science we need to shift from the current basic (Cartesian) "material-causal" scientific theoretical towards a singular (higher-ordered 'D2') *'Universal Consciousness Principle's a-causal computation'*:

This means that the current (Cartesian) 'material-causal' scientific framework assumes that any given'y'element(orvalue)canbeexplainedasaresultofits(directorindirect)'causal'interaction/ s with another (exhaustive hypothetical inductive or deductive) series of 'x' factor/s – which determines whether that 'y' element (or value) "exists" or "doesn't exist", thereby comprising a 'Self-Referential Ontological Computational System' (SROCS) (Bentwich: 2012a-d):

SROCS: PR{x,y}→ ['y' or 'not y']/di1…din.

But, since it was previously shown that such SROCS computational structure inevitably leads to both 'logical inconsistency' and 'computational indeterminacy' that were shown to be contradicted by robust empirical findings indicating the capacity of the major scientific SROCS paradigms to be capable of determining the "existence" or "non-existence" of the particular 'y' element, see Bentwich 2012b) – then the CUFT's 'Duality Principle' asserted the existence of the singular 'Universal Consciousness Principle' ('י ('which is capable of computing the "simultaneous co-occurrences" of any particular (exhaustive hypothetical) 'x-y' pairs series which are embedded within the Universal Computational/Consciousness Principle's rapid series of USCF's.

What this means is that both specifically for each of the (previously identified) key scientific SROCS paradigms as well as more generally for any hypothetical ('empirically knowable') inductive or deductive ('x-y') phenomenon, we must reformulate our scientific understanding in such a way which will allow us to present any such 'x-y' relationship/s as being computed by the singular Universal Consciousness Principle (e.g., as the computation of an exhaustivehypothetical "co-occurring" 'x-y' pairs' series); In that respect, this (novel) 'Universal Con‐ sciousness Principle's' scientific framework shifts Science from its current basic (Cartesian) assumption wherein all natural phenomena can be described as 'material-causal' ('x→y') relationships (e.g., comprising the apparent SROCS computational structure contradicted by the computational Duality Principle) – to an 'a-causal' singular Universal Consciousness Principle which computes the simultaneous "co-occurrences" of any inductive or deductive 'xy' pairs series comprising the various 'pixels' of the USCF's frames (e.g., produced by this Universal Consciousness Principle).

**b.** An exploration of the CUFT's Universal Consciousness Principle's ('י ('and Duality Principle's (Bentwich, 2003c, 2004, 2006) reformalization of all (apparent inductive or deductive) major SROCS computational paradigms (e.g., including: Darwin's 'Natural Selection Principle' (Darwin, 1859) and associated Genetic Encoding hypothesis, Neuro‐ science's Psychophysical Problem of human Consciousness and all inductive and deduc‐

**2. A singular 'a-causal' universal consciousness principle computation of**

We thus begin with an exploration of three potential theoretical ramifications of the CUFT's description of the operation of the (singular) Universal Consciousness Principle ('י ('which has been shown to compute an extremely rapid series of Universal Simultaneous Computational

The Universal Computational/Consciousness Principle was (previously) shown to encapsulate a singular higher-ordered 'D2' computation of an 'a-causal' computation of the "simultaneous co-occurrences" of all exhaustive hypothetical inductive or deductive (e.g., empirically knowable) 'x-y' pairs series; Therefore, the acceptance of the CUFT's description of the Universal Consciousness Principle necessarily implies that throughout the various (inductive or deductive) disciplines of Science we need to shift from the current basic (Cartesian) "material-causal" scientific theoretical towards a singular (higher-ordered 'D2') *'Universal*

This means that the current (Cartesian) 'material-causal' scientific framework assumes that any given'y'element(orvalue)canbeexplainedasaresultofits(directorindirect)'causal'interaction/ s with another (exhaustive hypothetical inductive or deductive) series of 'x' factor/s – which determines whether that 'y' element (or value) "exists" or "doesn't exist", thereby comprising a

But, since it was previously shown that such SROCS computational structure inevitably leads to both 'logical inconsistency' and 'computational indeterminacy' that were shown to be contradicted by robust empirical findings indicating the capacity of the major scientific SROCS paradigms to be capable of determining the "existence" or "non-existence" of the particular 'y' element, see Bentwich 2012b) – then the CUFT's 'Duality Principle' asserted the existence of the singular 'Universal Consciousness Principle' ('י ('which is capable of computing the "simultaneous co-occurrences" of any particular (exhaustive hypothetical) 'x-y' pairs series which are embedded within the Universal Computational/Consciousness Principle's rapid

What this means is that both specifically for each of the (previously identified) key scientific SROCS paradigms as well as more generally for any hypothetical ('empirically knowable')

'Self-Referential Ontological Computational System' (SROCS) (Bentwich: 2012a-d):

**c.** Theoretical Ramifications of the Universal Consciousness Principle.

**all inductive and deductive 'x-y' relationships**

tive Gödel-like SROCS paradigms).

*Consciousness Principle's a-causal computation'*:

SROCS: PR{x,y}→ ['y' or 'not y']/di1…din.

Frames (USCF's);

672 Advances in Quantum Mechanics

series of USCF's.

Finally, it should be noted that a key principle underlying this shift from the current 'materialcausal' (Cartesian) scientific framework towards the CUFT's (proven) higher-ordered singular Universal Consciousness Principle's ('י' ('a-causal' theoretical framework is the acceptance of the impossibility of the existence of any such 'material-causal' ('x-y') relationship/s – i.e., due to the impossibility of any 'physical' entity, attribute (or property) being transferred across any (two subsequent) 'USCF's frames: Thus, apart from the (previously shown) conceptual computational proof of the 'Duality Principle' wherein due to the inevitable 'logical inconsis‐ tency' and 'computational indeterminacy' arising from the SROCS computational structure (which is contradicted by empirical evidence indicating the capacity of these key scientific SROCS paradigms to compute the "existence" or "non-existence" of any particular 'y' element or value) – which points at the existence of the higher-ordered (singular) 'Universal Compu‐ tational/Consciousness Principle that computes the "simultaneous co-occurrences" of any (exhaustive-hypothetical) 'x-y' pairs' series; it is suggested that the inclusion of this computa‐ tional Duality Principle as one of the (seven) theoretical postulates of the CUFT (e.g., specifi‐ cally alongside the CUFT's 'Computational Invariance' and 'Universal Consciousness' postulates) unequivocally asserts that there cannot (in principle) exist any 'material-causal' effect/s (or relationship/s) being transferred across any (two subsequent) USCF's frames! This is because the CUFT's very definition of all four 'physical' properties of 'space', 'time', 'energy' and 'mass' – as secondary computational by-products of the (singular) Universal Computa‐ tional Consciousness' computation of (an extremely rapid series of) 'Universal Simultaneous Computational Frames' (USCF's); and moreover the CUFT's 'Computational Invariance' postulate indication that due to the 'computational variance' of these four (secondary compu‐ tational) 'physical' properties (e.g., as existing only "during" the appearance of the USCF frames but 'non-existence' "in-between" any two such subsequent frames, see Bentwich, 2012:c-d) as opposed to the 'computational invariance' of the 'Universal Consciousness Principle' ('י(', we need to regard only this singular (computationally invariant) 'Universal Consciousness Principle' as "real" whereas all four (secondary computationally variant) 'physical' properties must be regarded as merely 'phenomenal' (i.e., as being comprised in reality only from the singular Universal Consciousness Principle); Therefore, the CUFT's 'Universal Consciousness Principle' advocated that none of these four (secondary computationally variant) 'physical' properties (e.g., of 'space', 'time', 'energy' or 'mass') "really" exists – but rather that there is only this one singular Universal Consciousness Principle which exists (solely) "in-between" any (two subsequent) USCF's frames and also solely produces each of these USCF's derived four 'phenomenal physical' properties; Hence, it was evinced (by the CUFT's Universal Conscious‐ ness Principle) that there cannot be any 'transference' of any hypothetical 'material' or 'physical' entity, effect, or property across any (two subsequent) USCF's frames! We therefore reach the inevitable theoretical conclusion that the current scientific (Cartesian) ''material-causality' basic assumption underlying all key scientific SROCS paradigms as well as all (empirically knowable) 'Gödel-like' (inductive or deductive) SROCS 'x-y' relationships, wherein there exists a 'material-causal' effect/s (or relationship/s) between any given 'x' element and any (exhaus‐ tive hypothetical) 'y' series which determines the "existence" or "non-existence" of that (particular) 'y' element (or value) – is untenable! Instead, we must accept the CUFT's assertion that there can only exists one singular 'Universal Consciousness Principle' ('י ('which both (solely) produces- all (apparent) secondary computational 'physical' properties (of 'space', 'time', 'energy' and 'mass'), as well as computes the "simultaneous co-occurrences" of any (particular) exhaustive-hypothetical inductive or deductive 'x-y' pairs series (e.g., comprising the exhaustive USCF's frames).

Functional: **D2**: [{Cs(pp)fi

Phen.: **D2**: [{Cs(pp- fi)-Phi

Program";

Self: **D2**: [{Cs(pp- fi)Ph-Si, Na(pp- fi)Ph-Si

, Na(spp)fi}st-i ; … {Cs(pp)f(i+n), Na(spp)f(i+n)} st(i+n)]

Indeed, what may be seen from this singular description of all of these key scientific SROCS paradigms, is that it recognize the fact that all of these major (apparent) SROCS paradigms are computed simultaneously as different "co-occurring" 'x-y' pairs embedded within the same (single or multiple) USCF frame that is produced by the singular Universal Consciousness Principle ('י ;('What this means is that the recognition of the singularity of this Universal Consciousness Principle as the sole "reality" which computes the "simultaneous co-occurrenc‐ es" of all of these (particular) exhaustive hypothetical 'x-y' pairs series, and which also exists (solely) "in-between" any two such USCF's – forces us to transcend the 'narrow constraints' of the (current) Cartesian 'material-causal' theoretical framework (e.g., which assumes that any given 'y' entity (or phenomenon) is "caused" by its (direct or indirect) physical interaction/s with (an exhaustive hypothetical 'x' series); Instead, this singular Universal Consciousness Principle 'a-causal' computation asserts that it is the same singular Universal Consciousness Principle which computes- produces- retains- and evolves- all of these particular scientific

In other words, instead of the existence of any "real" material-causal relationship between any of these (particular SROCS) 'x→y' entities (e.g., Darwin's Natural Selection Principle's assumed 'material-causal' relationship between an organism's Environmental Factors, 'x', and own traits or behavior 'y'; or between any exhaustive hypothetical Genetic Factors and any given phenotypic behavior; or between Neuroscience's Psychophysical Problem of Human Con‐ sciousness' psychophysical stimulation, 'x', and Neural Activation, 'y'; or in fact between any hypothetical inductive or deductive Gödel-like SROCS 'x-y' factors); the CUFT's Universal Consciousness Principle offers an alternative singular (higher-ordered) computational mechanism which computes the "simultaneous co-occurrences" of any of these (exhaustive hypothetical) 'x-y' pairs' series – which are all produced- and embedded- within the Universal Consciousness Principle's computed USCF's frames… Indeed, the shift from the current 'material-causal' (Cartesian) scientific framework towards the Universal Consciousness Principle's singular computation of the "simultaneous co-occurrences" of all exhaustive hypothetical (inductive or deductive) 'x-y' pairs' series may lead the way for reformulating all of these key scientific SROCS paradigms (as well as any other hypothetical inductive or deductive 'x-y' series) within a basic "Universal Consciousness Principle Computational

Essentially, such a *'Universal Consciousness Principle's Computational Program'* is based upon the foundations of the CUFT's (abovementioned) three postulates of the 'Duality Principle', the 'Computational Invariance' principle and the 'Universal Consciousness Principle' – all pointing at the fact that all empirically computable (inductive or deductive) 'x-y' relationships must necessarily be based upon the singular (conceptually higher-ordered) Universal Con‐ sciousness Principle which is solely responsible for the computation of the "simultaneous co-

} st-i; …{Cs(pp- fi)-Ph(i+n), Na(spp-fi)-Ph} st-(i+n)]

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675

} st-i ; …{Cs(pp- fi)Ph-S(i+n), Na(pp- fi)Ph-S( i+n)} st-(i+n)]

, Na(spp-fi)-Phi

*GIT:***D2**: ([{S{1...n}, t}*i …* {S{1...n}, t}*z*], or [{x{1...n}*i*, y*i*} *…* {x{1...n}*z*, y*z*}])

(apparent) SROCS 'x-y' pairs series across a series of USCF's…

### **3. The "universal consciousness principle's computational program"**

Therefore, it follows that based on the recognition of the singularity of the Universal Con‐ sciousness Principle's 'a-casual' computation of the "simultaneous co-occurrences" of all (inductive or deductive) 'x-y' pairs' series (as comprising the exhaustive USCF's frames) – we need to be able to reformulate all of the previously mentioned key scientific SROCS paradigms (Bentwich, 2012b-d), including: Darwin's 'Natural Selection Principle' and associated 'Genetic Encoding' hypothesis, Neuroscience's Psychophysical Problem of human Consciousness, as well as all (exhaustive hypothetical) 'Gödel-like' (apparent) inductive or deductive SROCS computational paradigms based on this singular (higher-ordered) Universal Consciousness Principle's ('י' ('a-causal' USCF's computation;

Hence, what follows is a description of the principle theoretical ramifications of reformulating each of these key scientific (apparent) SROCS computational paradigms, as well as a more generalized description of a tentative 'Universal Consciousness Principle Program' (e.g., which may offer a successful alternative for 'Hilbert's Mathematical Program' to base all of our human scientific knowledge upon the foundations of the operation of the singular Universal Con‐ sciousness Principle). First, it may be worthwhile to rearticulate the reformalization of each of these key scientific (apparent) SROCS paradigms in terms of the operation of the singular Universal Consciousness Principle (as previously outlined: Bentwich, 2012b):

*S.:***D2**: [{E{1...n}, o}st1; {E{1...n}, o}st2... {E{1...n}, o}stn].

*G.F – P.S.:***D2**: [{G{1...n}, 'ph*i (o)*' }st1; {G{1...n}, 'ph*j (o)*' }sti;...{G{1...n}, 'ph*n(o)*' }stn].

*G.E. – P.S.:***D2**: [{G*e*{1...n}, p*i*-*synth* (o-phi)}st1; G*e*{1...n}, p*j*-*synth* (o-phi)}sti… ; G*e*{1...n}, p*nsynth* (o-phi)}stn]

*Psychophysical:***D2**: [{N(1…n) st-i, Cs-pp st-i}; … {N(1…n) st-i+n, Cs-pp st-i+n }]

Functional: **D2**: [{Cs(pp)fi , Na(spp)fi}st-i ; … {Cs(pp)f(i+n), Na(spp)f(i+n)} st(i+n)] Phen.: **D2**: [{Cs(pp- fi)-Phi , Na(spp-fi)-Phi } st-i; …{Cs(pp- fi)-Ph(i+n), Na(spp-fi)-Ph} st-(i+n)] Self: **D2**: [{Cs(pp- fi)Ph-Si, Na(pp- fi)Ph-Si } st-i ; …{Cs(pp- fi)Ph-S(i+n), Na(pp- fi)Ph-S( i+n)} st-(i+n)] *GIT:***D2**: ([{S{1...n}, t}*i …* {S{1...n}, t}*z*], or [{x{1...n}*i*, y*i*} *…* {x{1...n}*z*, y*z*}])

(two subsequent) USCF's frames and also solely produces each of these USCF's derived four 'phenomenal physical' properties; Hence, it was evinced (by the CUFT's Universal Conscious‐ ness Principle) that there cannot be any 'transference' of any hypothetical 'material' or 'physical' entity, effect, or property across any (two subsequent) USCF's frames! We therefore reach the inevitable theoretical conclusion that the current scientific (Cartesian) ''material-causality' basic assumption underlying all key scientific SROCS paradigms as well as all (empirically knowable) 'Gödel-like' (inductive or deductive) SROCS 'x-y' relationships, wherein there exists a 'material-causal' effect/s (or relationship/s) between any given 'x' element and any (exhaus‐ tive hypothetical) 'y' series which determines the "existence" or "non-existence" of that (particular) 'y' element (or value) – is untenable! Instead, we must accept the CUFT's assertion that there can only exists one singular 'Universal Consciousness Principle' ('י ('which both (solely) produces- all (apparent) secondary computational 'physical' properties (of 'space', 'time', 'energy' and 'mass'), as well as computes the "simultaneous co-occurrences" of any (particular) exhaustive-hypothetical inductive or deductive 'x-y' pairs series (e.g., comprising

**3. The "universal consciousness principle's computational program"**

Therefore, it follows that based on the recognition of the singularity of the Universal Con‐ sciousness Principle's 'a-casual' computation of the "simultaneous co-occurrences" of all (inductive or deductive) 'x-y' pairs' series (as comprising the exhaustive USCF's frames) – we need to be able to reformulate all of the previously mentioned key scientific SROCS paradigms (Bentwich, 2012b-d), including: Darwin's 'Natural Selection Principle' and associated 'Genetic Encoding' hypothesis, Neuroscience's Psychophysical Problem of human Consciousness, as well as all (exhaustive hypothetical) 'Gödel-like' (apparent) inductive or deductive SROCS computational paradigms based on this singular (higher-ordered) Universal Consciousness

Hence, what follows is a description of the principle theoretical ramifications of reformulating each of these key scientific (apparent) SROCS computational paradigms, as well as a more generalized description of a tentative 'Universal Consciousness Principle Program' (e.g., which may offer a successful alternative for 'Hilbert's Mathematical Program' to base all of our human scientific knowledge upon the foundations of the operation of the singular Universal Con‐ sciousness Principle). First, it may be worthwhile to rearticulate the reformalization of each of these key scientific (apparent) SROCS paradigms in terms of the operation of the singular

Universal Consciousness Principle (as previously outlined: Bentwich, 2012b):

*Psychophysical:***D2**: [{N(1…n) st-i, Cs-pp st-i}; … {N(1…n) st-i+n, Cs-pp st-i+n }]

*G.F – P.S.:***D2**: [{G{1...n}, 'ph*i (o)*' }st1; {G{1...n}, 'ph*j (o)*' }sti;...{G{1...n}, 'ph*n(o)*' }stn].

*G.E. – P.S.:***D2**: [{G*e*{1...n}, p*i*-*synth* (o-phi)}st1; G*e*{1...n}, p*j*-*synth* (o-phi)}sti… ; G*e*{1...n}, p*n*-

the exhaustive USCF's frames).

674 Advances in Quantum Mechanics

Principle's ('י' ('a-causal' USCF's computation;

*S.:***D2**: [{E{1...n}, o}st1; {E{1...n}, o}st2... {E{1...n}, o}stn].

*synth* (o-phi)}stn]

Indeed, what may be seen from this singular description of all of these key scientific SROCS paradigms, is that it recognize the fact that all of these major (apparent) SROCS paradigms are computed simultaneously as different "co-occurring" 'x-y' pairs embedded within the same (single or multiple) USCF frame that is produced by the singular Universal Consciousness Principle ('י ;('What this means is that the recognition of the singularity of this Universal Consciousness Principle as the sole "reality" which computes the "simultaneous co-occurrenc‐ es" of all of these (particular) exhaustive hypothetical 'x-y' pairs series, and which also exists (solely) "in-between" any two such USCF's – forces us to transcend the 'narrow constraints' of the (current) Cartesian 'material-causal' theoretical framework (e.g., which assumes that any given 'y' entity (or phenomenon) is "caused" by its (direct or indirect) physical interaction/s with (an exhaustive hypothetical 'x' series); Instead, this singular Universal Consciousness Principle 'a-causal' computation asserts that it is the same singular Universal Consciousness Principle which computes- produces- retains- and evolves- all of these particular scientific (apparent) SROCS 'x-y' pairs series across a series of USCF's…

In other words, instead of the existence of any "real" material-causal relationship between any of these (particular SROCS) 'x→y' entities (e.g., Darwin's Natural Selection Principle's assumed 'material-causal' relationship between an organism's Environmental Factors, 'x', and own traits or behavior 'y'; or between any exhaustive hypothetical Genetic Factors and any given phenotypic behavior; or between Neuroscience's Psychophysical Problem of Human Con‐ sciousness' psychophysical stimulation, 'x', and Neural Activation, 'y'; or in fact between any hypothetical inductive or deductive Gödel-like SROCS 'x-y' factors); the CUFT's Universal Consciousness Principle offers an alternative singular (higher-ordered) computational mechanism which computes the "simultaneous co-occurrences" of any of these (exhaustive hypothetical) 'x-y' pairs' series – which are all produced- and embedded- within the Universal Consciousness Principle's computed USCF's frames… Indeed, the shift from the current 'material-causal' (Cartesian) scientific framework towards the Universal Consciousness Principle's singular computation of the "simultaneous co-occurrences" of all exhaustive hypothetical (inductive or deductive) 'x-y' pairs' series may lead the way for reformulating all of these key scientific SROCS paradigms (as well as any other hypothetical inductive or deductive 'x-y' series) within a basic "Universal Consciousness Principle Computational Program";

Essentially, such a *'Universal Consciousness Principle's Computational Program'* is based upon the foundations of the CUFT's (abovementioned) three postulates of the 'Duality Principle', the 'Computational Invariance' principle and the 'Universal Consciousness Principle' – all pointing at the fact that all empirically computable (inductive or deductive) 'x-y' relationships must necessarily be based upon the singular (conceptually higher-ordered) Universal Con‐ sciousness Principle which is solely responsible for the computation of the "simultaneous cooccurrences" of all such (exhaustive hypothetical) inductive or deductive 'x-y' pairs series comprising the totality of the USCF's (single or multiple) frames…. Moreover, this singular Universal Consciousness Principle ('י ('was also shown to exist independently of any (secon‐ dary computational) 'physical properties' (e.g., of 'space', 'time', 'energy' and 'mass') and therefore constitute the only "reality" that exists invariantly (i.e., both as giving rise to the four 'phenomenal' physical properties and as existing solely "in-between" any two such subsequent USCF's frames).

means for computing the "simultaneous co-occurrences" of any (exhaustive hypothetical) 'xy' pairs series is carried out by the singular (higher-ordered) Universal Consciousness Principle ('י ...('Moreover, the (generalized format of the) Duality Principle goes farther to state that for all other (exhaustive hypothetical) inductive or deductive computational SROCS paradigms – *for which there exists a proven empirical capacity to determine the values of any particular 'x-y' pairs (e.g., empirically "known" or "knowable" 'x-y' pairs results)*- any of these (hypothetical) scientific SROCS computations must be carried out by the CUFT's identified singular Universal

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The (potential) significance of this generalized assertion made by the Computational Unified Field Theory's (CUFT): 'Duality Principle', 'Computational Invariance' principle and Universal

**a.** First, it narrows down the scope of (inductive or deductive) determinable scientific phenomena – to only those (inductive or deductive) 'x-y' relationships for which there is an empirical capacity to determine their "simultaneously co-occurring" values; essentially the *'Universal Consciousness Principle's Computational Program'* anchors itself in the Duality Principle's focus on only those inductive or deductive 'x-y' relationship/s or phenomenon for which there is an empirically 'known' or 'knowable' capacity to determine these 'x-y' pairs values. It is perhaps important to note (in this context) that all of the 'other' inductive or deductive 'x-y' relationship/s which cannot be (empirically) known – "naturally" lie outside the scope of our human (scientific) knowledge (and therefore should not be included, anyway within the scope of Science)… Nevertheless, the strict limitation imposed by the 'Universal Consciousness Principle Computational Program' – may indeed be significant, as it clearly defines the boundaries of "admissible scientific knowl‐ edge" to only that scientific knowledge which is based on empirically known or knowable results pertaining to the "simultaneous co-occurrences" of any 'x-y' relationship or phenomenon; (Needless to say that the strict insistence of the Universal Consciousness

**b.** Second, based on this strict definition of Science as dealing solely with 'empirical know‐ able' (simultaneously co-occurring) 'x-y' relationship/s or phenomenon – the 'Universal Consciousness Computational Program' may in fact offer a broader alternative to GIT (failing of Hilbert's 'Mathematical Program'); This is because once we accept the Universal Consciousness Principle's Computational Program's (above) strict 'empirical constrains', we are led to the Duality Principle's (generalized) conceptual computational proof that any (exhaustive hypothetical) inductive or deductive scientific SROCS' 'x-y' relationship must be determined by the singular Universal Consciousness Principle ('י ('computation of the "simultaneous co-occurrences" of any (exhaustive hypothetical) 'x-y' pairs series. This means that instead of GIT assertion that it is not possible (in principle) to construct a consistent Logical-Mathematical System which will be capable of computing any mathematical (or scientific) claim or theorem, the Universal Consciousness Computa‐ tional Program asserts that based on a strict definition of Science as dealing solely with empirically knowable 'x-y' relationship/s or phenomenon, we obtain a singular (higherordered) Universal Consciousness Principle which is solely responsible for computing the

Consciousness Principle ('י!('

Consciousness Principle ('י ('is twofold:

Computational Program upon dealing only with

In order to appreciate the full (potential) theoretical significance of such a 'Universal Con‐ sciousness Principle Computational Program' it may be worthwhile to reexamine Hillbert's famous 'Mathematical Program' to base Mathematics upon the foundations of Logic (e.g., and by extension also all of Science upon the foundations of Mathematics and Logic), and more specifically, to revisit 'Gödel's Incompleteness Theorem' (GIT) which delivered a critical blow to Hilbert's 'Mathematical Program'; It is a well-known that Hilbert's Mathematical Program sought to base Mathematics (e.g., and by extension also the rest of inductive and deductive Science) upon a logical foundation (e.g., of certain axiomatic definitions); It is also well known that Gödel's Incompleteness Theorem (GIT) has failed Hillbert's Mathematical Program due to its proof that there exists certain 'self-referential' logical-mathematical statements that cannot be determined as "true" or "false" (e.g., or logically 'consistent' or 'inconsistent') from within any hypothetical axiomatic logical-mathematical system… Previously (Bentwich, 2012c,d) it was suggested that perhaps scientific Gödel -like SROCS computational systems may in fact be constrained by the Duality Principle's (generalized) format, thus:

**i.** SROCS: PR{x,y}→['y' or 'not y']/di1…din

**ii.** SROCS CR{S,t}→ ['t' or 'not t']/di1…din

wherein it was shown that both inductive ('i') and deductive (ii) SROCS scientific computa‐ tional systems are necessarily constrained by the Duality Principle (e.g., as part of the broader CUFT). In other words, the Duality Principle's (generalized format) was shown to constrain all (exhaustive hypothetical) Gödel -like (inductive or deductive) scientific SROCS paradigms, thereby pointing at the existence of a singular (higher-ordered) Universal Consciousness Principle ('י ('which is solely capable of computing the "simultaneous co-occurrences" of any (exhaustive hypothetical) 'x-y' pairs series. It is important to note, however, that the conceptual computational constraint imposed upon all (Gödel -like) inductive or deductive scientific SROCS paradigms was shown to apply for all of those inductive or deductive (apparent) scientific SROCS paradigms – for which there is an empirically known (or 'knowable') 'x-y' pairs series results!

This latter assertion of the Duality Principle's (generalized proof) may be significant as it both narrows- and emphasized- the scope of the *'scientifically knowable domain'*; In other words, instead of the current 'materialistic-reductionistic' scientific framework which is anchored in a basic (inductive or deductive) SROCS computational format (see above) which inevitably leads to both 'logical inconsistency' and 'computational indeterminacy' that are contradicted by robust empirical findings (e.g., pertaining to the key scientific SROCS paradigms); The Duality Principle (e.g., as one of the postulates within the broader CUFT) proves that the only means for computing the "simultaneous co-occurrences" of any (exhaustive hypothetical) 'xy' pairs series is carried out by the singular (higher-ordered) Universal Consciousness Principle ('י ...('Moreover, the (generalized format of the) Duality Principle goes farther to state that for all other (exhaustive hypothetical) inductive or deductive computational SROCS paradigms – *for which there exists a proven empirical capacity to determine the values of any particular 'x-y' pairs (e.g., empirically "known" or "knowable" 'x-y' pairs results)*- any of these (hypothetical) scientific SROCS computations must be carried out by the CUFT's identified singular Universal Consciousness Principle ('י!('

occurrences" of all such (exhaustive hypothetical) inductive or deductive 'x-y' pairs series comprising the totality of the USCF's (single or multiple) frames…. Moreover, this singular Universal Consciousness Principle ('י ('was also shown to exist independently of any (secon‐ dary computational) 'physical properties' (e.g., of 'space', 'time', 'energy' and 'mass') and therefore constitute the only "reality" that exists invariantly (i.e., both as giving rise to the four 'phenomenal' physical properties and as existing solely "in-between" any two such subsequent

In order to appreciate the full (potential) theoretical significance of such a 'Universal Con‐ sciousness Principle Computational Program' it may be worthwhile to reexamine Hillbert's famous 'Mathematical Program' to base Mathematics upon the foundations of Logic (e.g., and by extension also all of Science upon the foundations of Mathematics and Logic), and more specifically, to revisit 'Gödel's Incompleteness Theorem' (GIT) which delivered a critical blow to Hilbert's 'Mathematical Program'; It is a well-known that Hilbert's Mathematical Program sought to base Mathematics (e.g., and by extension also the rest of inductive and deductive Science) upon a logical foundation (e.g., of certain axiomatic definitions); It is also well known that Gödel's Incompleteness Theorem (GIT) has failed Hillbert's Mathematical Program due to its proof that there exists certain 'self-referential' logical-mathematical statements that cannot be determined as "true" or "false" (e.g., or logically 'consistent' or 'inconsistent') from within any hypothetical axiomatic logical-mathematical system… Previously (Bentwich, 2012c,d) it was suggested that perhaps scientific Gödel -like SROCS computational systems

may in fact be constrained by the Duality Principle's (generalized) format, thus:

wherein it was shown that both inductive ('i') and deductive (ii) SROCS scientific computa‐ tional systems are necessarily constrained by the Duality Principle (e.g., as part of the broader CUFT). In other words, the Duality Principle's (generalized format) was shown to constrain all (exhaustive hypothetical) Gödel -like (inductive or deductive) scientific SROCS paradigms, thereby pointing at the existence of a singular (higher-ordered) Universal Consciousness Principle ('י ('which is solely capable of computing the "simultaneous co-occurrences" of any (exhaustive hypothetical) 'x-y' pairs series. It is important to note, however, that the conceptual computational constraint imposed upon all (Gödel -like) inductive or deductive scientific SROCS paradigms was shown to apply for all of those inductive or deductive (apparent) scientific SROCS paradigms – for which there is an empirically known (or 'knowable') 'x-y'

This latter assertion of the Duality Principle's (generalized proof) may be significant as it both narrows- and emphasized- the scope of the *'scientifically knowable domain'*; In other words, instead of the current 'materialistic-reductionistic' scientific framework which is anchored in a basic (inductive or deductive) SROCS computational format (see above) which inevitably leads to both 'logical inconsistency' and 'computational indeterminacy' that are contradicted by robust empirical findings (e.g., pertaining to the key scientific SROCS paradigms); The Duality Principle (e.g., as one of the postulates within the broader CUFT) proves that the only

**i.** SROCS: PR{x,y}→['y' or 'not y']/di1…din

**ii.** SROCS CR{S,t}→ ['t' or 'not t']/di1…din

USCF's frames).

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pairs series results!

The (potential) significance of this generalized assertion made by the Computational Unified Field Theory's (CUFT): 'Duality Principle', 'Computational Invariance' principle and Universal Consciousness Principle ('י ('is twofold:


"simultaneous co-occurrences" of any (exhaustive hypothetical) inductive or deductive 'xy' pairs series (e.g., which were shown by the CUFT to comprise the totality of any single or multiple USCF's frames that are solely produced by this Universal Consciousness Principle). In that sense, it may be said that the Universal Consciousness Principle Computational Program points at the existence of the singular (higher-ordered) Universal Consciousness Principle as constraining- and producing- all inductive or deductive scientific relationship/s or phenomena (e.g., which was also shown earlier and previously to constitute the only "reality" which both produces all USCF's derived secondary computational 'physical properties and also solely exists "in-between" any two such USCF's).

"pass" across any two subsequent USCF's, but only the computationally invariant "real" Universal Consciousness Principle which exists singularly – as solely producing all apparent secondary computational 'physical' properties as well as existing independently "in-between" any two such subsequent USCF's frames.) Indeed, the need to replace all apparent 'material-causal' 'x-y' SROCS relationships by a singular (higher-ordered) Universal Consciousness Principle computation of the 'simultaneous co-occurrences' of all possible inductive or deductive 'x-y' pairs series was shown to apply to all of the key (apparent) scientific SROCS paradigms (including: Darwin's Natural Selection Principle and associated Genetic Encoding hypothesis, Neuroscience's Psychophysical Problem of human Consciousness as well as to all Gödel-like hypothetical inductive or deductive SROCS paradigms; what this implies is that for all of these apparent SROCS scientific paradigms the sole "reality" of the Universal Consciousness Principle forces us to tran‐ scend each of the (particular) 'material-causal' x-y relationships in favor of the Universal Consciousness Principle's singular computation of all (exhaustive hypothetical) 'x-y' pairs series; Thus, for example, instead of Darwin's current 'Natural Selection Principle' SROCS material-causality thesis, which assumes that it is the direct (or indirect) physical inter‐ action between the organism and its Environmental Factors that causes that organism to 'survive' or be 'extinct', the adoption of the Universal Consciousness Principle (and Duality Principle) postulates brigs about a recognition that there is only a singular (Universal Consciousness based) conceptually higher-ordered 'a-causal' computation of the "simul‐ taneous co-occurrences" of an exhaustive hypothetical pairs series of 'organism' and 'Environmental Factors' (e.g., which are computed as part of the Universal Consciousness

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Principle's production of the series of USCF's frames).

**3. Possible Resolution of Physical (and Mathematical) Conundrums:** It is suggested that certain key Physical (and Mathematical) Conundrums including: Physics "dark energy", "dark matter" and "arrow of time" enigmas may be potentially resolved through the application of this singular 'Universal Consciousness Principle'; this is because according to the CUFT, all (four) 'physical' properties of 'space', 'time', 'energy' and 'mass' are (in reality) solely produced by the Universal Consciousness Principle (e.g., as secondary computational 'phenomenal' properties); Hence, the key enigma of "dark energy" and "dark matter" (e.g., the fact that based on the calculation of the totality of 'mass' and 'energy' in the observable cosmos the expansion of the universe should not be as rapid as is observed – which is currently interpreted as indicating that approximately 70-90% of the "energy" and "mass" in the universe in "dark", that is not yet observable) – may be explainable based on the CUFT's delineation of the Universal Consciousness Principle's (extremely rapid) computation of the series of USCF's. This is due to the fact that according to the Universal Consciousness Principle's (previously discovered: Bentwich, 2012a) 'Universal Computational Formula' the production of any "mass" or "energy" ("space" or "time") 'physical' properties – are entirely (and solely) produced through the Universal Consciousness Principle's computation of the degree of 'Consistency' (e.g., 'consistent' or 'inconsistent') across two other Computational Dimensions, i.e., 'Framework' ('frame' vs. 'object') and 'Locus' ('global' vs. 'local'): Thus, for instance it was shown that any "mass" measurement of any object in the universe is computed by the Universal Consciousness

### **4. Theoretical ramifications of the universal consciousness principle**

The discovery of the singular Universal Consciousness Principle (alongside its 'Universal Consciousness Computational Program') may bear a few significant theoretical ramifications:


"pass" across any two subsequent USCF's, but only the computationally invariant "real" Universal Consciousness Principle which exists singularly – as solely producing all apparent secondary computational 'physical' properties as well as existing independently "in-between" any two such subsequent USCF's frames.) Indeed, the need to replace all apparent 'material-causal' 'x-y' SROCS relationships by a singular (higher-ordered) Universal Consciousness Principle computation of the 'simultaneous co-occurrences' of all possible inductive or deductive 'x-y' pairs series was shown to apply to all of the key (apparent) scientific SROCS paradigms (including: Darwin's Natural Selection Principle and associated Genetic Encoding hypothesis, Neuroscience's Psychophysical Problem of human Consciousness as well as to all Gödel-like hypothetical inductive or deductive SROCS paradigms; what this implies is that for all of these apparent SROCS scientific paradigms the sole "reality" of the Universal Consciousness Principle forces us to tran‐ scend each of the (particular) 'material-causal' x-y relationships in favor of the Universal Consciousness Principle's singular computation of all (exhaustive hypothetical) 'x-y' pairs series; Thus, for example, instead of Darwin's current 'Natural Selection Principle' SROCS material-causality thesis, which assumes that it is the direct (or indirect) physical inter‐ action between the organism and its Environmental Factors that causes that organism to 'survive' or be 'extinct', the adoption of the Universal Consciousness Principle (and Duality Principle) postulates brigs about a recognition that there is only a singular (Universal Consciousness based) conceptually higher-ordered 'a-causal' computation of the "simul‐ taneous co-occurrences" of an exhaustive hypothetical pairs series of 'organism' and 'Environmental Factors' (e.g., which are computed as part of the Universal Consciousness Principle's production of the series of USCF's frames).

"simultaneous co-occurrences" of any (exhaustive hypothetical) inductive or deductive 'xy' pairs series (e.g., which were shown by the CUFT to comprise the totality of any single or multiple USCF's frames that are solely produced by this Universal Consciousness Principle). In that sense, it may be said that the Universal Consciousness Principle Computational Program points at the existence of the singular (higher-ordered) Universal Consciousness Principle as constraining- and producing- all inductive or deductive scientific relationship/s or phenomena (e.g., which was also shown earlier and previously to constitute the only "reality" which both produces all USCF's derived secondary computational 'physical properties and also solely exists "in-between" any two such

**4. Theoretical ramifications of the universal consciousness principle**

ordered) Universal Consciousness Principle "reality".

The discovery of the singular Universal Consciousness Principle (alongside its 'Universal Consciousness Computational Program') may bear a few significant theoretical ramifications: **1. The Sole "Reality" of the Universal Consciousness Principle:** As shown above, all scientific (inductive and deductive) disciplines need to be reformulated based on the recognition that there exists only a singular (higher-ordered) Universal Consciousness Principle ('י ('which solely produces- sustains- evolves (and constrains) all (apparent) SROCS (inductive or deductive) 'x-y' relationships; Moreover, this Universal Conscious‐ ness Principle is recognized as the sole "reality" that both produces- sustains- and evolvesany of the apparent (four) 'physical' properties of 'space', 'time', 'energy' and 'mass', as well as exists independently of any such 'physical' properties – and is therefore recognized as the only singular "reality", whereas these apparent 'physical' properties are seen as merely 'phenomenal' (secondary computational) manifestations of this singular (higher-

**2. The Transcendence of 'Material-Causality' by the Universal Consciousness Principle 'A-Causal' Computation:** As shown (above), the acceptance of the Universal Conscious‐ ness Principle ('י ('as the sole "reality" which both produces- (sustains- and evolves-) all USCF's (secondary computational) 'physical' properties, as well as exists independently "in-between" any (two subsequent) USCF's; (Alongside the Duality Principle's negation of any apparent SROCS' 'causal' relationships and the 'Computational Invariance' principle indication that only the 'computationally invariant' 'Universal Consciousness Principle' "really" exists whereas the secondary 'computationally variant' physical properties are only 'phenomenal'.) – point at the negation of any "real" material-causal ('xy') relationships, but instead indicate that there can only exist a singular (higher-ordered) Universal Consciousness Principle 'a-causal' computation of the "simultaneous cooccurrences" of any exhaustive hypothetical inductive or deductive 'x-y' pairs' series… (As shown earlier, the strict negation of the existence of any "real" 'material-causal' 'x→y' relationships was evinced by the simple fact that according to the CUFT's model there cannot exist any "real" computationally variant 'physical' or 'material' property that can

USCF's).

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**3. Possible Resolution of Physical (and Mathematical) Conundrums:** It is suggested that certain key Physical (and Mathematical) Conundrums including: Physics "dark energy", "dark matter" and "arrow of time" enigmas may be potentially resolved through the application of this singular 'Universal Consciousness Principle'; this is because according to the CUFT, all (four) 'physical' properties of 'space', 'time', 'energy' and 'mass' are (in reality) solely produced by the Universal Consciousness Principle (e.g., as secondary computational 'phenomenal' properties); Hence, the key enigma of "dark energy" and "dark matter" (e.g., the fact that based on the calculation of the totality of 'mass' and 'energy' in the observable cosmos the expansion of the universe should not be as rapid as is observed – which is currently interpreted as indicating that approximately 70-90% of the "energy" and "mass" in the universe in "dark", that is not yet observable) – may be explainable based on the CUFT's delineation of the Universal Consciousness Principle's (extremely rapid) computation of the series of USCF's. This is due to the fact that according to the Universal Consciousness Principle's (previously discovered: Bentwich, 2012a) 'Universal Computational Formula' the production of any "mass" or "energy" ("space" or "time") 'physical' properties – are entirely (and solely) produced through the Universal Consciousness Principle's computation of the degree of 'Consistency' (e.g., 'consistent' or 'inconsistent') across two other Computational Dimensions, i.e., 'Framework' ('frame' vs. 'object') and 'Locus' ('global' vs. 'local'): Thus, for instance it was shown that any "mass" measurement of any object in the universe is computed by the Universal Consciousness Principle ('י ('as the degree of 'consistent-object' measurement (of that particular) object across a series of USCF frames.

However, it is suggested that according to one of the CUFT critical empirical predictions (previously outlined: Bentwich, 2012b) this "arrow of time" Physical conundrum may be resolved: This is because one (of three) critical empirical predictions of the CUFT assert the possibility of reversing any spatial-temporal sequence associated with any given 'electromag‐ netic spatial pixel' through the appropriate manipulation of that object's (or event's) electro‐ magnetic spatial pixel values (across a series of USCF's): It was thus indicated that if we were to accurately record the spatial electromagnetic pixels' values of any particular object (e.g., such as an amoeba or any other living organism for instance) across a series of USCF's frames (e.g., or even through a certain sampling from a series of USCF's), and to the extent that we could appropriately manipulate these various electromagnetic spatial pixels' values in such a manner which allows us to reproduce that objects' electromagnetic spatial pixels' values (across the measured series of USCF's) – in the reversed spatial-temporal sequence, then it may be possible to reverse the "flow of time" (e.g., spatial-temporal electromagnetic pixels' sequence). In this way it should be possible (according to one of the critical predictions of the CUFT) to actually "reverse" the "arrow of time" (e.g., at least for particular object/s or event/s: such as for instance, bring about a situation in which a "broken glass cup may in fact be reintegrated"…)

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[1] Bentwich, J. (2003a). From Cartesian Logical-Empiricism to the'Cognitive Reality': A Paradigmatic Shift, *Proceedings of Inscriptions in the Sand, Sixth International Literature*

[2] Bentwich, J. (2003b). The Duality Principle's resolution of the Evolutionary Natural Selection Principle; The Cognitive 'Duality Principle': A resolution of the 'Liar Paradox'

[3] Empiricism to the 'Cognitive Reality: A paradigmatic shift *Proceedings of 12th Interna‐ tional Congress of Logic, Methodology and Philosophy of Science,* August Oviedo, Spain [4] Bentwich, J. (2003c). The cognitive'Duality Principle': a resolution of the'Liar Paradox' and'Gödel's Incompleteness Theorem' conundrums, *Proceedings of Logic Colloquium,*

[5] Bentwich, J. (2004). The Cognitive Duality Principle: A resolution of major scientific conundrums, *Proceedings of The international Interdisciplinary Conference,* Calcutta,

and 'Gödel's Incompleteness Theorem' conundrums; From Cartesian Logical-

**Author details**

Jonathan Bentwich\*

**References**

Brain Perfection LTD, Israel

*and Humanities Conference*, Cyprus

Helsinki, Finland, August 2003

January

Hence, by extension, the totality of the "mass" measured across the entire physical universe should be a measure of the degree of consistent-object/s values across a series of USCF's! Note, however, that based on the abovementioned recognition that in "reality" – only the Universal Consciousness Principle ('י" ('exists" (e.g., both as producing any of the USCF's derived four secondary computational 'physical' properties as well as existing independently "in-between" any two such USCF's frames), and therefore that only this Universal Consciousness Principle "really" produces all of the (apparent) "mass" and "energy" in the 'physical' universe (e.g., rather than the "energy" and "mass" in the 'physical' universe being "caused" by the "material" objects in the cosmos)… Hence, also all of the "energy" in the physical universe is solely produced by this (singular) Universal Consciousness Principle, e.g., as a measure of the degree of 'incon‐ sistent-frame' (changes) of all of the objects (I the universe) across a series of USCF's frames. Therefore, according to the CUFT, the explanation of all of the "mass" and "energy" values observed in the 'physical' universe – should be solely attributed to the operation of the Universal Computational Principle, i.e., through its (extremely rapid) computation of the rapid series of USCF's (respective secondary computational measures of the abovementioned degree of 'consistent-object': "mass", or 'inconsistent-frame': "energy"). We therefore obtain that the (accelerated) rate of expansion of the physical universe – should be explained (according to the CUFT) based on the Universal Consciousness (extremely rapid) computation of the USCF's (e.g., which gives rise to the apparent secondary computational 'physical' measures of 'consistent-object': "mass" or 'inconsistent-frame': "energy"), rather than arise from any 'material-causal' effects of any (strictly hypothetical) "dark mass" or "dark energy"… (Once again, it may be worth pointing at the abovementioned conceptual computational proof that there cannot be any transference of any "physical" property (entity or effect etc.) across any (two subsequent) USCF's frames, but only the retention- or evolution- of all of the spatial pixels' "physical" properties by the singular Universal Consciousness Principle across the series of USCF's – which therefore also precludes the possibility of any "real" "material" effects exerted by any "dark" mass or energy on the expansion of the 'physical' universe across a series of USCF frames.)

Similarly, the "arrow of time" conundrum in modern Physics, e.g., which essentially points at the fact that according to the laws of Physics, there should not be any difference between the physical pathways of say the "breaking of a glass cup into a (thousand) small glass' pieces" and the "re-integration of these thousand glass' pieces into a unitary glass cup"! In other words, according to the strict laws of Physics, there should not be any preference for us seeing "glasses" break into a thousand pieces – over our seeing of the thousand pieces become "reintegrated" into whole glass cups (again), which is obviously contradicted by our (everyday) phenomenal experiences (as well as by our empirical scientific observations)… Hence, according to the current state of (quantum and relativistic) models of Physical reality – there is no reasonable explanation for this "arrow of time" apparent empirical "preference" for the "glass breaking into pieces" scenario over the "reintegration of the glass pieces" scenario…

However, it is suggested that according to one of the CUFT critical empirical predictions (previously outlined: Bentwich, 2012b) this "arrow of time" Physical conundrum may be resolved: This is because one (of three) critical empirical predictions of the CUFT assert the possibility of reversing any spatial-temporal sequence associated with any given 'electromag‐ netic spatial pixel' through the appropriate manipulation of that object's (or event's) electro‐ magnetic spatial pixel values (across a series of USCF's): It was thus indicated that if we were to accurately record the spatial electromagnetic pixels' values of any particular object (e.g., such as an amoeba or any other living organism for instance) across a series of USCF's frames (e.g., or even through a certain sampling from a series of USCF's), and to the extent that we could appropriately manipulate these various electromagnetic spatial pixels' values in such a manner which allows us to reproduce that objects' electromagnetic spatial pixels' values (across the measured series of USCF's) – in the reversed spatial-temporal sequence, then it may be possible to reverse the "flow of time" (e.g., spatial-temporal electromagnetic pixels' sequence). In this way it should be possible (according to one of the critical predictions of the CUFT) to actually "reverse" the "arrow of time" (e.g., at least for particular object/s or event/s: such as for instance, bring about a situation in which a "broken glass cup may in fact be reintegrated"…)

### **Author details**

Principle ('י ('as the degree of 'consistent-object' measurement (of that particular) object

Hence, by extension, the totality of the "mass" measured across the entire physical universe should be a measure of the degree of consistent-object/s values across a series of USCF's! Note, however, that based on the abovementioned recognition that in "reality" – only the Universal Consciousness Principle ('י" ('exists" (e.g., both as producing any of the USCF's derived four secondary computational 'physical' properties as well as existing independently "in-between" any two such USCF's frames), and therefore that only this Universal Consciousness Principle "really" produces all of the (apparent) "mass" and "energy" in the 'physical' universe (e.g., rather than the "energy" and "mass" in the 'physical' universe being "caused" by the "material" objects in the cosmos)… Hence, also all of the "energy" in the physical universe is solely produced by this (singular) Universal Consciousness Principle, e.g., as a measure of the degree of 'incon‐ sistent-frame' (changes) of all of the objects (I the universe) across a series of USCF's frames. Therefore, according to the CUFT, the explanation of all of the "mass" and "energy" values observed in the 'physical' universe – should be solely attributed to the operation of the Universal Computational Principle, i.e., through its (extremely rapid) computation of the rapid series of USCF's (respective secondary computational measures of the abovementioned degree of 'consistent-object': "mass", or 'inconsistent-frame': "energy"). We therefore obtain that the (accelerated) rate of expansion of the physical universe – should be explained (according to the CUFT) based on the Universal Consciousness (extremely rapid) computation of the USCF's (e.g., which gives rise to the apparent secondary computational 'physical' measures of 'consistent-object': "mass" or 'inconsistent-frame': "energy"), rather than arise from any 'material-causal' effects of any (strictly hypothetical) "dark mass" or "dark energy"… (Once again, it may be worth pointing at the abovementioned conceptual computational proof that there cannot be any transference of any "physical" property (entity or effect etc.) across any (two subsequent) USCF's frames, but only the retention- or evolution- of all of the spatial pixels' "physical" properties by the singular Universal Consciousness Principle across the series of USCF's – which therefore also precludes the possibility of any "real" "material" effects exerted by any "dark" mass or energy on the expansion of the 'physical' universe across a series of

Similarly, the "arrow of time" conundrum in modern Physics, e.g., which essentially points at the fact that according to the laws of Physics, there should not be any difference between the physical pathways of say the "breaking of a glass cup into a (thousand) small glass' pieces" and the "re-integration of these thousand glass' pieces into a unitary glass cup"! In other words, according to the strict laws of Physics, there should not be any preference for us seeing "glasses" break into a thousand pieces – over our seeing of the thousand pieces become "reintegrated" into whole glass cups (again), which is obviously contradicted by our (everyday) phenomenal experiences (as well as by our empirical scientific observations)… Hence, according to the current state of (quantum and relativistic) models of Physical reality – there is no reasonable explanation for this "arrow of time" apparent empirical "preference" for the "glass breaking

into pieces" scenario over the "reintegration of the glass pieces" scenario…

across a series of USCF frames.

680 Advances in Quantum Mechanics

USCF frames.)

Jonathan Bentwich\*

Brain Perfection LTD, Israel

### **References**


[6] Bentwich, J. (2006). The 'Duality Principle': Irreducibility of sub-threshold psycho‐ physical computation to neuronal brain activation. *Synthese*, , 153(3), 451-455.

**Chapter 29**

**Provisional chapter**

**Shannon Informational Entropies and Chemical**

**Shannon Informational Entropies and Chemical**

During the last decade, researchers around the world have shown that Information Theory [7] is probably one of the most important models in modern science. This model has given rise to applications and reinterpretations of concepts in Physics, Chemistry, Biology, Mathematics, Telecommunications and many other areas that are not, in principle, related to Information Theory. In the 90's, E. R. Frieden showed that important results such as the Schrödinger equation, the Maxwell-Boltzmann and Boltzmann distributions, the Dirac equation can be

Indded, information is a general concept that is perfectly applicable to any case. It is possible to ask what is the importance of the concept of information measure in quantum mechanics? What do they have in common the codes used to send messages from a communications' satellite have in common with the bases in a DNA molecule? How does the second law of Thermodynamics relate to Communication, to the extent that it is possible to speak of the entropy of a musical theme? How do the intricate problems of probability relate to the way we express ourselves orally or in writing? The answer to these questions can be found in *information*, and the fact that this concept can link very different ideas reveals its great

In Chemistry, Information Theory has been applied to the characterization of chemical systems and chemical processes [10–13]. It has been shown that it is possible to use informational entropies to characterize processes such as bond breaking and bond formation. Shannon's entropy can be regarded as a general measure of information that can be used to obtain the Fukui function, which is a parameter of chemical reactivity in atomic and simple molecular systems [6]. In practice, the formal development of applications of Shannon's entropies in Density Functional Theory (DFT) is a fertile area of research. In this context the

> ©2012 Flores-Gallegos, 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. © 2013 Flores-Gallegos; licensee InTech. This is an open access article 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.

© 2013 Flores-Gallegos; licensee InTech. This is a paper 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.

**Reactivity**

10.5772/54329

**1. Introduction**

generality and power.

**Reactivity**

Nelson Flores-Gallegos

Nelson Flores-Gallegos

http://dx.doi.org/10.5772/54329

Additional information is available at the end of the chapter

derived from principles of Information Theory [1–5].

Additional information is available at the end of the chapter


**Provisional chapter**

### **Shannon Informational Entropies and Chemical Reactivity Reactivity**

**Shannon Informational Entropies and Chemical**

Nelson Flores-Gallegos Additional information is available at the end of the chapter

Nelson Flores-Gallegos

[6] Bentwich, J. (2006). The 'Duality Principle': Irreducibility of sub-threshold psycho‐ physical computation to neuronal brain activation. *Synthese*, , 153(3), 451-455.

[7] Bentwich, J. (2012a). Quantum Mechanics / Book 1 (979-9-53307-377-3Chapter title: The'Computational Unified Field Theory' (CUFT): Harmonizing Quantum Mechanics

[8] Bentwich, J. (2012b). Quantum Mechanics / Book 1 (979-9-53307-377-3Chapter 23, Theoretical Validation of the Computational Unified Field Theory., 551-598.

[9] Bentwich, J. (2012c). Quantum Mechanics (In Press) Chapter title: The'Computational

[10] Born, M. (1954). The statistical interpretation of quantum mechanics, *Nobel Lecture,*

[12] Darwin, C. (1859). *On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life* John Murray, London; modern reprint Charles Darwin, Julian Huxley (2003). *On The Origin of Species*. Signet Classics. 0-45152-906-5

[13] Ellis, J. (1986). The Superstring: Theory of Everything, or of Nothing? *Nature,* , 23 (6089),

Unified Field Theory' (CUFT): A Candidate Theory of Everything.

[11] Brumfiel, G. (2006). Our Universe: Outrageous fortune. *Nature*, , 439, 10-12.

[14] Greene, B. (2003). *The Elegant Universe,* Vintage Books, New York

and Relativity Theory.

682 Advances in Quantum Mechanics

*December 11, 1954*

595-598.

Additional information is available at the end of the chapter 10.5772/54329

http://dx.doi.org/10.5772/54329

### **1. Introduction**

During the last decade, researchers around the world have shown that Information Theory [7] is probably one of the most important models in modern science. This model has given rise to applications and reinterpretations of concepts in Physics, Chemistry, Biology, Mathematics, Telecommunications and many other areas that are not, in principle, related to Information Theory. In the 90's, E. R. Frieden showed that important results such as the Schrödinger equation, the Maxwell-Boltzmann and Boltzmann distributions, the Dirac equation can be derived from principles of Information Theory [1–5].

Indded, information is a general concept that is perfectly applicable to any case. It is possible to ask what is the importance of the concept of information measure in quantum mechanics? What do they have in common the codes used to send messages from a communications' satellite have in common with the bases in a DNA molecule? How does the second law of Thermodynamics relate to Communication, to the extent that it is possible to speak of the entropy of a musical theme? How do the intricate problems of probability relate to the way we express ourselves orally or in writing? The answer to these questions can be found in *information*, and the fact that this concept can link very different ideas reveals its great generality and power.

In Chemistry, Information Theory has been applied to the characterization of chemical systems and chemical processes [10–13]. It has been shown that it is possible to use informational entropies to characterize processes such as bond breaking and bond formation. Shannon's entropy can be regarded as a general measure of information that can be used to obtain the Fukui function, which is a parameter of chemical reactivity in atomic and simple molecular systems [6]. In practice, the formal development of applications of Shannon's entropies in Density Functional Theory (DFT) is a fertile area of research. In this context 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. © 2013 Flores-Gallegos; licensee InTech. This is an open access article 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. © 2013 Flores-Gallegos; licensee InTech. This is a paper 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.

©2012 Flores-Gallegos, licensee InTech. This is an open access chapter distributed under the terms of the

maximum entropy method has been applied to DFT [33]. Following this line of research, we have initiated a new investigation to derive formal relationships between Information Theory concepts and Theoretical Chemistry.

In this work, we consider the idea that Shannon's entropy can be directly related to some fundamental DFT concepts. To show this, we present some simple mathematical derivations to prove that the first derivative of the Shannon entropy is directly related to DFT reactivity parameters such as the Fukui function, the hardness, softness, and chemical potential and that it might even be possible to obtain a formal relationship between Shannon's entropy and the electron energy. Finally, chemical applications are presented in which the relationships obtained in this work are used in two case studies involving a simple chemical reaction and a conformational analysis.

### **2. The first derivatives of Shannon's entropy and their relationship to chemical reactivity**

For purposes of this chapter we take as a starting point the definition of the unnormalized Shannon's entropy in terms of the electron density in position space, which is defined as:

$$s(\mathbf{r}) = -\int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r},\tag{1}$$

10.5772/54329

685

*<sup>N</sup>* , (4)

http://dx.doi.org/10.5772/54329

, (5)

ln *ρ*(**r**)*d***r**

ln *ρ*(**r**)*d***r**, (7)

(6)

 *∂S*(**r**) *∂N*

Also, we can set up the follow relationship

 *∂N ∂µ ν*(**r**)

The term *∂ρ*(**r**)

 *∂S*(**r**) *∂N*

simplify,

density,

 *ν*(**r**)

 *∂S*(**r**) *∂µ*

*ν*(**r**) = 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

In this expression, we recognize the term *∂ρ*(**r**)

[33].

*∂N ν*(**r**)  *ν*(**r**) <sup>=</sup> <sup>−</sup>*s*(**r**)

 *∂ρ*(**r**) *∂µ*

> <sup>=</sup> <sup>−</sup>*s*(**r**) *N*2

> > + 1 *N*

 *∂S*(**r**) *∂ρ*(**r**)

for the total Shannon entropy [14], which is defined as

*ν*(**r**)

*<sup>N</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> *N*

*ν*(**r**) =  *∂ρ*(**r**) *∂N*

 *∂ρ*(**r**) *∂N*

and obtain the variation of Shannon's entropy with respect to the chemical potential:

 *∂N ∂µ ν*(**r**)

> *∂µ ν*(**r**)

Also, is it interesting to obtain the variation of Shannon's entropy with respect to the electron

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 

A similar procedure can be used to obtain the above relations in momentum space as well as

where *S*(**r**) and *S*(**p**) are Shannon's entropies in position and momentum spaces, respectively. The variation of the total Shannon entropy with respect to electron number would be:

 *∂N ∂µ ν*(**r**)

 *∂N ∂µ ν*(**r**)  *ν*(**r**)

− 1 *N*

> − 1 *N*

 *ν*(**r**)

, is the Fukui function, which is a chemical reactivity parameter in DFT

 *∂N ∂µ ν*(**r**)

 *∂ρ*(**r**) *∂N*

> *∂ρ*(**r**) *∂µ*

 *ν*(**r**)  *∂N ∂µ ν*(**r**)

*ν*(**r**)

1 + ln *ρ*(**r**)*d***r**. (8)

which is the local softness.

*ST* = *S*(**r**) + *S*(**p**), (9)

ln *ρ*(**r**)*d***r** +

1

Shannon Informational Entropies and Chemical Reactivity

If normalized electron densities are used, *<sup>ρ</sup>*(**r**) *<sup>N</sup>* , Shannon's entropy becomes positive: *S*(**r**) ≥ 0 and the normalized Shannon entropy is

$$S(\mathbf{r}) = -\int \frac{\rho(\mathbf{r})}{N} \ln \frac{\rho(\mathbf{r})}{N} d\mathbf{r},\tag{2}$$

where *N* is the number of electrons. This expression can be rewritten as:

$$\begin{split} S(\mathbf{r}) &= -\frac{1}{N} \int \rho(\mathbf{r}) \ln \left( \frac{\rho(\mathbf{r})}{N} \right) d\mathbf{r} \\ &= -\frac{1}{N} \int \rho(\mathbf{r}) \left[ \ln \rho(\mathbf{r}) - \ln N \right] d\mathbf{r} \\ &= -\frac{1}{N} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{N} \ln N \underbrace{\int \rho(\mathbf{r}) d\mathbf{r}}\_{=N} \\ &= -\frac{1}{N} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \ln N \\ &= -\frac{s(\mathbf{r})}{N} + \ln N. \end{split} \tag{3}$$

Differentiating the entropy *S*(**r**) with respect to *N* at constant external potential *ν*(**r**):

$$
\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} = -\frac{s(\mathbf{r})}{N^2} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{N'} \tag{4}
$$

The term *∂ρ*(**r**) *∂N ν*(**r**) , is the Fukui function, which is a chemical reactivity parameter in DFT [33].

Also, we can set up the follow relationship

$$
\left(\frac{\partial\rho(\mathbf{r})}{\partial\mu}\right)\_{\nu(\mathbf{r})} = \left(\frac{\partial\rho(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} \left(\frac{\partial N}{\partial\mu}\right)\_{\nu(\mathbf{r})}\tag{5}
$$

and obtain the variation of Shannon's entropy with respect to the chemical potential:

$$
\begin{split}
\left(\frac{\partial \mathbf{S}(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \left(\frac{\partial N}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= -\frac{\mathbf{s}(\mathbf{r})}{N^2} \left(\frac{\partial N}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \left(\frac{\partial N}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} \\ &+ \frac{1}{N} \left(\frac{\partial N}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})}
\end{split}
\tag{6}
$$

simplify,

2 Quantum Mechanics

concepts and Theoretical Chemistry.

a conformational analysis.

**chemical reactivity**

If normalized electron densities are used, *<sup>ρ</sup>*(**r**)

and the normalized Shannon entropy is

maximum entropy method has been applied to DFT [33]. Following this line of research, we have initiated a new investigation to derive formal relationships between Information Theory

In this work, we consider the idea that Shannon's entropy can be directly related to some fundamental DFT concepts. To show this, we present some simple mathematical derivations to prove that the first derivative of the Shannon entropy is directly related to DFT reactivity parameters such as the Fukui function, the hardness, softness, and chemical potential and that it might even be possible to obtain a formal relationship between Shannon's entropy and the electron energy. Finally, chemical applications are presented in which the relationships obtained in this work are used in two case studies involving a simple chemical reaction and

**2. The first derivatives of Shannon's entropy and their relationship to**

*s*(**r**) = −

*S*(**r**) = −

where *N* is the number of electrons. This expression can be rewritten as:

*ρ*(**r**)ln

*<sup>S</sup>*(**r**) = <sup>−</sup> <sup>1</sup>

*N* 

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 

<sup>=</sup> <sup>−</sup>*s*(**r**)

For purposes of this chapter we take as a starting point the definition of the unnormalized Shannon's entropy in terms of the electron density in position space, which is defined as:

*ρ*(**r**)

 *ρ*(**r**) *N d***r**

*ρ*(**r**)[ln *ρ*(**r**) − ln *N*] *d***r**

*ρ*(**r**)ln *ρ*(**r**)*d***r** + ln *N*

1 *<sup>N</sup>* ln *<sup>N</sup>* 

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

Differentiating the entropy *S*(**r**) with respect to *N* at constant external potential *ν*(**r**):

*<sup>N</sup>* ln *<sup>ρ</sup>*(**r**)

*ρ*(**r**)ln *ρ*(**r**)*d***r**, (1)

*<sup>N</sup>* , Shannon's entropy becomes positive: *S*(**r**) ≥ 0

*ρ*(**r**)*d***r** =*N*

*<sup>N</sup>* <sup>+</sup> ln *<sup>N</sup>*. (3)

*<sup>N</sup> <sup>d</sup>***r**, (2)

$$
\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} = \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r},\tag{7}$$

In this expression, we recognize the term *∂ρ*(**r**) *∂µ ν*(**r**) which is the local softness.

Also, is it interesting to obtain the variation of Shannon's entropy with respect to the electron density,

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int 1 + \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{8}$$

A similar procedure can be used to obtain the above relations in momentum space as well as for the total Shannon entropy [14], which is defined as

$$S\_T = S(\mathbf{r}) + S(\mathbf{p}),\tag{9}$$

where *S*(**r**) and *S*(**p**) are Shannon's entropies in position and momentum spaces, respectively. The variation of the total Shannon entropy with respect to electron number would be:

$$\frac{d\mathbf{S}\_T}{dN} = \left(\frac{\partial \mathbf{S}(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} + \left(\frac{\partial \mathbf{S}(\mathbf{p})}{\partial N}\right)\_{\nu(\mathbf{p})} \tag{10}$$

687

**2.1. The second derivatives of Shannon's entropy**

density, how a first case, we take the Eq. (11),

*ν*(**r**)

<sup>=</sup> *<sup>∂</sup> ∂ρ*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*2 

> − 1 *N*

> > 1

 *∂N ∂µ ν*(**r**) 

 *∂s ∂ρ*(**r**)

 <sup>−</sup>*s*(**r**) *<sup>N</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> *N*

 *∂ f*(**r**) *∂ρ*(**r**)

*<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

*ν*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

**2.2. The change of Shannon entropy respect to the electron energy using**

 *∂E ∂N ν*(**r**)

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂N*

where *f*(**r**) is it the Fukui function.

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂µ*

where *s* is the softness.

**Parr-Gadre-Bartolotti model**

consider the chemical potential *µ* =

Finally, for Eq. (13),

How a second case, consider the Eq. (12),

*ν*(**r**)

<sup>=</sup> *<sup>∂</sup> ∂ρ*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*2

−

 *∂*2*S*(**r**) *∂ρ*(**r**)<sup>2</sup>

*ν*(**r**)

In this section we obtain the second derivatives of the Shannon entropy respect to the electron

1 + ln *ρ*(**r**)*d***r**

*ν*(**r**)

 *∂N ∂µ ν*(**r**) −

1 + ln *ρ*(**r**)*d***r**

*ρ*(**r**)

*∂ρ*(**r**) (<sup>1</sup> <sup>+</sup> ln *<sup>ρ</sup>*(**r**)) *<sup>d</sup>***<sup>r</sup>**

ln *<sup>ρ</sup>*(**r**) + *<sup>s</sup>*

*∂*

 *d***r** *ρ*(**r**)

Now, we obtain the variation of the Shannon entropy respect to the energy, in this case,

and the Eq. (4),

 *∂ρ*(**r**) *∂N*

ln *<sup>ρ</sup>*(**r**) + *<sup>f</sup>*(**r**)

 *ν*(**r**)

*ρ*(**r**)

 *∂ρ*(**r**) *∂µ*

*ν*(**r**)

*d***r**. (19)

. (20)

ln *ρ*(**r**)*d***r** +

Shannon Informational Entropies and Chemical Reactivity

1 *N* 

http://dx.doi.org/10.5772/54329

*d***r**. (18)

ln *ρ*(**r**)*d***r**

this permit us, open a door, to the study of this kind of derivatives and chemical descriptors, in momentum space. Results about, the chemical reactivity parameters in momentum space, will be present in other work.

Summarizing, the relations obtained are:

i)

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} = -\frac{s(\mathbf{r})}{N^2} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{N}.\tag{11}$$

ii)

$$
\left(\frac{\partial S(\mathbf{r})}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} = \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial \boldsymbol{\mu}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{12}
$$

iii)

$$\left(\frac{\partial S(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \mathbf{1} + \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{13}$$

In the formalism of DFT, the reactivity parameters are defined as:

$$\text{Chemical potential} = \mu = \left(\frac{\partial E}{\partial \mathbf{N}}\right)\_{\nu(\mathbf{r})}.\tag{14}$$

$$\text{Hardness} = \eta = \left(\frac{\partial \mu}{\partial N}\right)\_{\nu(\mathbf{r})} = \left(\frac{\partial^2 E}{\partial N^2}\right)\_{\nu(\mathbf{r})}.\tag{15}$$

$$\text{Softness} = s = \left(\frac{\partial N}{\partial \mu}\right)\_{\nu(\mathbf{r})} = \left(\frac{\partial \rho(\mathbf{r})}{\partial \mu}\right)\_{\nu(\mathbf{r})}.\tag{16}$$

$$\text{Fukui function} = f(\mathbf{r}) = \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})}.\tag{17}$$

where *µ* is the Chemical potential, *η* is the Hardness, *s* is the Softness and *f*(**r**) is the Fukui Function. Each one of these equations has a specific interpretation in Chemistry. The Chemical potential, *µ*, is a measure the escaping tendency of an electron, which is constant, through all space, for the ground state of an atom, molecule, or solid [16]. The Hardness, *η*, is related to the polarizability [15] and the Fukui function [23–25], *∂ρ*(**r**) *∂N ν*(**r**) , is interpreted as a measure of the sensitivity of the chemical potential with respect to an external perturbation at a particular point. These properties have been included in the chemical vocabulary since the early 1950s.

#### **2.1. The second derivatives of Shannon's entropy**

In this section we obtain the second derivatives of the Shannon entropy respect to the electron density, how a first case, we take the Eq. (11),

$$
\begin{split}
\left(\frac{\partial^2 \mathcal{S}(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= \frac{\partial}{\partial \rho(\mathbf{r})} \left\{ -\frac{s(\mathbf{r})}{N^2} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{N} \right\} \\ &= -\frac{1}{N^2} \int 1 + \ln \rho(\mathbf{r}) d\mathbf{r} \\ &\quad - \frac{1}{N} \int \left(\frac{\partial f(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) + \frac{f(\mathbf{r})}{\rho(\mathbf{r})} d\mathbf{r}.\end{split} \tag{18}
$$

where *f*(**r**) is it the Fukui function.

4 Quantum Mechanics

i)

ii)

iii)

the early 1950s.

will be present in other work.

Summarizing, the relations obtained are:

 *∂S*(**r**) *∂µ*

 *∂S*(**r**) *∂N*

*ν*(**r**) = 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

 *ν*(**r**)

*dST dN* <sup>=</sup>  *∂S*(**r**) *∂N*

<sup>=</sup> <sup>−</sup>*s*(**r**)

 *∂S*(**r**) *∂ρ*(**r**)

In the formalism of DFT, the reactivity parameters are defined as:

Chemical potential = *µ* =

Hardness = *η* =

Softness = *s* =

Fukui function = *f*(**r**) =

related to the polarizability [15] and the Fukui function [23–25],

*ν*(**r**)

*<sup>N</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> *N*

> *∂N ∂µ*

> > <sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

> > > *∂E ∂N ν*(**r**)

 *∂µ ∂N ν*(**r**) =

 *∂N ∂µ* 

where *µ* is the Chemical potential, *η* is the Hardness, *s* is the Softness and *f*(**r**) is the Fukui Function. Each one of these equations has a specific interpretation in Chemistry. The Chemical potential, *µ*, is a measure the escaping tendency of an electron, which is constant, through all space, for the ground state of an atom, molecule, or solid [16]. The Hardness, *η*, is

a measure of the sensitivity of the chemical potential with respect to an external perturbation at a particular point. These properties have been included in the chemical vocabulary since

*ν*(**r**) =

> *ν*(**r**)

 *∂ρ*(**r**) *∂N*

 *ν*(**r**) +

this permit us, open a door, to the study of this kind of derivatives and chemical descriptors, in momentum space. Results about, the chemical reactivity parameters in momentum space,

 *∂S*(**p**) *∂N*

 *∂ρ*(**r**) *∂N*

*ν*(**r**)

− 1 *N*

 *ν*(**r**)

ln *ρ*(**r**)*d***r** +

*ν*(**r**)

1 + ln *ρ*(**r**)*d***r**. (13)

. (14)

 *∂ρ*(**r**) *∂µ*

> *∂*2*E ∂N*<sup>2</sup> *ν*(**r**)

 *∂ρ*(**r**) *∂µ*

 *∂ρ*(**r**) *∂N ν*(**r**)

*ν*(**r**)

1

*<sup>N</sup>* . (11)

ln *ρ*(**r**)*d***r**. (12)

. (15)

. (16)

, is interpreted as

. (17)

*ν*(**p**)

, (10)

How a second case, consider the Eq. (12),

$$
\begin{split}
\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= \frac{\partial}{\partial \rho(\mathbf{r})} \left\{ \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} - \int \left(\frac{\partial \rho(\mathbf{r})}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} \right\} \\ &= -\frac{1}{N^2} \left(\frac{\partial N}{\partial \mu}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \int 1 + \ln \rho(\mathbf{r}) d\mathbf{r} \\ &\quad - \int \left(\frac{\partial s}{\partial \rho(\mathbf{r})}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) + \frac{s}{\rho(\mathbf{r})} d\mathbf{r}.\end{split} \tag{19}
$$

where *s* is the softness.

Finally, for Eq. (13),

$$
\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r})^2}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \frac{\partial}{\partial \rho(\mathbf{r})} \left(1 + \ln \rho(\mathbf{r})\right) d\mathbf{r}
$$

$$
= -\frac{1}{N} \int \frac{d\mathbf{r}}{\rho(\mathbf{r})}.\tag{20}
$$

#### **2.2. The change of Shannon entropy respect to the electron energy using Parr-Gadre-Bartolotti model**

Now, we obtain the variation of the Shannon entropy respect to the energy, in this case, consider the chemical potential *µ* = *∂E ∂N ν*(**r**) and the Eq. (4),

$$
\begin{split}
\left(\frac{\partial \mathbf{S}(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= -\frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} \\ &+ \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \\ &= -\frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} \\ &+ \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})},
\end{split} \tag{21}
$$

689

http://dx.doi.org/10.5772/54329

with this and the Eq. (23),

now consider the Eq. (21),

*ν*(**r**)

first we use the property (24)

now, we use the Eq. (21)

 *∂S*(**r**) *<sup>∂</sup>ETF k*

*ν*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

<sup>=</sup> <sup>−</sup> <sup>3</sup> 5*NCF*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

*dETF*

 *∂S*(**r**) *<sup>∂</sup>EPGB*

**energy**

and,

 *∂EPGB ∂ρ*(**r**)

 *∂ρ*(**r**) *<sup>∂</sup>EPGB*

5

*ν*(**r**)

<sup>=</sup> *<sup>∂</sup> ∂ρ*(**r**)

<sup>=</sup> <sup>5</sup> 3

*ν*(**r**)

5

ln *ρ*(**r**)

<sup>3</sup>*Cρ*(**r**)2/3 <sup>+</sup> *<sup>ν</sup>*(**r**) + <sup>4</sup>

based on a formal derivation rather than on a phenomenological interpretation.

**2.3. The variation of Shannon's entropy with respect to the electron kinetic**

In this case, we consider the famous Thomas-Fermi kinetic energy functional, defined as

*<sup>k</sup>* [*ρ*(**r**)] = *CFρ*(**r**)5/3*d***r**, *CF* <sup>=</sup> <sup>3</sup>

 ln *ρ*(**r**) *CF* 5

ln *ρ*(**r**)

*ν*(**r**)

<sup>3</sup> *<sup>ρ</sup>*(**r**)2/3 *<sup>d</sup>***<sup>r</sup>** <sup>+</sup>

*<sup>ρ</sup>*(**r**)2/3 *<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup>

 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2 1 *µ* ,

*µN*<sup>2</sup> 

= *CF* 5 3

 *<sup>∂</sup>ETF k ∂ρ*(**r**)

*<sup>C</sup>ρ*(**r**)2/3 <sup>+</sup> *<sup>ν</sup>*(**r**) + <sup>4</sup>

<sup>=</sup> <sup>1</sup>

<sup>3</sup>*Cρ*(**r**)2/3 <sup>+</sup> *<sup>ν</sup>*(**r**) + <sup>4</sup>

The importance of this results, resides in that for the first time we prove that there exist a formal relationship between concepts of the information theory and the chemical reactivity,

*<sup>C</sup>ρ*(**r**)5/3 <sup>+</sup> *<sup>ρ</sup>*(**r**)*ν*(**r**) + *BN*2/3*ρ*(**r**)4/3

*BN*2/3*ρ*(**r**)1/3 (25)

Shannon Informational Entropies and Chemical Reactivity

<sup>3</sup> *BN*2/3*ρ*(**r**)1/3 , (26)

<sup>10</sup> (3*π*2)2/3, (28)

1

*<sup>µ</sup><sup>N</sup>* . (30)

*ρ*(**r**)2/3, (29)

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

 *∂N ∂E ν*(**r**) . (27)

 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

3

<sup>3</sup> *BN*2/3*ρ*(**r**)1/3 *<sup>d</sup>***<sup>r</sup>** <sup>+</sup>

note the importance of this relation, we can establish a formal relation between the Shannon entropy with the electronic energy1.

In a similar way, we can obtain the second derivative of the last equation respect to electron density.

$$
\begin{split}
\left(\frac{\partial^2 \mathcal{S}(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= \frac{\partial}{\partial \rho(\mathbf{r})} \left\{ -\frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \right\} \\ &= -\frac{1}{N} \int \left(\frac{\partial \mu(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) + \frac{\mu(\mathbf{r})}{\rho(\mathbf{r})} d\mathbf{r} \\ &\quad - \frac{1}{N^2} \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \int 1 + \ln \rho(\mathbf{r}) d\mathbf{r}. \end{split} \tag{22}
$$

where *µ*(**r**) it is a local chemical potential.

For obtain a direct application of the last result in the DFT model, we selected the Parr-Gadre-Bartolotti model [26], PGB, this is a local model based in the Thomas-Fermi model [17, 18]. The expression for the energy in the PGB model is,

$$d\left[E\right]\_{PGB} = \mathbb{C}\rho(\mathbf{r})^{5/3} + \rho(\mathbf{r})\nu(\mathbf{r}) + \mathcal{B}N^{2/3}\rho(\mathbf{r})^{4/3}d\mathbf{r},\tag{23}$$

where *B* = 0.7544 and *C* = 3.8738.

Considering Eq. (21), and Eq. (23), the first step is it obtain *∂ρ*(**r**) *∂E ν*(**r**) , for do this, consider

$$\left(\frac{\partial\rho(\mathbf{r})}{\partial E\_{PGB}}\right)\_{\nu(\mathbf{r})}^{-1} = \left(\frac{\partial E\_{PGB}}{\partial\rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})}\tag{24}$$

<sup>1</sup> A more complete study about of the formal relation between electron energy and Shannon entropy will present in other work.

with this and the Eq. (23),

$$\begin{split} \left(\frac{\partial E\_{\rm PGB}}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} &= \frac{\partial}{\partial \rho(\mathbf{r})} \left\{ \mathbb{C} \rho(\mathbf{r})^{5/3} + \rho(\mathbf{r}) \nu(\mathbf{r}) + \mathcal{B} N^{2/3} \rho(\mathbf{r})^{4/3} \right\} \\ &= \frac{5}{3} \mathcal{C} \rho(\mathbf{r})^{2/3} + \nu(\mathbf{r}) + \frac{4}{3} \mathcal{B} N^{2/3} \rho(\mathbf{r})^{1/3} \end{split} \tag{25}$$

and,

6 Quantum Mechanics

density.

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂E*

 *∂S*(**r**) *∂N*

entropy with the electronic energy1.

*ν*(**r**)

<sup>=</sup> *<sup>∂</sup> ∂ρ*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

where *µ*(**r**) it is a local chemical potential.

where *B* = 0.7544 and *C* = 3.8738.

other work.

− 1 *N*2  − 1 *N*

 *∂µ*(**r**) *∂ρ*(**r**)

model [17, 18]. The expression for the energy in the PGB model is,

 *∂ρ*(**r**) *∂EPGB* <sup>−</sup><sup>1</sup>

*ν*(**r**) =

<sup>1</sup> A more complete study about of the formal relation between electron energy and Shannon entropy will present in

Considering Eq. (21), and Eq. (23), the first step is it obtain

 *∂N ∂E ν*(**r**) 

 *ν*(**r**)  *∂N ∂E ν*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

> + 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

> + 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

 *∂ρ*(**r**) *∂E*

*ν*(**r**)

note the importance of this relation, we can establish a formal relation between the Shannon

In a similar way, we can obtain the second derivative of the last equation respect to electron

 *ν*(**r**)

For obtain a direct application of the last result in the DFT model, we selected the Parr-Gadre-Bartolotti model [26], PGB, this is a local model based in the Thomas-Fermi

ln *<sup>ρ</sup>*(**r**) + *<sup>µ</sup>*(**r**)

 *∂ρ*(**r**) *∂N*

 *∂ρ*(**r**) *∂E*

 *ν*(**r**)

 *∂N ∂E ν*(**r**)

> *ν*(**r**)

 *∂N ∂E ν*(**r**)

ln *ρ*(**r**)*d***r** +

*ρ*(**r**) *d***r**

*d* [*E*]*PGB* = *Cρ*(**r**)5/3 + *ρ*(**r**)*ν*(**r**) + *BN*2/3*ρ*(**r**)4/3*d***r**, (23)

*ν*(**r**)

 *∂EPGB ∂ρ*(**r**)

 *∂ρ*(**r**) *∂E ν*(**r**)

 *∂N ∂E ν*(**r**)

ln *ρ*(**r**)*d***r**

 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

1 + ln *ρ*(**r**)*d***r**. (22)

ln *ρ*(**r**)*d***r**

, (21)

 *∂N ∂E ν*(**r**)

, for do this, consider

(24)

$$\left(\frac{\partial \rho(\mathbf{r})}{\partial E\_{\rm PGB}}\right)\_{\nu(\mathbf{r})} = \frac{1}{\frac{5}{3}C\rho(\mathbf{r})^{2/3} + \nu(\mathbf{r}) + \frac{4}{3}\mathcal{B}\mathcal{N}^{2/3}\rho(\mathbf{r})^{1/3}},\tag{26}$$

now consider the Eq. (21),

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial \mathcal{E}\_{\rm FGB}}\right)\_{\mathbf{v}(\mathbf{r})} = -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\frac{5}{3} \mathcal{C} \rho(\mathbf{r})^{2/3} + \nu(\mathbf{r}) + \frac{4}{3} \mathcal{B} \mathcal{N}^{2/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} + \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial \mathcal{N}}{\partial E}\right)\_{\mathbf{v}(\mathbf{r})}.\tag{27}$$

The importance of this results, resides in that for the first time we prove that there exist a formal relationship between concepts of the information theory and the chemical reactivity, based on a formal derivation rather than on a phenomenological interpretation.

#### **2.3. The variation of Shannon's entropy with respect to the electron kinetic energy**

In this case, we consider the famous Thomas-Fermi kinetic energy functional, defined as

$$dE\_k^{TF}[\rho(\mathbf{r})] = \mathbb{C}\_F \rho(\mathbf{r})^{5/3} d\mathbf{r}, \qquad \qquad \mathbb{C}\_F = \frac{3}{10} (3\pi^2)^{2/3} \text{.} \tag{28}$$

first we use the property (24)

$$
\left(\frac{\partial E\_k^{TF}}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} = C\_F \frac{5}{3} \rho(\mathbf{r})^{2/3} \text{.}\tag{29}
$$

now, we use the Eq. (21)

$$
\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E\_k^{TF}}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\mathcal{C}\_F \frac{5}{3} \rho(\mathbf{r})^{2/3}} d\mathbf{r} + \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \frac{1}{\mu'}
$$

$$
= -\frac{3}{5N \mathcal{C}\_F} \int \frac{\ln \rho(\mathbf{r})}{\rho(\mathbf{r})^{2/3}} d\mathbf{r} - \frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N}.\tag{30}
$$

#### **2.4. The variation of Shannon's entropy with respect to the exchange energy in LDA**

For simplicity we take only the exchange energy, in the Local Density Approximation (LDA), in this approximation the total energy of a system can be write as the sum of the correlation and exchange energy: *εcx* = *εx* + *εc*. The correlation part, *εc* has been calculated and the results obtained were expressed like complicated expression of *ρ*(**r**) [27]. In our case, we only consider the exchange term, defined as

$$dE\_\mathbf{x}^{LDA} \left[\rho(\mathbf{r})\right] = -\frac{3}{4} \left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{4/3} d\mathbf{r} \tag{31}$$

10.5772/54329

691

ln *ρ*(**r**)*d***r**

http://dx.doi.org/10.5772/54329

ln *ρ*(**r**)*d***r** + *BN*, (34)

− 1 *µN*<sup>2</sup> 

− 1 *µN*<sup>2</sup> 

> *A*

> > *<sup>A</sup>* <sup>=</sup> <sup>3</sup> 5*NCF*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 3 5*CF* +

= −

**2.6. Summary of relationships obtained**

where

and

Functional Theory.

 *∂S*(**r**) *∂N*

 *∂S*(**r**) *∂µ*

 *∂S*(**r**) *∂ρ*(**r**)

 *∂S*(**r**) *∂E*

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂N*

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂µ*

> *∂*2*S*(**r**) *∂ρ*(**r**)<sup>2</sup>

 *ν*(**r**)

 *ν*(**r**)

*ν*(**r**) = 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

*ν*(**r**)

*ν*(**r**)

*ν*(**r**)

*ν*(**r**)

<sup>=</sup> <sup>−</sup>*s*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N* 

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

= −

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

*<sup>N</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> *N*

> *∂ρ*(**r**) *∂E*

 *∂ f*(**r**) *∂ρ*(**r**)

 *∂s ∂ρ*(**r**)

> *d***r** *ρ*(**r**)

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

1 + *ρ*(**r**)1/3 *<sup>ρ</sup>*(**r**)2/3

+

*<sup>B</sup>* <sup>=</sup> <sup>1</sup> *<sup>µ</sup>N*<sup>2</sup> .

Finally, we present a summary of the different relations obtained in this work. Whit this results, is possible say that the information theory is a model that is subjacent to the Density

In the follow section we show some results of this relations applied to some chemical process.

 *∂ρ*(**r**) *∂N*

 *∂N ∂µ ν*(**r**)

> *ν*(**r**)

*ν*(**r**)

*ν*(**r**)

 *ν*(**r**)

> − 1 *N*

ln *ρ*(**r**)*d***r** +

ln *<sup>ρ</sup>*(**r**) + *<sup>s</sup>*

ln *<sup>ρ</sup>*(**r**) + *<sup>f</sup>*(**r**)

*ρ*(**r**)

ln *ρ*(**r**)*d***r** +

 *∂ρ*(**r**) *∂µ*

> 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

*ρ*(**r**)

1 + ln *ρ*(**r**)*d***r**. (37)

. (41)

1

*ν*(**r**)

 *∂N ∂E ν*(**r**)

*d***r**. (39)

*d***r**. (40)

*<sup>N</sup>* . (35)

ln *ρ*(**r**)*d***r**. (36)

. (38)

1 (3/*π*)1/3

1 *µN*

1 *µN*

− *Bρ*(**r**)

1 *<sup>N</sup>*(3/*π*)2/3 , 1 + *ρ*(**r**)1/3 *<sup>ρ</sup>*(**r**)2/3

Shannon Informational Entropies and Chemical Reactivity

again,

$$\left(\frac{\partial E\_{\mathbf{x}}^{LDA}}{\partial \rho(\mathbf{r})}\right)\_{\mathbf{v}(\mathbf{r})} = -\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3} \text{.}\tag{32}$$

and

$$\left(\frac{\partial \mathbf{S}(\mathbf{r})}{\partial E\_{\mathbf{x}}^{LDA}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} = -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} + \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})} $$

$$= -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} - \frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N}.\tag{33}$$

#### **2.5. Variation of Shannon's entropy with respect to the energy, considering the kinetic and exchange effects.**

Now, we can take the previous results to obtain the variation on Shannon's entropy considering the kinetic and exchange effects using local models. This derivative will be

$$
\begin{split} \left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E\_{k,\mathbf{r}}^{\mathrm{TF},\mathrm{L}DA}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= \left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E\_{k}^{\mathrm{TF}}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} + \left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E\_{\mathrm{T}}^{\mathrm{L}DA}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \\ &= -\frac{3}{5\mathcal{N}\mathcal{C}\_{F}} \int \frac{\ln \rho(\mathbf{r})}{\rho(\mathbf{r})^{2/3}} d\mathbf{r} - \frac{1}{\mu N^{2}} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N} \\ &\quad - \frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} - \frac{1}{\mu N^{2}} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N} \\ &= \left[ -\frac{3}{5N\mathcal{C}\_{F}} \int \frac{\ln \rho(\mathbf{r})}{\rho(\mathbf{r})^{2/3}} d\mathbf{r} - \frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} \right] \end{split}
$$

<sup>690</sup> Advances in Quantum Mechanics Shannon Informational Entropies and Chemical Reactivity 9 10.5772/54329 Shannon Informational Entropies and Chemical Reactivity http://dx.doi.org/10.5772/54329 691

$$\begin{split} & -\frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N} \\ &= -\frac{1}{N} \left( \frac{3}{5\mathsf{C}\_F} + \frac{1}{(3/\pi)^{1/3}} \right) \int \left( \frac{1 + \rho(\mathbf{r})^{1/3}}{\rho(\mathbf{r})^{2/3}} \right) \ln \rho(\mathbf{r}) d\mathbf{r} \\ & - \frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N} \\ &= -\int \left[ A \left( \frac{1 + \rho(\mathbf{r})^{1/3}}{\rho(\mathbf{r})^{2/3}} \right) - B\rho(\mathbf{r}) \right] \ln \rho(\mathbf{r}) d\mathbf{r} + BN\_{\prime} \end{split} \tag{34}$$

where

8 Quantum Mechanics

only consider the exchange term, defined as

*dELDA*

� *<sup>∂</sup>ELDA x ∂ρ*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

*<sup>x</sup>* [*ρ*(**r**)] <sup>=</sup> <sup>−</sup><sup>3</sup>

�

� ln *ρ*(**r**) � 3 *π* �1/3

� ln *ρ*(**r**) � 3 *π* �1/3

�

*ν*(**r**) +

� ln *ρ*(**r**) � 3 *π* �1/3

� ln *ρ*(**r**)

� ln *ρ*(**r**)

*ν*(**r**)

**LDA**

again,

and

� *∂S*(**r**) *<sup>∂</sup>ELDA x*

**kinetic and exchange effects.**

*∂S*(**r**) *<sup>∂</sup>ETF*,*LDA k*,*x*

�

*ν*(**r**)

= � *∂S*(**r**) *<sup>∂</sup>ETF k*

= <sup>−</sup> <sup>3</sup> 5*NCF*

<sup>=</sup> <sup>−</sup> <sup>3</sup> 5*NCF*

> − 1 *N*

�

�

*ν*(**r**)

**2.4. The variation of Shannon's entropy with respect to the exchange energy in**

For simplicity we take only the exchange energy, in the Local Density Approximation (LDA), in this approximation the total energy of a system can be write as the sum of the correlation and exchange energy: *εcx* = *εx* + *εc*. The correlation part, *εc* has been calculated and the results obtained were expressed like complicated expression of *ρ*(**r**) [27]. In our case, we

> 4 � 3 *π*

= − � 3 *π*

*ρ*(**r**)1/3

*ρ*(**r**)1/3

**2.5. Variation of Shannon's entropy with respect to the energy, considering the**

Now, we can take the previous results to obtain the variation on Shannon's entropy considering the kinetic and exchange effects using local models. This derivative will be

> � *∂S*(**r**) *<sup>∂</sup>ELDA x*

*<sup>ρ</sup>*(**r**)2/3 *<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup>

*ρ*(**r**)1/3

*<sup>ρ</sup>*(**r**)2/3 *<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup>

�

*µN*<sup>2</sup> �

*<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup> *µN*<sup>2</sup> �

*N*

*ν*(**r**)

�1/3

�1/3

*d***r** + � 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

*<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup> *µN*<sup>2</sup> �

*ρ*(**r**)4/3*d***r** (31)

*ρ*(**r**)1/3, (32)

1

*<sup>µ</sup><sup>N</sup>* . (33)

� � *∂N ∂E* � *ν*(**r**)

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

� ln *ρ*(**r**) � 3 *π* �1/3

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

*ρ*(**r**)1/3

*d***r** 

1 *µN*

> 1 *µN*

$$A = \frac{3}{5\mathcal{N}\mathcal{C}\_F} + \frac{1}{N(3/\pi)^{2/3}}\prime$$

and

$$B = \frac{1}{\mu N^2}.$$

#### **2.6. Summary of relationships obtained**

Finally, we present a summary of the different relations obtained in this work. Whit this results, is possible say that the information theory is a model that is subjacent to the Density Functional Theory.

In the follow section we show some results of this relations applied to some chemical process.

$$
\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} = -\frac{\boldsymbol{s}(\mathbf{r})}{N^2} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial N}\right)\_{\boldsymbol{\nu}(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{N}.\tag{35}
$$

$$
\left(\frac{\partial S(\mathbf{r})}{\partial \mu}\right)\_{\nu(\mathbf{r})} = \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial \mu}\right)\_{\nu(\mathbf{r})} - \frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial \mu}\right)\_{\nu(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{36}
$$

$$
\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \mathbf{1} + \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{37}
$$

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \left(\frac{\partial \rho(\mathbf{r})}{\partial E}\right)\_{\nu(\mathbf{r})} \ln \rho(\mathbf{r}) d\mathbf{r} + \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\nu(\mathbf{r})}.\tag{38}$$

$$\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial N}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \left(\frac{\partial f(\mathbf{r})}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} \ln \rho(\mathbf{r}) + \frac{f(\mathbf{r})}{\rho(\mathbf{r})} d\mathbf{r}.\tag{39}$$

$$\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial \mu}\right)\_{\nu(\mathbf{r})} = -\int \left(\frac{\partial s}{\partial \rho(\mathbf{r})}\right)\_{\nu(\mathbf{r})} \ln \rho(\mathbf{r}) + \frac{s}{\rho(\mathbf{r})} d\mathbf{r}.\tag{40}$$

$$\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r})^2}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \frac{d\mathbf{r}}{\rho(\mathbf{r})}.\tag{41}$$

$$
\left(\frac{\partial^2 S(\mathbf{r})}{\partial \rho(\mathbf{r}) \partial E}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N^2} \left(\frac{\partial N}{\partial E}\right)\_{\nu(\mathbf{r})} \int 1 + \ln \rho(\mathbf{r}) d\mathbf{r}.\tag{42}
$$

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Shannon Informational Entropies and Chemical Reactivity

is large. Similarly, a molecule is susceptible of an electrophilic attack at sites where *<sup>f</sup>*(**r**)<sup>−</sup> is large, because these are the regions where electron removal destabilizes the molecule the least. In chemical density functional theory, the Fukui functions are the key regioselectivity indicators for electron-transfer controlled reactions. In order to use these expressions we have chosen the Natural Population Analysis, NPA, which involves only matrix diagonalization of small subsets of the density matrix, and also requires a negligible amount of computer time. Although it is more involved than a Mulliken or Löwdin analysis, for a theoretical analysis

In Figure (1(a)) we present the electron energy profile of the reaction as a function of the reaction coordinate RX in Å. Trends in the condensed Fukui function are shown in Figure (1(b)). Analogously, trends in Hardness and chemical potential are shown in Figures (1(c))

In Figures (2(a)) and (2(b)) we present the trends in the kinetic and exchange energies as a

Energy values show that the reaction is exothermic, with the following energy values at stationary points: E(reactants)=-217.629652 A.U.; E(transition state)= -217.615808 A.U. and E(products)= -217.674564 A.U. One of the main points of interest for our purposes is the analysis of the structures along the reaction path in terms of descriptors that are related to chemical reactivity. In this specific case, we consider the kinetic and exchange energies (see Figs. (2(a)) and (2(b))). For the kinetic energy we note that there exists a region limited by a maximum at *RX* = −0.64230 and a minimum at *RX* = 1.44550, in which important chemical changes occurs. It is associated with bond forming, in the Figures (2(c))-(2(l)) present the Molecular Electrostatic Potential of this process, probably this result permit us establish a kinetically classification criteria of this reaction, that is, the principal parameter that govern this process it is the kinetic. The minima in *RX* = 1.44550 is it in relation to important chemical changes that occurs in the frontier orbitals, see the Figures (2(m)), (2(n)) (2(o)) and

In the case of the Hardness, we note that not are a correspondence between the maximum or minima of the electron, kinetic or exchange energy with the minimum and maximum of the Hardness. This points, again are related at with the structural changes that occurs in this zone, also, of course, this changes involved important changes in the frontier orbitals. A similar aspect occurs with the condensed Fukui functions, Figure (1(b)), where we note that in the point *RX* = 0.32119 exist a equality *<sup>f</sup>*(**r**)<sup>+</sup> ≃ *<sup>f</sup>*(**r**)<sup>−</sup> ≃ *<sup>f</sup>*(**r**) ≃ 0.5 with this numerical result we can note a parallelism with the relation of a hard/soft acids/basis, proposed by Pearson. That in terms of the condensed Fukui functions would be, when a chemical process occurs, exist a point where the active sites of the structures have a equalization of a kind of chemical attack, nucleophilic or electrophilic. See, the Figures (3(a)), (3(b)) and (3(c)).

In this sense, is important to note that, in this case we suspect that the more important changes in the parameters and their equalization occurs in the transition state, this can be suspect by chemical intuition, but how we can see in the different graphics this not occurs, in fact, the hardness, Fig. (1(c)), that exist a minima in *RX* = −0.64230 and a maximum in *RX* = 3.21128 this zone permit us define as a zone where the process occurs via nucleophilic

using von Neumann entropies, NPA is an attractive method [29, 30, 34].

(2(p)) where present the isosurfaces of the HOMO and LUMO orbitals.

and (1(d)), respectively.

attack.

function of the reaction coordinate.

$$\left(\frac{\partial S(\mathbf{r})}{\partial E\_k^{TF}}\right)\_{\nu(\mathbf{r})} = -\frac{3}{5\mathcal{N}\mathcal{C}\_F} \int \frac{\ln \rho(\mathbf{r})}{\rho(\mathbf{r})^{2/3}} d\mathbf{r} - \frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N}.\tag{43}$$

$$\left(\frac{\partial S(\mathbf{r})}{\partial E\_{\mathbf{x}}^{LDA}}\right)\_{\nu(\mathbf{r})} = -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\left(\frac{3}{\pi}\right)^{1/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} - \frac{1}{\mu N^2} \int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r} + \frac{1}{\mu N}.\tag{44}$$

$$
\left(\frac{S(\mathbf{r})}{E\_{k,\mathbf{r}}^{TF,LDA}}\right)\_{\mathbf{v}(\mathbf{r})} = -\int \left[A\left(\frac{1+\rho(\mathbf{r})^{1/3}}{\rho(\mathbf{r})^{2/3}}\right) - B\rho(\mathbf{r})\right] \ln \rho(\mathbf{r})d\mathbf{r} + BN. \tag{45}
$$

$$
\begin{split} \left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E^{PGB}}\right)\_{\boldsymbol{\nu}(\mathbf{r})} &= -\frac{1}{N} \int \frac{\ln \rho(\mathbf{r})}{\frac{5}{3} \mathbb{C}\rho(\mathbf{r})^{2/3} + \nu(\mathbf{r}) + \frac{4}{3} \mathcal{B}N^{2/3} \rho(\mathbf{r})^{1/3}} d\mathbf{r} \\ &+ \left[\frac{1}{N} - \frac{s(\mathbf{r})}{N^2}\right] \left(\frac{\partial N}{\partial E}\right)\_{\boldsymbol{\nu}(\mathbf{r})}. \end{split} \tag{46}
$$

#### **3. Description of a simple chemical process**

### **3.1. Reaction** *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*<sup>2</sup>

To show the application of the relations obtained, we have selected the following radical-molecule chemical reaction: *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

Structures and energies have been obtained along the reaction path using m062x/6-311++G(d,p) density functional method with Gaussian 09 [31]. The electron density was calculated with Pérez-Jordá's algorithms [8] and a D-Grid 4.6 [9]. Molecular Electrostatic Potential (MEP) isosurfaces were obtained with Molden 5.0

[22]. The Fukui function condensed was calculated using natural atomics orbitals obtained in a Natural Population Analysis [28].

The condensed Fukui function were calculated according to the following approximations:

$$f(\mathbf{r})^+ = \left|\phi(\mathbf{r})\_{\text{LUMO}}\right|^2 + \sum\_{i=1}^n \frac{\partial}{\partial N} \left|\phi(\mathbf{r})\right|^2. \tag{47}$$

$$\left|f(\mathbf{r})^{-}=\left|\phi(\mathbf{r})\_{\text{HOMO}}\right|^{2}+\sum\_{i=1}^{n}\frac{\partial}{\partial N}\left|\phi(\mathbf{r})\right|^{2}.\tag{48}$$

$$f(\mathbf{r}) = \frac{1}{2} \left( f(\mathbf{r})^+ + f(\mathbf{r})^- \right). \tag{49}$$

where *φ*(**r**)LUMO correspond to LUMO electron density and *φ*(**r**)HOMO to HOMO electron density. When a molecule accepts electrons, the electrons tend to go to places where *f*(**r**)<sup>+</sup> is large because it is at these locations that the molecule is most able to stabilize additional electrons. Therefore a molecule is susceptible of a nucleophilic attack at sites where *f*(**r**)<sup>+</sup>

is large. Similarly, a molecule is susceptible of an electrophilic attack at sites where *<sup>f</sup>*(**r**)<sup>−</sup> is large, because these are the regions where electron removal destabilizes the molecule the least. In chemical density functional theory, the Fukui functions are the key regioselectivity indicators for electron-transfer controlled reactions. In order to use these expressions we have chosen the Natural Population Analysis, NPA, which involves only matrix diagonalization of small subsets of the density matrix, and also requires a negligible amount of computer time. Although it is more involved than a Mulliken or Löwdin analysis, for a theoretical analysis using von Neumann entropies, NPA is an attractive method [29, 30, 34].

10 Quantum Mechanics

 *∂*2*S*(**r**) *∂ρ*(**r**)*∂E*

> *∂S*(**r**) *<sup>∂</sup>ETF k*

 *∂S*(**r**) *<sup>∂</sup>ELDA x*

*S*(**r**) *ETF*,*LDA k*,*x*

 *∂S*(**r**) *∂EPGB*  *ν*(**r**)

in a Natural Population Analysis [28].

*ν*(**r**)

*ν*(**r**)

*ν*(**r**)

*ν*(**r**)

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*2

<sup>=</sup> <sup>−</sup> <sup>3</sup> 5*NCF*

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

= −

<sup>=</sup> <sup>−</sup> <sup>1</sup> *N*

> + 1 *<sup>N</sup>* <sup>−</sup> *<sup>s</sup>*(**r**) *N*2

**3. Description of a simple chemical process**

**3.1. Reaction** *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*<sup>2</sup>

 *A* 

5

radical-molecule chemical reaction: *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

Electrostatic Potential (MEP) isosurfaces were obtained with Molden 5.0

*<sup>f</sup>*(**r**)<sup>+</sup> = |*φ*(**r**)LUMO|

*<sup>f</sup>*(**r**)<sup>−</sup> <sup>=</sup> <sup>|</sup>*φ*(**r**)HOMO<sup>|</sup>

*<sup>f</sup>*(**r**) = <sup>1</sup> 2 

 *∂N ∂E ν*(**r**) 

ln *ρ*(**r**)

 ln *ρ*(**r**) 3 *π* 1/3

*<sup>ρ</sup>*(**r**)2/3 *<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup>

*ρ*(**r**)1/3

1 + *ρ*(**r**)1/3 *ρ*(**r**)2/3

ln *ρ*(**r**)

 *∂N ∂E ν*(**r**)

<sup>3</sup>*Cρ*(**r**)2/3 <sup>+</sup> *<sup>ν</sup>*(**r**) + <sup>4</sup>

To show the application of the relations obtained, we have selected the following

Structures and energies have been obtained along the reaction path using m062x/6-311++G(d,p) density functional method with Gaussian 09 [31]. The electron density was calculated with Pérez-Jordá's algorithms [8] and a D-Grid 4.6 [9]. Molecular

[22]. The Fukui function condensed was calculated using natural atomics orbitals obtained

The condensed Fukui function were calculated according to the following approximations:

<sup>2</sup> + *n* ∑ *i*=1

<sup>2</sup> + *n* ∑ *i*=1

*<sup>f</sup>*(**r**)<sup>+</sup> <sup>+</sup> *<sup>f</sup>*(**r**)−

where *φ*(**r**)LUMO correspond to LUMO electron density and *φ*(**r**)HOMO to HOMO electron density. When a molecule accepts electrons, the electrons tend to go to places where *f*(**r**)<sup>+</sup> is large because it is at these locations that the molecule is most able to stabilize additional electrons. Therefore a molecule is susceptible of a nucleophilic attack at sites where *f*(**r**)<sup>+</sup>

*∂ <sup>∂</sup><sup>N</sup>* <sup>|</sup>*φ*(**r**)<sup>|</sup>

*∂ <sup>∂</sup><sup>N</sup>* <sup>|</sup>*φ*(**r**)<sup>|</sup>

*µN*<sup>2</sup> 

*<sup>d</sup>***<sup>r</sup>** <sup>−</sup> <sup>1</sup> *µN*<sup>2</sup> 

− *Bρ*(**r**)

<sup>3</sup> *BN*2/3*ρ*(**r**)1/3 *<sup>d</sup>***<sup>r</sup>**

1 + ln *ρ*(**r**)*d***r**. (42)

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

. (46)

1

ln *ρ*(**r**)*d***r** + *BN*. (45)

<sup>2</sup> . (47)

<sup>2</sup> . (48)

. (49)

*<sup>µ</sup><sup>N</sup>* . (43)

*<sup>µ</sup><sup>N</sup>* . (44)

1

*ρ*(**r**)ln *ρ*(**r**)*d***r** +

In Figure (1(a)) we present the electron energy profile of the reaction as a function of the reaction coordinate RX in Å. Trends in the condensed Fukui function are shown in Figure (1(b)). Analogously, trends in Hardness and chemical potential are shown in Figures (1(c)) and (1(d)), respectively.

In Figures (2(a)) and (2(b)) we present the trends in the kinetic and exchange energies as a function of the reaction coordinate.

Energy values show that the reaction is exothermic, with the following energy values at stationary points: E(reactants)=-217.629652 A.U.; E(transition state)= -217.615808 A.U. and E(products)= -217.674564 A.U. One of the main points of interest for our purposes is the analysis of the structures along the reaction path in terms of descriptors that are related to chemical reactivity. In this specific case, we consider the kinetic and exchange energies (see Figs. (2(a)) and (2(b))). For the kinetic energy we note that there exists a region limited by a maximum at *RX* = −0.64230 and a minimum at *RX* = 1.44550, in which important chemical changes occurs. It is associated with bond forming, in the Figures (2(c))-(2(l)) present the Molecular Electrostatic Potential of this process, probably this result permit us establish a kinetically classification criteria of this reaction, that is, the principal parameter that govern this process it is the kinetic. The minima in *RX* = 1.44550 is it in relation to important chemical changes that occurs in the frontier orbitals, see the Figures (2(m)), (2(n)) (2(o)) and (2(p)) where present the isosurfaces of the HOMO and LUMO orbitals.

In the case of the Hardness, we note that not are a correspondence between the maximum or minima of the electron, kinetic or exchange energy with the minimum and maximum of the Hardness. This points, again are related at with the structural changes that occurs in this zone, also, of course, this changes involved important changes in the frontier orbitals. A similar aspect occurs with the condensed Fukui functions, Figure (1(b)), where we note that in the point *RX* = 0.32119 exist a equality *<sup>f</sup>*(**r**)<sup>+</sup> ≃ *<sup>f</sup>*(**r**)<sup>−</sup> ≃ *<sup>f</sup>*(**r**) ≃ 0.5 with this numerical result we can note a parallelism with the relation of a hard/soft acids/basis, proposed by Pearson. That in terms of the condensed Fukui functions would be, when a chemical process occurs, exist a point where the active sites of the structures have a equalization of a kind of chemical attack, nucleophilic or electrophilic. See, the Figures (3(a)), (3(b)) and (3(c)).

In this sense, is important to note that, in this case we suspect that the more important changes in the parameters and their equalization occurs in the transition state, this can be suspect by chemical intuition, but how we can see in the different graphics this not occurs, in fact, the hardness, Fig. (1(c)), that exist a minima in *RX* = −0.64230 and a maximum in *RX* = 3.21128 this zone permit us define as a zone where the process occurs via nucleophilic attack.

In Figure (3(e)) and (3(d)) we present, the tendencies of Shannon entropy and the electron energy with the LDA approximation, *ELDA*. In this case, the *ELDA* increase basically in a linear way, but in the point *RX* = 2.73035 have a maximum, this maximum appears to in the chemical potential tendency. Respect to the Shannon entropy we note more structure than *ELDA*, and in this case, we note than the Shannon entropy have a similar tendency than the electron energy, in the same form than the energy the Shannon entropy can detect the transition state, see the Figure (3(f)).

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695

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energy exchange and increased kinetic energy. It is relevant note, that the Shannon entropy, see Figure (4(b)), can detect three zones where the frequencies are negative, the first of them is between −3.5 ≤ *RX* ≤ −2.2 the second in −0.5 ≤ *RX*0.5 and the last in 3.8 ≤ *RX* ≤ 4.0. In the first and third zone, the Shannon entropy exhibits a change in their curvature, also, note that this last observation have a correspondence with the other parameters such, hardness,

In Figure (5(b)) we compare the trend of the derivative of Snannon's entropy with respect

are not the same everywhere, there is a perfect coincidence in the region close to the transition

A similar situation occurs with the derivative of Shannon's entropy with respect to the

<sup>=</sup> <sup>−</sup> <sup>1</sup> 3 *π*

Finally, a plot of the derivative of Shannon's entropy with respect to the electronic energy

(Figure (5(f)).

The relations that were suggested above based on numerical evidence can be summarized

, and the inverse of the Exchange energy,

1/3 *<sup>ρ</sup>*(**r**)4/3*d***<sup>r</sup>**

as a function of RX, behaves in a manner that is remarkably similar to that of the

1/3

*<sup>ν</sup>*(**r**) as a function of RX, with that of the DFT chemical

Shannon Informational Entropies and Chemical Reactivity

, which is associated to hardness. Even though absolute values

,

. (50)

. (52)

*ρ*(**r**)4/3*d***r**. (51)

softness, chemical potential and with some Shannon entropy derivatives.

*∂N* 

state, in terms of the RX position as well as in the absolute values.

1 *ELDA i*

*∂µ ν*(**r**)

 *∂S*(**r**) *∂N*

 *∂S*(**r**) *∂µ*

 *∂S*(**r**) *∂E*

 *ν*(**r**) ≃ *∂µ ∂N ν*(**r**)

 *ν*(**r**) ≃ *∂N ∂µ ν*(**r**)

*ν*(**r**)

≃ −3 4 3 *π*

From the numerical results obtained from Eqs. (50) and (52), we have been able to establish a linear relationship between the parameters involved, and the following expressions have

to the number of electrons: *<sup>∂</sup>S*(**r**)

*∂N ν*(**r**)

*∂µ ν*(**r**)

reactivity descriptor: *∂µ*

chemical potential: *<sup>∂</sup>S*(**r**)

DFT descriptor for softness: *<sup>∂</sup><sup>N</sup>*

see Figure (5(d)).

been obtained:

 *<sup>∂</sup>S*(**r**) *∂E ν*(**r**)

as:

By comparison between the Shannon entropy and electron energy profiles we can note that the Shannon entropy, have a zone delimited in the region −0.5 ≤ *RX* ≤ 0.5, see Figure (3(f)), that have a correspondence with the transition state zone, this is an important observation, because, the Shannon entropy can detect a transition state zone, where occurs a transfer and redistribution of electron density. Comparing, the tendencies of condensed Fukui function, with the Shannon entropy tendency, we note that this zone, have a relation with a charge transfer between the methyl and the molecule and is in this zone where occurs the process of bond forming. In the same way that the condensed Fukui function, this zone is where the subtle interactions among frontier orbitals occurs, and permit to the molecular system start a complex process of the chemical bond forming, in Figure (4(a)) we present the tendency of a normal mode of vibration, and we note that, the zone predicted by the Shannon entropy exhibit a correspondence with a zone of the negative values of the frequencies, consequently, is possible that we can not speak of a specific point in the reaction path where occurs a bond breaking or bond forming. A similar argumentation, can be applied to the description of the transition state zone. Note the importance of this, is possible that, still with the modern techniques of the experimental chemistry, such the femtochemistry [19–21], we can not detect just a transition state structure, or else, a zone of transition. This zone of transition, is accoted by a zone where the Shannon entropy tendency have a slope approximately to zero, see the Figure (4(b)) accordingly, we can say that the Shannon entropy can detect and predict the zone where the most important chemical changes will be occurs, so, this kind of entropy permit us reveals some chemical aspects that are subjacent in a chemical process. Other important, observation is that the maximum of the electron energy, not correspond to the minimum of the frequencies, it is represents, probability, a conflict between the convectional interpretation, in the sense that the maximum of the energy correspond at one possible transition state structure that, in general, have the most negative value of the frequencies, but how we show in this case, this not occurs, by other hand, the minimum value of the frequencies it is related at one equality with the condensed Fukui functions, see the Figure (4(c)).

In general terms, is important to note, that the frequencies tendencies have a tree zones of negative frequencies, these Hessian values represent the transition vector which show maxima at the vicinity of the transition state. Several features are worth mentioning, the TS corresponds indeed to a saddle point, maxima at the Hessian correspond to high kinetic energy values (largest frequencies for the energy cleavage reservoirs) since they fit with maximal values in the entropy profile, and the Hessian is minimal at the TS, where the kinetic energy is the lowest (minimal molecular frequency) and it corresponds to a saddle point. In this case, the analysis of frequencies can give us, a general idea about that this mechanics occurs in tree steps, in each one, occurs some structural rearrays at expense of a decrease in energy exchange and increased kinetic energy. It is relevant note, that the Shannon entropy, see Figure (4(b)), can detect three zones where the frequencies are negative, the first of them is between −3.5 ≤ *RX* ≤ −2.2 the second in −0.5 ≤ *RX*0.5 and the last in 3.8 ≤ *RX* ≤ 4.0. In the first and third zone, the Shannon entropy exhibits a change in their curvature, also, note that this last observation have a correspondence with the other parameters such, hardness, softness, chemical potential and with some Shannon entropy derivatives.

In Figure (5(b)) we compare the trend of the derivative of Snannon's entropy with respect to the number of electrons: *<sup>∂</sup>S*(**r**) *∂N <sup>ν</sup>*(**r**) as a function of RX, with that of the DFT chemical reactivity descriptor: *∂µ ∂N ν*(**r**) , which is associated to hardness. Even though absolute values are not the same everywhere, there is a perfect coincidence in the region close to the transition state, in terms of the RX position as well as in the absolute values.

A similar situation occurs with the derivative of Shannon's entropy with respect to the chemical potential: *<sup>∂</sup>S*(**r**) *∂µ ν*(**r**) , and the inverse of the Exchange energy,

$$\frac{1}{E\_i^{LDA}} = -\frac{1}{\left(\frac{3}{\pi}\right)^{1/3} \int \rho(\mathbf{r})^{4/3} d\mathbf{r}} \mathbf{r}$$

see Figure (5(d)).

12 Quantum Mechanics

(4(c)).

transition state, see the Figure (3(f)).

In Figure (3(e)) and (3(d)) we present, the tendencies of Shannon entropy and the electron energy with the LDA approximation, *ELDA*. In this case, the *ELDA* increase basically in a linear way, but in the point *RX* = 2.73035 have a maximum, this maximum appears to in the chemical potential tendency. Respect to the Shannon entropy we note more structure than *ELDA*, and in this case, we note than the Shannon entropy have a similar tendency than the electron energy, in the same form than the energy the Shannon entropy can detect the

By comparison between the Shannon entropy and electron energy profiles we can note that the Shannon entropy, have a zone delimited in the region −0.5 ≤ *RX* ≤ 0.5, see Figure (3(f)), that have a correspondence with the transition state zone, this is an important observation, because, the Shannon entropy can detect a transition state zone, where occurs a transfer and redistribution of electron density. Comparing, the tendencies of condensed Fukui function, with the Shannon entropy tendency, we note that this zone, have a relation with a charge transfer between the methyl and the molecule and is in this zone where occurs the process of bond forming. In the same way that the condensed Fukui function, this zone is where the subtle interactions among frontier orbitals occurs, and permit to the molecular system start a complex process of the chemical bond forming, in Figure (4(a)) we present the tendency of a normal mode of vibration, and we note that, the zone predicted by the Shannon entropy exhibit a correspondence with a zone of the negative values of the frequencies, consequently, is possible that we can not speak of a specific point in the reaction path where occurs a bond breaking or bond forming. A similar argumentation, can be applied to the description of the transition state zone. Note the importance of this, is possible that, still with the modern techniques of the experimental chemistry, such the femtochemistry [19–21], we can not detect just a transition state structure, or else, a zone of transition. This zone of transition, is accoted by a zone where the Shannon entropy tendency have a slope approximately to zero, see the Figure (4(b)) accordingly, we can say that the Shannon entropy can detect and predict the zone where the most important chemical changes will be occurs, so, this kind of entropy permit us reveals some chemical aspects that are subjacent in a chemical process. Other important, observation is that the maximum of the electron energy, not correspond to the minimum of the frequencies, it is represents, probability, a conflict between the convectional interpretation, in the sense that the maximum of the energy correspond at one possible transition state structure that, in general, have the most negative value of the frequencies, but how we show in this case, this not occurs, by other hand, the minimum value of the frequencies it is related at one equality with the condensed Fukui functions, see the Figure

In general terms, is important to note, that the frequencies tendencies have a tree zones of negative frequencies, these Hessian values represent the transition vector which show maxima at the vicinity of the transition state. Several features are worth mentioning, the TS corresponds indeed to a saddle point, maxima at the Hessian correspond to high kinetic energy values (largest frequencies for the energy cleavage reservoirs) since they fit with maximal values in the entropy profile, and the Hessian is minimal at the TS, where the kinetic energy is the lowest (minimal molecular frequency) and it corresponds to a saddle point. In this case, the analysis of frequencies can give us, a general idea about that this mechanics occurs in tree steps, in each one, occurs some structural rearrays at expense of a decrease in Finally, a plot of the derivative of Shannon's entropy with respect to the electronic energy *<sup>∂</sup>S*(**r**) *∂E ν*(**r**) as a function of RX, behaves in a manner that is remarkably similar to that of the DFT descriptor for softness: *<sup>∂</sup><sup>N</sup> ∂µ ν*(**r**) (Figure (5(f)).

The relations that were suggested above based on numerical evidence can be summarized as:

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} \simeq \left(\frac{\partial \mu}{\partial N}\right)\_{\nu(\mathbf{r})}.\tag{50}$$

$$
\left(\frac{\partial S(\mathbf{r})}{\partial \mu}\right)\_{\nu(\mathbf{r})} \simeq -\frac{3}{4} \left(\frac{3}{\pi}\right)^{1/3} \int \rho(\mathbf{r})^{4/3} d\mathbf{r}.\tag{51}
$$

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial E}\right)\_{\nu(\mathbf{r})} \simeq \left(\frac{\partial N}{\partial \mu}\right)\_{\nu(\mathbf{r})}.\tag{52}$$

From the numerical results obtained from Eqs. (50) and (52), we have been able to establish a linear relationship between the parameters involved, and the following expressions have been obtained:

$$\text{Hardness} = \left(\frac{\partial \mu}{\partial N}\right)\_{\nu(\mathbf{r})} = 28.3141 \left(\frac{\partial S(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} + 0.0226,\tag{53}$$

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$$\text{Softness} = \left(\frac{\partial N}{\partial \mu}\right)\_{\nu(\mathbf{r})} = 24.1088 \left(\frac{\partial S(\mathbf{r})}{\partial E}\right)\_{\nu(\mathbf{r})} + 0.5876,\tag{54}$$

196.05



 **C H H H**

 **C H H H**

 **C H**

 **F**

(e) *RX* = −0.16058

 **C H**

 **F**

(i) *RX* = 0.64242

 **C H**

 **C H**

 **F**

 **C H H**

 **C H H H**

(m) HOMO in *RX* = 2.08795

> **C H H H**

(p) LUMO in *RX* = 2.24856

 **F**

 **C H H**

 **C H H**

 **C H H**

Exchange Energy [A.U.]

 **C H**

 **F**

(d) *RX* = −0.32112

 **C H**

 **F**

(h) *RX* = 0.48180

 **C H**

(l) *RX* = 1.44550

 **C H**

 **C H H H**

(o) HOMO in *RX* = 2.24856

**Figure 2.** Isosurfaces of the Molecular Electrostatic Potential, in −0.64230 ≤ *RX* ≤ 1.44550.

 **F**

 **C H H**

 **F**

 **C H H**

 **C H H**

 **C H H H**

 **C H H H**

 **C H H H**

 **C H H** -8 -6 -4 -2 0 2 4

Shannon Informational Entropies and Chemical Reactivity

RX (b) Exchange Energy.

> **C H**

 **F**

(f) *RX* = 0.00000

 **C H**

(j) *RX* = 0.96366

 **C H**

 **F**

 **C H H**

 **F**

 **C H H**

 **C H H H**

 **C H H H**

 **C H H H**

(n) LUMO in *RX* = 2.08795  **C H H**

RX (a) Kinetic Energy.

> **C H**

 **F**

(c) *RX* = −0.64230

 **C H**

 **F**

(g) *RX* = 0.32119

 **C H**

(k) *RX* = 1.12427

 **F**

 **C H H**

 **C H H**

 **C H H H**

 **C H H H**

 **C H H H**

 **C H H**

196.1

196.15

Kinetic Energy [A.U.]

196.2

196.25

196.3

Thus, the hardness and softness values of this chemical reaction can be obtained with a good level of accuracy from the derivatives of Shannon's entropy.

**Figure 1.** Trends of the reaction *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

14 Quantum Mechanics


 0.152 0.154 0.156 0.158 0.16 0.162 0.164 0.166 0.168 0.17

Hardness [A.U.]

E [A.U.]

Hardness =

Softness =

 *∂µ ∂N ν*(**r**)

 *∂N ∂µ* 

level of accuracy from the derivatives of Shannon's entropy.


RX (a) Electron energy profile.


RX (c) Hardness.

**Figure 1.** Trends of the reaction *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

*ν*(**r**)

= 28.3141

= 24.1088

Thus, the hardness and softness values of this chemical reaction can be obtained with a good

 *∂S*(**r**) *∂N*

 *∂S*(**r**) *∂E*

 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

subsystem *CH*3.


Chemical Potential [A.U.]

f(r)

 *ν*(**r**)

 *ν*(**r**) + 0.0226, (53)

+ 0.5876, (54)


RX A1 A2 A3 B1 B2 B3 (b) Condensed Fukui function profile *A*1 : *f*(**r**)−, *A*2 : *f*(**r**)+, *A*3 : *f*(**r**), *B*1 : *f*(**r**)−, *B*2 : *f*(**r**)+, *B*3 : *f*(**r**), where *A* denote the subsystem *CH*2*CHF* and *B* is the


RX (d) Chemical potential.

**Figure 2.** Isosurfaces of the Molecular Electrostatic Potential, in −0.64230 ≤ *RX* ≤ 1.44550.

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entropy.

f(r)

 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequencies.

subsystem *CH*<sup>3</sup>

**Figure 4.** Comparison among Frequencies, Shannon entropy and Fukui function.

A1 A2 A3

Frequencies [cm^-1]

Frequencies [cm^-1]


RX (a) Frequencies.


RX Frequencies S(r) (b) Comparison between the frequencies and the Shannon


RX

B1 B2 B3

(c) Comparison between the condensed Fukui function and the

Where *A*1 : *f*(**r**)−, *A*2 : *f*(**r**)+, *A*3 : *f*(**r**), *B*1 : *f*(**r**)−, *B*2 : *f*(**r**)+, *B*3 : *f*(**r**), where *A* denote the subsystem *CH*2*CHF* and *B* is the

 4.575 4.58 4.585 4.59 4.595 4.6 4.605 4.61 4.615 4.62

> -1000 -800 -600 -400 -200 0 200 400

Frequencies

Frequencies [cm^-1]

S(r)

Shannon Informational Entropies and Chemical Reactivity

**Figure 3.** Trends of the reaction *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

16 Quantum Mechanics

 **C H**

 **0.3717**

 **-0.0724**

(a) HOMO, in *RX* = 0.32119

 **C H**

(c) MEP, in *RX* = 0.32119


RX (e) Shannon entropy

**Figure 3.** Trends of the reaction *CH*2*CHF* + *CH*<sup>3</sup> → *CH*3*CHFCH*2.

 **F**

 **C H H**

 **H**


S(r)

 4.575 4.58 4.585 4.59 4.595 4.6 4.605 4.61 4.615 4.62

Energy LDA [A.U.]

 **C H H**

 4.575 4.58 4.585 4.59 4.595 4.6 4.605 4.61 4.615 4.62

S(r)

 **-0.4049**

 **F**

 **C H H**

 **0.2286 0.0727**

 **-0.1476**

 **-0.0570**

 **H**

 **C H**

(b) LUMO, in *RX* = 0.32119


RX (d) Exchange energy with LDA


RX S(r) E (f) Shannon entropy and the Electron energy


E

 **0.0478**

 **-0.1565 -0.0968 -0.2435**

 **F**

 **0.2967**

 **C H H**

 **0.0719 0.1061**

 **-0.0986**

 **-0.1356**

 **0.0117**

 **H**

 **0.1226**

 **C H H**

 **0.1546**

 **0.0407**

 **-0.0564**

 **-0.0073**

 **0.0364 0.0365**

 **C H H**

 **0.0807 0.0768**

 **-0.1580**

 **-0.0641**

 **-0.0531**

(b) Comparison between the frequencies and the Shannon entropy.

(c) Comparison between the condensed Fukui function and the Frequencies.

Where *A*1 : *f*(**r**)−, *A*2 : *f*(**r**)+, *A*3 : *f*(**r**), *B*1 : *f*(**r**)−, *B*2 : *f*(**r**)+, *B*3 : *f*(**r**), where *A* denote the subsystem *CH*2*CHF* and *B* is the subsystem *CH*<sup>3</sup>

**Figure 4.** Comparison among Frequencies, Shannon entropy and Fukui function.

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Shannon Informational Entropies and Chemical Reactivity

conformer, and an Intrinsic Reaction Coordinate analysis (IRC) has been performed, with ten degrees steps in the dihedral angle. This system was calculated in Gaussian 03 [32], using

In Figure (6(a)) we present the electron energy and Shannon entropy profiles. An excellent agreement is observed between the two curves. Furthermore, a clear similarity is also observed between the behavior, along the reaction path, of the hardness and of the first derivative of Shannon's entropy with respect to the number of electrons. This is shown in

In Figure (6(c)) we show the kinetic and exchange energies as a function of RX. Both quantities exhibit inverse behaviors,the position of the minima in one of them coinciding

One of the most important results is shown in Figure (6(d)), where the exchange energy is plotted against Shannon's entropy along the reaction path. A perfect linear correlation is

i) The Shannon entropy in position space can be used as a measure of the exchange effects

ii) There exists a direct relationship between the Shannon entropy and the exchange energy.

Based in these conclusions and results, a new research line is being developed, to construct

. (55)

*Ee* = *AS*(**r**) − *B*, (57)

*ρ*(**r**)4/3*d***r**. (56)

observed between these apparently unconnected quantities, leading to conclude that

The numerical results obtained above cab be summarized in the following manner:

*<sup>S</sup>*(**r**) ≃ −<sup>3</sup>

4 3 *π*

The last equation suggests that there may be a fundamental connection between Chemical Density Functional Theory and Information Theory. In Table (1), we present the numerical results of Shannon's entropy and exchange energy. Using these results and a least square regression, is it possible to obtain the following linear relation between Shannon entropy and

where *A* = 2.2127 and *B* = 21.0061. With this equation, we can reproduce the Exchange

1/3

B3LYP/cc-pVTZ.

Figure (6(b)).

with the maxima of the other.

in molecular systems.

exchange energy:

Energy with a precision of 1 × <sup>10</sup><sup>−</sup>6.

a functional based on an entropic criterion.

 *∂S*(**r**) *∂N*

 *ν*(**r**) ≃ *∂N ∂µ ν*(**r**)

**Figure 5.** Trends of the first derivatives of Shannon entropy.

#### **3.2. Ethane conformational analysis**

In this section we present a conformational analysis of ethane, as a function of the dihedral rotation angle between the two methyl groups. The initial point corresponds to the eclipsed conformer, and an Intrinsic Reaction Coordinate analysis (IRC) has been performed, with ten degrees steps in the dihedral angle. This system was calculated in Gaussian 03 [32], using B3LYP/cc-pVTZ.

18 Quantum Mechanics

 0.0038 0.0039 0.004 0.0041 0.0042 0.0043 0.0044 0.0045 0.0046


 0.245 0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295

De

Dc

Df


 0.0038 0.0039 0.004 0.0041 0.0042 0.0043 0.0044 0.0045 0.0046

(b) Trend of Df =


(d) Trend of Dp =

exchange energy.

 0.245 0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295

(f) Trend of

 *<sup>∂</sup>S*(**r**) *∂E ν*(**r**)

De

In this section we present a conformational analysis of ethane, as a function of the dihedral rotation angle between the two methyl groups. The initial point corresponds to the eclipsed

Dp

Df


RX Df Hardness


RX Dp E.E.

> *<sup>∂</sup>S*(**r**) *∂µ ν*(**r**)


RX De Softness

and Softness =

and Hardness =

 *<sup>∂</sup>S*(**r**) *∂N ν*(**r**)  0.13 0.132 0.134 0.136 0.138 0.14 0.142 0.144 0.146 0.148 0.15

 *∂µ ∂N ν*(**r**) .


and the inverse of

 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

 *∂N ∂µ ν*(**r**) .

Softness

1/E.E.

Hardness

RX

 *<sup>∂</sup>S*(**r**) *∂N ν*(**r**) .


RX


RX

 *<sup>∂</sup>S*(**r**) *∂E ν*(**r**) .

(e) Trend of

**Figure 5.** Trends of the first derivatives of Shannon entropy.

**3.2. Ethane conformational analysis**

 *<sup>∂</sup>S*(**r**) *∂µ ν*(**r**) .

(a) Trend of

(c) Trend of Dp =

In Figure (6(a)) we present the electron energy and Shannon entropy profiles. An excellent agreement is observed between the two curves. Furthermore, a clear similarity is also observed between the behavior, along the reaction path, of the hardness and of the first derivative of Shannon's entropy with respect to the number of electrons. This is shown in Figure (6(b)).

In Figure (6(c)) we show the kinetic and exchange energies as a function of RX. Both quantities exhibit inverse behaviors,the position of the minima in one of them coinciding with the maxima of the other.

One of the most important results is shown in Figure (6(d)), where the exchange energy is plotted against Shannon's entropy along the reaction path. A perfect linear correlation is observed between these apparently unconnected quantities, leading to conclude that


Based in these conclusions and results, a new research line is being developed, to construct a functional based on an entropic criterion.

The numerical results obtained above cab be summarized in the following manner:

$$\left(\frac{\partial \mathcal{S}(\mathbf{r})}{\partial N}\right)\_{\nu(\mathbf{r})} \simeq \left(\frac{\partial N}{\partial \mu}\right)\_{\nu(\mathbf{r})}.\tag{55}$$

$$\mathcal{S}(\mathbf{r}) \simeq -\frac{3}{4} \left(\frac{3}{\pi}\right)^{1/3} \int \rho(\mathbf{r})^{4/3} d\mathbf{r}.\tag{56}$$

The last equation suggests that there may be a fundamental connection between Chemical Density Functional Theory and Information Theory. In Table (1), we present the numerical results of Shannon's entropy and exchange energy. Using these results and a least square regression, is it possible to obtain the following linear relation between Shannon entropy and exchange energy:

$$E\_{\mathcal{E}} = AS(\mathbf{r}) - B\_{\prime} \tag{57}$$

where *A* = 2.2127 and *B* = 21.0061. With this equation, we can reproduce the Exchange Energy with a precision of 1 × <sup>10</sup><sup>−</sup>6.


**Table 1.** Numerical values of Shannon's entropy and Exchange Energy, in the ethane conformational analysis, where *A* = 2.2127 and *B* = 21.0061.

(c) Exchange Energy and Kinetic Energy profiles

(d) Exchange Energy and Shannon entropy profiles

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Shannon Informational Entropies and Chemical Reactivity

**4. Conclusion**

Shannon entropy.

In this work, I have derived relationships that connect Shannon's entropy and its derivatives, with well-known concepts in Density Functional Theory. Numerical applications of these

This has been permit us start a new investigation line about it, and with some this results we continue whit the study of the formalism for the construction of a functional based in a principles of physics, and the information theory, also, we pretend continue the develop of the some models that permit us find the direct relation between electron energy with the

By other hand, the application of the concepts of the information theory permit do a description more accurate than the description based in only a energetically criteria, and continue with the spirit of some works of Frieden, we speculate that is possible define or found a form that derived the DFT from some fundamental expression, that come form the Information Theory, as Frieden derivations of the fundamental equations of the

Finally, with this example we have tried to link information from a system that is subjected to a process with the physical and chemical changes. Thus, we have linked the concept of *information*, which is an epistemological concept completely with ontological concepts and the solution concepts or interpretation of the results allows us feedback on these concepts in

By other part, is probable that today do not exist a ortodoxical 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

So, there is no doubt that both knowledge and the praxis and reality as knowledge scientific understanding and also, is it clear that information concept and the model itself is interdisciplinary or transdisciplinary. The concept and moreover, the model itself, promotes a systematic relation with causal analogies and parallelism with scientific knowledge, which transcends the framework of the source domain and extend in various directions, thus making the knowledge acquires an unusual resonance, as this, we believe it is feasible to

This model is not intended that the manner of the old school, that using metaphysical substance, the particularities of the processes reveal themselves to us in the end as a progressive manifestation of homogeneous order or a unitary whole and absolute. It is simply to promote and implement a partnership scheme which promises analog route and

I wish to thank José María Pérez Jordá and M. Kohout for kindly providing their numerical codes. I acknowledge financial support from Prof. Annik Vivier-Bunge through project "Red

de Química Teórica para el Medio Ambiente y Salud" and for helpful discussions.

relationships have been performed for two simple problems.

Thermodynamics, or the derivation of the Scrhödinger equation.

ontological terms, according to the author, abstract and more general.

need be discussed to try of establish a formal relation between concepts.

complement the explanations of natural processes and natural systems.

cover knowledge in a way easier.

**Acknowledgments**

**Figure 6.** Tendencies obtained for the Ethane Conformational Analysis.

### **4. Conclusion**

20 Quantum Mechanics

*A* = 2.2127 and *B* = 21.0061.


 71.402 71.404 71.406 71.408 71.41 71.412 71.414 71.416

K.E. [A.U.]

E [A.U.]

0 20 40 60 80 100 120 140 160 180

Angle [Degrees] E S(r) (a) Energy and Shannon entropy profiles

0 20 40 60 80 100 120 140 160 180

Angle [Degrees] K.E. E.E. (c) Exchange Energy and Kinetic Energy profiles

**Figure 6.** Tendencies obtained for the Ethane Conformational Analysis.

**Angle** *S*(**r**) *E*.*E*. *Ee* = *AS*(**r**) − *B* **precision [nats] [A.U.] [A.U.]** 0 4.5902766 -10.849190 -10.8492232214 0.9999 10 4.5901580 -10.849449 -10.8494856465 0.9999 20 4.5898836 -10.850050 -10.8500928087 0.9999 30 4.5895363 -10.850815 -10.8508612761 0.9999 40 4.5892201 -10.851516 -10.8515609289 0.9999 50 4.5890276 -10.851953 -10.8519868718 0.9999 60 4.5889557 -10.852113 -10.8521459642 0.9999 70 4.5890276 -10.851953 -10.8519868718 0.9999 80 4.5892201 -10.851516 -10.8515609289 0.9999 90 4.5895363 -10.850815 -10.8508612761 0.9999 100 4.5898836 -10.850050 -10.8500928087 0.9999 110 4.5901580 -10.849449 -10.8494856465 0.9999 120 4.5902766 -10.849190 -10.8492232214 0.9999 130 4.5901580 -10.849449 -10.8494856465 0.9999 140 4.5898836 -10.850050 -10.8500928087 0.9999 150 4.5895363 -10.850815 -10.8508612761 0.9999 160 4.5892201 -10.851516 -10.8515609289 0.9999 170 4.5890276 -10.851953 -10.8519868718 0.9999 180 4.5889557 -10.852113 -10.8521459642 0.9999

**Table 1.** Numerical values of Shannon's entropy and Exchange Energy, in the ethane conformational analysis, where

S(r) [nats]

dS/dN

 0.44016 0.44017 0.44018 0.44019 0.4402 0.44021 0.44022 0.44023 0.44024

 4.5888 4.589 4.5892 4.5894 4.5896 4.5898 4.59 4.5902 4.5904 0 20 40 60 80 100 120 140 160 180

Angle dS/dN Softness (b) Derivative of Shannon entropy and Softness profile


E.E. [A.U.] (d) Exchange Energy and Shannon entropy profiles

 0.098045 0.098046 0.098047 0.098048 0.098049 0.09805 0.098051 0.098052

Softness [a.u.]

 4.5888 4.589 4.5892 4.5894 4.5896 4.5898 4.59 4.5902 4.5904


E.E. [A.U.]

S(r) [nats]

In this work, I have derived relationships that connect Shannon's entropy and its derivatives, with well-known concepts in Density Functional Theory. Numerical applications of these relationships have been performed for two simple problems.

This has been permit us start a new investigation line about it, and with some this results we continue whit the study of the formalism for the construction of a functional based in a principles of physics, and the information theory, also, we pretend continue the develop of the some models that permit us find the direct relation between electron energy with the Shannon entropy.

By other hand, the application of the concepts of the information theory permit do a description more accurate than the description based in only a energetically criteria, and continue with the spirit of some works of Frieden, we speculate that is possible define or found a form that derived the DFT from some fundamental expression, that come form the Information Theory, as Frieden derivations of the fundamental equations of the Thermodynamics, or the derivation of the Scrhödinger equation.

Finally, with this example we have tried to link information from a system that is subjected to a process with the physical and chemical changes. Thus, we have linked the concept of *information*, which is an epistemological concept completely with ontological concepts and the solution concepts or interpretation of the results allows us feedback on these concepts in ontological terms, according to the author, abstract and more general.

By other part, is probable that today do not exist a ortodoxical 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 establish a formal relation between concepts.

So, there is no doubt that both knowledge and the praxis and reality as knowledge scientific understanding and also, is it clear that information concept and the model itself is interdisciplinary or transdisciplinary. The concept and moreover, the model itself, promotes a systematic relation with causal analogies and parallelism with scientific knowledge, which transcends the framework of the source domain and extend in various directions, thus making the knowledge acquires an unusual resonance, as this, we believe it is feasible to complement the explanations of natural processes and natural systems.

This model is not intended that the manner of the old school, that using metaphysical substance, the particularities of the processes reveal themselves to us in the end as a progressive manifestation of homogeneous order or a unitary whole and absolute. It is simply to promote and implement a partnership scheme which promises analog route and cover knowledge in a way easier.

### **Acknowledgments**

I wish to thank José María Pérez Jordá and M. Kohout for kindly providing their numerical codes. I acknowledge financial support from Prof. Annik Vivier-Bunge through project "Red de Química Teórica para el Medio Ambiente y Salud" and for helpful discussions.

### **Author details**

Nelson Flores-Gallegos

Universidad Autónoma Metropolitana-Iztapalapa, México

### **References**

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Shannon Informational Entropies and Chemical Reactivity

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[22] Schaftenaar, G. and Noordik, J. H., MOLDEN: a pre- and post-processing program for molecular and electronic structures. *J. Comput. Aided Mol. Design.*, *2000*, 123-134.

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[24] K. Fukui, T. Yonezawa and H. Shingu. A molecular orbital theory of reactivity in

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[26] R. G. Parr, S. R. Gadre and L. J. Bartolotti. Local density functional theory of atoms and

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[29] Nelson Flores-Gallegos *Teoría de información cuántica como lenguaje conceptual en Química*. Tesis Doctoral. Universidad Autónoma Metropolitana-Iztapalapa. México D.F. (2010).

the chemical reactivity. *J. Am. Chem. Soc.*, *1984*, 106, 4049-4050.

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22 Quantum Mechanics

**References**

**Author details**

Nelson Flores-Gallegos

Universidad Autónoma Metropolitana-Iztapalapa, México

physics. *Phys. Rev. A* 1990, *41*, 4265-4276.

information. *Phys. Rev. E* 1994, *49*, 2644-2649.

of physical laws *Phys. Rev. E* 1996, *54*, 257-260.

[9] M. Kohout. DGrid, version 4.6, Radebeul, 2011.

Diatomic Molecules *J. Mex. Chem. Soc.*, 2008, *52*, 19-30.

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*J. Phys.* 1989, *11*, 1004-1008.

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[2] B. Roy Frieden. Fisher information, disorder, and the equilibrium distributions of

[3] B. Roy Frieden and Roy J. Hughes. Spectral 1/ *f* noise derived from extremized physical

[4] B. Roy Frieden and Bernard H. Soffer. Lagrangians of physics and the game of

[5] B. Roy Frieden and W. J. Cocke. Foundation for Fisher-information-based derivations

[6] Parveen Fazal, S., Sen, K.D., Gutierrez G., and Fuentealba, P. Shannon entropy of 1-normalized electron density. *Indian Journal of Chemistry.* 2000, *39A*, 48-49.

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[8] Pérez-Jordá, Jose M., Becke, Axel D. and San-Fabian, Emilio. Automatic numerical integration techniques for polyatomic molecules. *J. Chem. Phys.* 1994, *100*, 6520-6534.

[10] Nelson Flores-Gallegos and Rodolfo O. Esquivel. von Neumann Entropies Analysis in Hilbert Space for the Dissociation Processes of Homonuclear and Heteronuclear

[11] Rodolfo O. Esquivel, Nelson Flores-Gallegos, Cristina Iuga, Edmundo Carrera, Juan Carlos Angulo and Juan Antolín. Phenomenological description of the transition state, and the bond breaking and bond forming processes of selected elementary chemical reactions: An information-theoretic study. *Theoretical Chemistry Accounts.*, 2009, *124*,

[12] R.O. Esquivel, N. Flores-Gallegos, C. Iuga, E. Carrera, J.C. Angulo and J. Antolín. Phenomenological description of selected elementary chemical reaction mechanisms:

[13] Rodolfo O. Esquivel, Juan Carlos Angulo, Juan Antolín, Jesús S. Dehesa, Sheila López-Rosa and Nelson Flores-Gallegos. Complexity analysis of selected molecules

An information-theoretic study. *Physics Letters A*, *2010* , 374, 948-951.


[30] Edmundo M. Carrera, Nelson Flores-Gallegos, Rodolfo O. Esquivel. Natural atomic probabilities in quantum information theory. *Journal of Computational and Applied Mathematics.*, Vol. 233, 1483-1490 (2010).

**Chapter 30**

**Provisional chapter**

**A Novel Isospectral Deformation Chain in**

**Supersymmetric Quantum Mechanics**

**A Novel Isospectral Deformation Chain in**

Supersymmetric quantum mechanics (SUSYQM) has turned out to be surprisingly fertile field which is also able to successfully address challenges in traditional quantum mechanics and beyond. It has its roots in the works of Schrödinger, Infeld and Hull [1] on factorization methods of the Schrödinger equation. The term *supersymmetric* is due to a work by Witten [2] which brought these methods in contact with contemporary ideas in high energy physics. He showed in particular that the a factorized one-dimensional Schrödinger equation can accompany a super-Lie algebra thus providing a rich toy-model where features and concepts in supersymmetric quantum field theories can be studied in a greatly simplified context. A key ingredient in supersymmetric theories is that every bosonic state has a fermionic superpartner with all properties equal except the spin. In SUSYQM these states emerge as bosonic doublet states. The bosonic and the fermionic states are described in terms of the Schrödinger equation, but they interact with different physical potentials. These potentials are called partner potentials. Not completely surprising, knowing in advance the energy-eigenvalues and functions of the bosonic (fermionic) states the theory provides a map to the fermionic (bosonic) states with exactly the same energy-eigenvalues. Of key interest to us is that the physical partner potentials are expressed in terms of the same superpotential. These expressions are in general not unique. Different superpotentials can give rise to a particular physical potential in the fermionic (bosonic) sector. This does not imply that when the superpotential is changed (deformed) in such a way that the physical potential in the fermionic (bosonic) sector stays unchanged that the physical potential stays invariant in the bosonic (fermionic) sector. Whenever we deform a superpotential in the fermionic (bosonic) sector such that the fermionic (bosonic) potential is invariant the bosonic (fermionic) potential will generally change, but the theory nevertheless assures that the energy-eigenvalues in the bosonic (fermionic) sector stays the same. Such deformations are called isospectral

> ©2012 Jensen, 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. © 2013 Jensen; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Jensen; licensee InTech. This is a paper 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.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Supersymmetric Quantum Mechanics**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

deformations. They are the subject of this chapter.

Bjørn Jensen

**1. Introduction**

Bjørn Jensen

http://dx.doi.org/10.5772/53845


**Provisional chapter**

### **A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics Supersymmetric Quantum Mechanics**

**A Novel Isospectral Deformation Chain in**

Bjørn Jensen Additional information is available at the end of the chapter

Bjørn Jensen

24 Quantum Mechanics

706 Advances in Quantum Mechanics

[30] Edmundo M. Carrera, Nelson Flores-Gallegos, Rodolfo O. Esquivel. Natural atomic probabilities in quantum information theory. *Journal of Computational and Applied*

[31] Gaussian 09, Revision B.01, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J.

V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2010.

[32] Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Wallingford CT, 2004.

[33] Parr, R.G. and Weitao, Y. *Density Functional Theory of Atoms and Molecules*; Oxford

[34] Nelson Flores-Gallegos and Carmen Salazar-Hernández. *Some Applications of Quantum Mechanics; Chapter 10, Flows of Information and Informational Trajectories in Chemical*

University Press: New York, 1989; pp. 87-104.

*Processes.* InTech, 2012; pp. 233-256.

*Mathematics.*, Vol. 233, 1483-1490 (2010).

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53845

### **1. Introduction**

Supersymmetric quantum mechanics (SUSYQM) has turned out to be surprisingly fertile field which is also able to successfully address challenges in traditional quantum mechanics and beyond. It has its roots in the works of Schrödinger, Infeld and Hull [1] on factorization methods of the Schrödinger equation. The term *supersymmetric* is due to a work by Witten [2] which brought these methods in contact with contemporary ideas in high energy physics. He showed in particular that the a factorized one-dimensional Schrödinger equation can accompany a super-Lie algebra thus providing a rich toy-model where features and concepts in supersymmetric quantum field theories can be studied in a greatly simplified context. A key ingredient in supersymmetric theories is that every bosonic state has a fermionic superpartner with all properties equal except the spin. In SUSYQM these states emerge as bosonic doublet states. The bosonic and the fermionic states are described in terms of the Schrödinger equation, but they interact with different physical potentials. These potentials are called partner potentials. Not completely surprising, knowing in advance the energy-eigenvalues and functions of the bosonic (fermionic) states the theory provides a map to the fermionic (bosonic) states with exactly the same energy-eigenvalues. Of key interest to us is that the physical partner potentials are expressed in terms of the same superpotential. These expressions are in general not unique. Different superpotentials can give rise to a particular physical potential in the fermionic (bosonic) sector. This does not imply that when the superpotential is changed (deformed) in such a way that the physical potential in the fermionic (bosonic) sector stays unchanged that the physical potential stays invariant in the bosonic (fermionic) sector. Whenever we deform a superpotential in the fermionic (bosonic) sector such that the fermionic (bosonic) potential is invariant the bosonic (fermionic) potential will generally change, but the theory nevertheless assures that the energy-eigenvalues in the bosonic (fermionic) sector stays the same. Such deformations are called isospectral deformations. They are the subject of this chapter.

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. © 2013 Jensen; licensee InTech. This is an open access article 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. © 2013 Jensen; licensee InTech. This is a paper 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.

©2012 Jensen, licensee InTech. This is an open access chapter distributed under the terms of the Creative

Isospectral transformations in the context of SUSYQM has a long history exhibiting methods dating all the way back to Darboux [3]. The dominating approach is to study isospectral Hamilton operators. Different operator methods exist, but the main ones was brought under a single unifying principle by Pursey [4] with the use of isometric operators. A second approach to the study of isospectral transformations is what has been called deformation theory (see [5], e.g.). This is a more direct approach compared with the operator approach in that one studies deformations of the superpotential as described briefly above. It is rather surprising to note that this second approach has not been given much attention in the literature. To the knowledge of this author only one of the simplest deformations possible has been discussed to some extend. In a previous work [6] we initiated a work with the aim to remedy this situation. In [6] we showed that the isospectral deformation which has been considered in previous works is part of a more general deformation scheme. In this work we extend our results in [6]. We explicitly construct an in principle infinite recursively defined isospectral deformation chain where the deformation scheme in [6] emerges as the root of the chain.

The ground state eigenfunction is thus simply given by

to a matrix form. Defining

It is straightforward to verify that

representation

It is clear that

symmetry the supersymmetry of the system.

*ψ*± <sup>0</sup> (*x*) <sup>∼</sup> *<sup>e</sup>*

*Q*<sup>−</sup> ≡

*H* ≡

 0 0 *<sup>A</sup>*<sup>−</sup> <sup>0</sup> 

we find that we naturally can construct a matrix-valued Hamiltonian *H* given by

 =

This constitutes what is called a super-Lie algebra in contrast to an ordinary Lie algebra which only contains commutators. The commutator in Eq.(7) shows that *<sup>Q</sup>*<sup>±</sup> are generators of a symmetry which is left intact under time-translations generated by *H*. We call this

The matrices above naturally act on a two-dimensional vector space with the natural

 0 *ψ*+ *<sup>n</sup>*−1(*x*)  *<sup>ψ</sup>*<sup>−</sup> *<sup>n</sup>* (*x*) *ψ*+ *<sup>n</sup>*−1(*x*)

> =

Hence, *<sup>Q</sup>*<sup>±</sup> relate states with the same eigenvalue of *<sup>H</sup>*; the energy states are in other words

*Q*<sup>−</sup>

*Q*+

 *<sup>ψ</sup>*<sup>−</sup> *<sup>n</sup>* (*x*) 0

 0 *ψ*<sup>+</sup> *<sup>n</sup>* (*x*) = *<sup>ψ</sup>*<sup>−</sup> *<sup>n</sup>*+1(*x*) 0

degenerate. An orthogonal basis can naturally be taken to be states on the form

 *α*(*x*) 0 , 0 *β*(*x*) 

 *H*<sup>−</sup> 0 0 *H*<sup>+</sup>

<sup>±</sup> *<sup>x</sup> <sup>W</sup>*(*x*)*dx* ∼ 1

, *Q*<sup>+</sup> ≡

 *<sup>A</sup>*<sup>+</sup> *<sup>A</sup>*<sup>−</sup> <sup>0</sup> <sup>0</sup> *<sup>A</sup>*<sup>−</sup> *<sup>A</sup>*<sup>+</sup>

*<sup>H</sup>* <sup>=</sup> *<sup>Q</sup>*−*Q*<sup>+</sup> <sup>+</sup> *<sup>Q</sup>*+*Q*<sup>−</sup> ≡ {*Q*−, *<sup>Q</sup>*+} , [*Q*±, *<sup>H</sup>*] = 0 , (*Q*±)<sup>2</sup> <sup>=</sup> 0 . (7)

The factorization in Eq.(1) carries a symmetry which is not manifestly present in the usual form of the Schrödinger equation. This symmetry is made manifest when Eq.(1) is brought

*ψ*∓ <sup>0</sup> (*x*)

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

 0 *A*<sup>+</sup> 0 0

. (4)

http://dx.doi.org/10.5772/53845

. (6)

. (8)

, (9)

. (10)

. (11)

(5)

709

This chapter is organized as follows. In the next section we very briefly review some of the basics of SUSYQM, mainly in order to fix notation. We define the notions of partner potentials, superpotential, isospectrality and supersymmetry. In section 2 we briefly remain ourselves about the results in [6]. In section 3 we define the recursive deformation scheme. We also discuss other various deformation schemes but show that a number of other canonical deformation schemes defined along the lines of our recursive scheme either do not allow a recursive structure or either reduces to our scheme. We apply our apparently rather unique recursive deformation scheme to the Coulomb potential and calculate several novel potentials. We summarize our findings and conclude in the last section. No attempt has been made to give an in depth review of the relevant literature due to its immense size. The works which have been acknowledged in the list of references have been so only because of their utility to this author.

### **2. SUSYQM - A very brief introduction**

SUSYQM can in its most basic formulation be thought of as the following two factorizations of the Hamiltonian in the stationary Schrödinger equation in appropriate units

$$\begin{cases} (-\partial + W(\mathfrak{x}))(\partial + W(\mathfrak{x})) \equiv A^+ A^- \equiv H\_- \\ (\partial + W(\mathfrak{x}))(-\partial + W(\mathfrak{x})) \equiv A^- A^+ \equiv H\_+ \end{cases} \tag{1}$$

Here *∂* is short hand for differentiation with respect to the single spatial coordinate *x*, and *W*(*x*) is the so called superpotential. Both of these factorizations give rise to a Schrödinger equation, but with different potentials *V*−(*x*) and *V*+(*x*) (the so called superpotentials) given by

$$W\_{\pm}(\mathbf{x}) = W^2(\mathbf{x}) \pm \partial W(\mathbf{x}) \,. \tag{2}$$

Let us denote the energy-eigenvalues and eigenstates associated with *<sup>H</sup>*<sup>±</sup> by *<sup>E</sup>*<sup>±</sup> *<sup>n</sup>* and *<sup>ψ</sup>*<sup>±</sup> *<sup>n</sup>* (*x*), respectively. Let *n* = 0 denote the ground state. We note that

$$A^{\pm} \psi\_0^{\pm}(\mathbf{x}) = 0 \Rightarrow H\_{\pm} \psi\_0^{\pm}(\mathbf{x}) = 0 \,. \tag{3}$$

The ground state eigenfunction is thus simply given by

$$
\psi\_0^{\pm}(\mathbf{x}) \sim e^{\pm \int^\chi \mathcal{W}(\mathbf{x}) d\mathbf{x}} \sim \frac{1}{\psi\_0^{\mp}(\mathbf{x})}.\tag{4}
$$

The factorization in Eq.(1) carries a symmetry which is not manifestly present in the usual form of the Schrödinger equation. This symmetry is made manifest when Eq.(1) is brought to a matrix form. Defining

$$\mathcal{Q}^- \equiv \begin{pmatrix} 0 & 0 \\ A^- & 0 \end{pmatrix} \mathcal{Q}^+ \equiv \begin{pmatrix} 0 \ A^+ \\ 0 & 0 \end{pmatrix} \tag{5}$$

we find that we naturally can construct a matrix-valued Hamiltonian *H* given by

$$H \equiv \begin{pmatrix} H\_- & 0 \\ 0 & H\_+ \end{pmatrix} = \begin{pmatrix} A^+ A^- & 0 \\ 0 & A^- A^+ \end{pmatrix} . \tag{6}$$

It is straightforward to verify that

$$H = Q^-Q^+ + Q^+Q^- \equiv \{Q^-, Q^+\} \,, \left[Q^\pm, H\right] = 0 \,, (Q^\pm)^2 = 0 \,. \tag{7}$$

This constitutes what is called a super-Lie algebra in contrast to an ordinary Lie algebra which only contains commutators. The commutator in Eq.(7) shows that *<sup>Q</sup>*<sup>±</sup> are generators of a symmetry which is left intact under time-translations generated by *H*. We call this symmetry the supersymmetry of the system.

The matrices above naturally act on a two-dimensional vector space with the natural representation

> *<sup>ψ</sup>*<sup>−</sup> *<sup>n</sup>* (*x*) *ψ*+ *<sup>n</sup>*−1(*x*) . (8)

It is clear that

2 Quantum Mechanics

the chain.

by

of their utility to this author.

**2. SUSYQM - A very brief introduction**

Isospectral transformations in the context of SUSYQM has a long history exhibiting methods dating all the way back to Darboux [3]. The dominating approach is to study isospectral Hamilton operators. Different operator methods exist, but the main ones was brought under a single unifying principle by Pursey [4] with the use of isometric operators. A second approach to the study of isospectral transformations is what has been called deformation theory (see [5], e.g.). This is a more direct approach compared with the operator approach in that one studies deformations of the superpotential as described briefly above. It is rather surprising to note that this second approach has not been given much attention in the literature. To the knowledge of this author only one of the simplest deformations possible has been discussed to some extend. In a previous work [6] we initiated a work with the aim to remedy this situation. In [6] we showed that the isospectral deformation which has been considered in previous works is part of a more general deformation scheme. In this work we extend our results in [6]. We explicitly construct an in principle infinite recursively defined isospectral deformation chain where the deformation scheme in [6] emerges as the root of

This chapter is organized as follows. In the next section we very briefly review some of the basics of SUSYQM, mainly in order to fix notation. We define the notions of partner potentials, superpotential, isospectrality and supersymmetry. In section 2 we briefly remain ourselves about the results in [6]. In section 3 we define the recursive deformation scheme. We also discuss other various deformation schemes but show that a number of other canonical deformation schemes defined along the lines of our recursive scheme either do not allow a recursive structure or either reduces to our scheme. We apply our apparently rather unique recursive deformation scheme to the Coulomb potential and calculate several novel potentials. We summarize our findings and conclude in the last section. No attempt has been made to give an in depth review of the relevant literature due to its immense size. The works which have been acknowledged in the list of references have been so only because

SUSYQM can in its most basic formulation be thought of as the following two factorizations

(−*<sup>∂</sup>* <sup>+</sup> *<sup>W</sup>*(*x*))(*<sup>∂</sup>* <sup>+</sup> *<sup>W</sup>*(*x*)) <sup>≡</sup> *<sup>A</sup>*<sup>+</sup> *<sup>A</sup>*<sup>−</sup> <sup>≡</sup> *<sup>H</sup>*<sup>−</sup> ,

Here *∂* is short hand for differentiation with respect to the single spatial coordinate *x*, and *W*(*x*) is the so called superpotential. Both of these factorizations give rise to a Schrödinger equation, but with different potentials *V*−(*x*) and *V*+(*x*) (the so called superpotentials) given

(*<sup>∂</sup>* <sup>+</sup> *<sup>W</sup>*(*x*))(−*<sup>∂</sup>* <sup>+</sup> *<sup>W</sup>*(*x*)) <sup>≡</sup> *<sup>A</sup>*<sup>−</sup> *<sup>A</sup>*<sup>+</sup> <sup>≡</sup> *<sup>H</sup>*<sup>+</sup> . (1)

*V*±(*x*) = *W*2(*x*) ± *∂W*(*x*). (2)

<sup>0</sup> (*x*) = 0 . (3)

*<sup>n</sup>* and *<sup>ψ</sup>*<sup>±</sup>

*<sup>n</sup>* (*x*),

of the Hamiltonian in the stationary Schrödinger equation in appropriate units

Let us denote the energy-eigenvalues and eigenstates associated with *<sup>H</sup>*<sup>±</sup> by *<sup>E</sup>*<sup>±</sup>

<sup>0</sup> (*x*) = <sup>0</sup> <sup>⇒</sup> *<sup>H</sup>*±*ψ*<sup>±</sup>

respectively. Let *n* = 0 denote the ground state. We note that

*<sup>A</sup>*±*ψ*<sup>±</sup>

$$Q^{-}\begin{pmatrix}\psi\_{n}^{-}(\mathbf{x})\\0\end{pmatrix} = \begin{pmatrix}0\\\psi\_{n-1}^{+}(\mathbf{x})\end{pmatrix} . \tag{9}$$

$$Q^+\begin{pmatrix} 0\\\psi\_n^+(\mathbf{x}) \end{pmatrix} = \begin{pmatrix} \psi\_{n+1}^-(\mathbf{x})\\0 \end{pmatrix}.\tag{10}$$

Hence, *<sup>Q</sup>*<sup>±</sup> relate states with the same eigenvalue of *<sup>H</sup>*; the energy states are in other words degenerate. An orthogonal basis can naturally be taken to be states on the form

$$
\begin{pmatrix} \mathfrak{a}(\mathfrak{x}) \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ \beta(\mathfrak{x}) \end{pmatrix}. \tag{11}
$$

It is customary, due to the intimate relation to supersymmetric quantum field theory, to say that the first vector belongs to the bosonic sector and the other to the fermionic sector, even though no fermions appear in this theory. That *<sup>Q</sup>*<sup>±</sup> relate states corresponding to the same energy eigenvalue of the *H*-operator can also be seen on the level of the *H*<sup>±</sup> operators by noting that

$$H\_+(A^-\psi\_n^-) = E\_n^-(A^-\psi\_n^-)\,. \tag{12}$$

This is the generalized Riccati equation [7] . If one particular solution *F*00(*x*) of Eq.(16) is

*<sup>F</sup>*0(*x*) = *<sup>F</sup>*00(*x*) + <sup>1</sup>

Eq.(18) can be solved by elementary means. The resulting superpotential *W*ˆ <sup>0</sup>(*x*) is given by

*<sup>C</sup>*<sup>01</sup> <sup>+</sup> *<sup>x</sup> <sup>e</sup>*−<sup>2</sup>

*C*<sup>01</sup> is an integration constant, which we will assume to be real. We have explicitly introduced upper integration limits in Eq.(19) in order to avoid sign ambiguities. This explains the difference in the sign in the denominator in Eq.(19) compared with Eq.(2.5) in [6] where the reverse order of integration in one of the integrals was implicitly assumed. We do not specify the lower integration limits in Eq.(19). These are not important, of course, since the values of the integrals there can essentially be absorbed into *C*01. We can by simple inspection see that the particular solution *F*0(*x*) = 1, the *identity deformation*, solves Eq.(16). With *F*00(*x*) = 1 we identically rederive Eq.(15) and the corresponding expression discussed in [5]. The identity deformation corresponds to the limit *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> in Eq.(19) with *<sup>F</sup>*00(*x*) = 1. In the limit *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> we generally get *<sup>W</sup>*<sup>ˆ</sup> <sup>0</sup>(*x*) = *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*). This deformation will play a pivotal role in this work; it will represent the base of a recursive scheme for generating novel isospectral deformations. We will therefore refer to a particular *W*ˆ <sup>00</sup>(*x*) as *a base deformation* in the

In order to expand the space of concrete isospectral deformations further we transform

*d*

*W*(*x*)

This substitution gives rise to the following linear homogeneous second order differential

Eq.(16) into an ordinary second order differential equation by the substitution

*<sup>F</sup>*0(*x*) = <sup>1</sup>

<sup>−</sup> *<sup>d</sup>*<sup>2</sup>

)*W*(*x*) <sup>≡</sup> *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) + <sup>1</sup>

*<sup>x</sup> <sup>F</sup>*00(*t*)*W*(*t*)*dt*

*X*0(*x*)

<sup>=</sup> *<sup>F</sup>*00(*x*)*W*(*x*) + *<sup>e</sup>*−<sup>2</sup>

*<sup>X</sup>*0(*x*) , (17)

http://dx.doi.org/10.5772/53845

711

*W*(*x*) =

*<sup>u</sup> <sup>F</sup>*00(*t*)*W*(*t*)*dtdu* . (19)

*dx* ln *<sup>U</sup>*0(*x*). (20)

*dx*<sup>2</sup> *<sup>U</sup>*0(*x*) + *<sup>V</sup>*+(*x*)*U*0(*x*) = 0 . (21)

*dx* ln *<sup>W</sup>*(*x*) + <sup>2</sup>*F*00(*x*)*W*(*x*))*X*0(*x*) = *<sup>W</sup>*(*x*). (18)

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

*X*0(*x*)

known another solution is given by [8]

where *X*0(*x*) solves the equation

[6]

following.

equation

*d*

*dx <sup>X</sup>*0(*x*) <sup>−</sup> ( *<sup>d</sup>*

*<sup>W</sup>*<sup>ˆ</sup> <sup>0</sup>(*x*)=(*F*00(*x*) + <sup>1</sup>

Hence, given an eigenstate *<sup>ψ</sup>*<sup>−</sup> *<sup>n</sup>* of *<sup>H</sup>*<sup>−</sup> with energy eigenvalue *<sup>E</sup>*<sup>−</sup> *<sup>n</sup>* , the state *<sup>A</sup>*−*ψ*<sup>−</sup> *n* is an eigenstate of *<sup>H</sup>*<sup>+</sup> with energy eigenvalue *<sup>E</sup>*<sup>−</sup> *n* . There is thus a one-to-one correspondence between bosonic and fermionic states with the same energy eigenvalue (Eq.(9-10) above). We call this property the isospectrality of SUSYQM. Much more can be said about SUSYQM, such as the role played by the vacuum in connection with isospectrality. However, for the purpose of this chapter this very brief exposition of some of the basics of SUSYQM is sufficient to fix notation and certain concepts.

#### **3. A novel isospectral deformation chain**

In [6] we introduced within the framework of SUSYQM an isospectral deformation on the form

$$W(\mathbf{x}) \to \hat{W}\_0(\mathbf{x}) = F\_0(\mathbf{x})W(\mathbf{x})\,. \tag{13}$$

where *W*(*x*) is some known superpotential and *F*0(*x*) some function to be determined by the isospectrality condition

$$
\hat{\mathcal{W}}\_{+}(\mathbf{x}) \equiv \hat{\mathcal{W}}\_{0}^{2}(\mathbf{x}) + \hat{\mathcal{W}}\_{0}'(\mathbf{x}) = \mathcal{W}^{2}(\mathbf{x}) + \mathcal{W}'(\mathbf{x}) \equiv V\_{+}(\mathbf{x}) \,. \tag{14}
$$

It was shown that Eq.(13) includes the only previously explored deformation of this kind, which has the form [5]

$$\mathcal{W}(\mathbf{x}) \to \hat{\mathcal{W}}\_0(\mathbf{x}) = \mathcal{W}(\mathbf{x}) + f(\mathbf{x}) \,. \tag{15}$$

*f*(*x*) is some function which is determined by Eq.(14). In this work we expand the deformation Eq.(13) in various directions and study the implications drawn from the isospectrality condition. We show in particular that the deformation Eq.(13) is the root of an infinitely long and recursively generated chain of deformations. Let us next briefly review some of the findings in [6].

#### **3.1. Base deformations**

The deformation Eq.(1) implies the following differential equation for *F*0(*x*) [6]<sup>1</sup>

$$\frac{d}{d\mathbf{x}}F\_0(\mathbf{x}) + (\frac{d}{d\mathbf{x}}\ln\mathcal{W}(\mathbf{x}))F\_0(\mathbf{x}) + \mathcal{W}(\mathbf{x})F\_0^2(\mathbf{x}) = \mathcal{W}(\mathbf{x}) + \frac{d}{d\mathbf{x}}\ln\mathcal{W}(\mathbf{x})\,. \tag{16}$$

<sup>1</sup> We will often rewrite fractions on the form *W*′ (*x*)/*W*(*x*) as the logarithmic derivative of *W*(*x*) as a formal tool. Caution must of course be exercised when using the corresponding expressions in actual computations.

This is the generalized Riccati equation [7] . If one particular solution *F*00(*x*) of Eq.(16) is known another solution is given by [8]

$$F\_0(\mathbf{x}) = F\_{00}(\mathbf{x}) + \frac{1}{X\_0(\mathbf{x})} \, ^\prime \tag{17}$$

where *X*0(*x*) solves the equation

4 Quantum Mechanics

noting that

form

Hence, given an eigenstate *<sup>ψ</sup>*<sup>−</sup>

notation and certain concepts.

isospectrality condition

which has the form [5]

some of the findings in [6].

**3.1. Base deformations**

*d*

*dx <sup>F</sup>*0(*x*)+( *<sup>d</sup>*

<sup>1</sup> We will often rewrite fractions on the form *W*′

*V*ˆ

+(*x*) ≡ *W*ˆ <sup>2</sup>

<sup>0</sup> (*x*) + *<sup>W</sup>*<sup>ˆ</sup> ′

eigenstate of *<sup>H</sup>*<sup>+</sup> with energy eigenvalue *<sup>E</sup>*<sup>−</sup>

**3. A novel isospectral deformation chain**

It is customary, due to the intimate relation to supersymmetric quantum field theory, to say that the first vector belongs to the bosonic sector and the other to the fermionic sector, even though no fermions appear in this theory. That *<sup>Q</sup>*<sup>±</sup> relate states corresponding to the same energy eigenvalue of the *H*-operator can also be seen on the level of the *H*<sup>±</sup> operators by

*<sup>n</sup>* ) = *<sup>E</sup>*<sup>−</sup>

between bosonic and fermionic states with the same energy eigenvalue (Eq.(9-10) above). We call this property the isospectrality of SUSYQM. Much more can be said about SUSYQM, such as the role played by the vacuum in connection with isospectrality. However, for the purpose of this chapter this very brief exposition of some of the basics of SUSYQM is sufficient to fix

In [6] we introduced within the framework of SUSYQM an isospectral deformation on the

where *W*(*x*) is some known superpotential and *F*0(*x*) some function to be determined by the

It was shown that Eq.(13) includes the only previously explored deformation of this kind,

*f*(*x*) is some function which is determined by Eq.(14). In this work we expand the deformation Eq.(13) in various directions and study the implications drawn from the isospectrality condition. We show in particular that the deformation Eq.(13) is the root of an infinitely long and recursively generated chain of deformations. Let us next briefly review

The deformation Eq.(1) implies the following differential equation for *F*0(*x*) [6]<sup>1</sup>

*dx* ln *<sup>W</sup>*(*x*))*F*0(*x*) + *<sup>W</sup>*(*x*)*F*<sup>2</sup>

Caution must of course be exercised when using the corresponding expressions in actual computations.

<sup>0</sup>(*x*) = *<sup>W</sup>*2(*x*) + *<sup>W</sup>*′

*<sup>n</sup>* of *<sup>H</sup>*<sup>−</sup> with energy eigenvalue *<sup>E</sup>*<sup>−</sup>

*<sup>n</sup>* (*A*−*ψ*<sup>−</sup>

*n* ). (12)

*n* . There is thus a one-to-one correspondence

*<sup>W</sup>*(*x*) → *<sup>W</sup>*<sup>ˆ</sup> <sup>0</sup>(*x*) = *<sup>F</sup>*0(*x*)*W*(*x*), (13)

*<sup>W</sup>*(*x*) → *<sup>W</sup>*<sup>ˆ</sup> <sup>0</sup>(*x*) = *<sup>W</sup>*(*x*) + *<sup>f</sup>*(*x*). (15)

<sup>0</sup> (*x*) = *<sup>W</sup>*(*x*) + *<sup>d</sup>*

(*x*)/*W*(*x*) as the logarithmic derivative of *W*(*x*) as a formal tool.

*<sup>n</sup>* , the state *<sup>A</sup>*−*ψ*<sup>−</sup>

(*x*) ≡ *V*+(*x*). (14)

*dx* ln *<sup>W</sup>*(*x*). (16)

*n* is an

*<sup>H</sup>*+(*A*−*ψ*<sup>−</sup>

$$\frac{d}{d\mathbf{x}}X\_0(\mathbf{x}) - \left(\frac{d}{d\mathbf{x}}\ln W(\mathbf{x}) + 2F\_{00}(\mathbf{x})W(\mathbf{x})\right)X\_0(\mathbf{x}) = W(\mathbf{x})\,. \tag{18}$$

Eq.(18) can be solved by elementary means. The resulting superpotential *W*ˆ <sup>0</sup>(*x*) is given by [6]

$$\begin{split} \hat{\mathcal{W}}\_{0}(\mathbf{x}) &= (F\_{00}(\mathbf{x}) + \frac{1}{\mathcal{X}\_{0}(\mathbf{x})}) \mathcal{W}(\mathbf{x}) \equiv \hat{\mathcal{W}}\_{00}(\mathbf{x}) + \frac{1}{\mathcal{X}\_{0}(\mathbf{x})} \mathcal{W}(\mathbf{x}) = \\ &= F\_{00}(\mathbf{x}) \mathcal{W}(\mathbf{x}) + \frac{e^{-2 \int^{x} F\_{00}(t) \mathcal{W}(t) dt}}{\mathcal{C}\_{01} + \int^{x} e^{-2 \int^{x} F\_{00}(t) \mathcal{W}(t) dt} du} . \end{split} \tag{19}$$

*C*<sup>01</sup> is an integration constant, which we will assume to be real. We have explicitly introduced upper integration limits in Eq.(19) in order to avoid sign ambiguities. This explains the difference in the sign in the denominator in Eq.(19) compared with Eq.(2.5) in [6] where the reverse order of integration in one of the integrals was implicitly assumed. We do not specify the lower integration limits in Eq.(19). These are not important, of course, since the values of the integrals there can essentially be absorbed into *C*01. We can by simple inspection see that the particular solution *F*0(*x*) = 1, the *identity deformation*, solves Eq.(16). With *F*00(*x*) = 1 we identically rederive Eq.(15) and the corresponding expression discussed in [5]. The identity deformation corresponds to the limit *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> in Eq.(19) with *<sup>F</sup>*00(*x*) = 1. In the limit *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> we generally get *<sup>W</sup>*<sup>ˆ</sup> <sup>0</sup>(*x*) = *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*). This deformation will play a pivotal role in this work; it will represent the base of a recursive scheme for generating novel isospectral deformations. We will therefore refer to a particular *W*ˆ <sup>00</sup>(*x*) as *a base deformation* in the following.

In order to expand the space of concrete isospectral deformations further we transform Eq.(16) into an ordinary second order differential equation by the substitution

$$F\_0(\mathbf{x}) = \frac{1}{\mathcal{W}(\mathbf{x})} \frac{d}{d\mathbf{x}} \ln \mathcal{U}\_0(\mathbf{x}) \,. \tag{20}$$

This substitution gives rise to the following linear homogeneous second order differential equation

$$-\frac{d^2}{dx^2}\mathcal{U}\_0(\mathbf{x}) + V\_+(\mathbf{x})\mathcal{U}\_0(\mathbf{x}) = 0.\tag{21}$$

This equation coincides of course with the zero-energy eigenfunction equation. However, keep in mind that *U*<sup>0</sup> is *not* in general to be identified with the eigenfunction of the system. This is of particular importance to remember in light of Eq.(16). The special solution *F*0(*x*) = 1 is generated by the solution

$$\mathcal{U}\_0(\mathbf{x}) \sim e^{\int^\mathbf{x} W(t)dt}.\tag{22}$$

The *λi*'s are assumed to be independent real constants. Starting with a known superpotential *m* consecutive applications of the isospectrality condition yields the following set of equations

The first equation in Eq.(26) coincides of course per definition with Eq.(16). Note that *Fj*0(*x*) = 1 only solves the first equation in Eq.(26). Let us consider an arbitrary iteration

*W*(*x*)

This equation corresponds to Eq.(21) in the case when *n* = 0. It reduces in general to an ordinary linear differential equation only when *λ<sup>n</sup>* = 1 , ∀*n* �= 0. We will focus on this

The general solution of Eq.(28) for arbitrary *n* �= 0, and with *λ<sup>n</sup>* set to unity, can be found by

−2

� *<sup>x</sup> e* −2

*Cn*<sup>1</sup> and *Cn*<sup>2</sup> are integration constants, which we assume to be real. We can reduce the number of integration constants to one at each iteration level, but we will stick to the habit of explicitly writing down the actual number of constants in order to make it easier to compare the various formulas we deduce, which stem from both second and first order differential equations. We also note that the structure of *Fn*0(*x*) implies that previous deformations are not regenerated in general. Of course, this does not exclude this possibility to arise, as we will see in Section 5. Hence, *m* in Eq.(25) has in principle no natural upper bound. From Eq.(25) and Eq.(29) we get the following expression for the superpotential at iteration level *m*

*Un*(*x*) <sup>+</sup> <sup>2</sup>*W*<sup>ˆ</sup> (*n*−1)0(*x*)*U*′

� *<sup>x</sup> <sup>W</sup>*<sup>ˆ</sup> (*n*−1)0(*t*)*dt*

� *<sup>x</sup> e* −2

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> (*n*−1)0(*t*)*dtdu*

=

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> (*n*−1)0(*t*)*dtdu*). (29)

<sup>00</sup>(*x*) = *<sup>W</sup>*(*x*)+(ln *<sup>W</sup>*(*x*))′ ,

<sup>10</sup>(*x*) = 0 ,

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

*<sup>m</sup>*0(*x*) = 0 .

(ln *Un*(*x*))′ . (27)

*<sup>n</sup>*(*x*) = 0 . (28)

<sup>20</sup>(*x*) = 0 ,

http://dx.doi.org/10.5772/53845

(26)

713

*F*00(*x*) + *W*(*x*)*F*<sup>2</sup>

<sup>10</sup>(*x*) + [(ln *<sup>W</sup>*(*x*))′ <sup>+</sup> <sup>2</sup>*F*00(*x*)*W*(*x*)]*F*10(*x*) + *<sup>λ</sup>*1*W*(*x*)*F*<sup>2</sup>

*<sup>m</sup>*0(*x*) + [(ln *<sup>W</sup>*(*x*))′ <sup>+</sup> <sup>2</sup>*W*<sup>ˆ</sup> (*m*−1)0(*x*))]*Fm*0(*x*) + *<sup>λ</sup>mW*(*x*)*F*<sup>2</sup>

*Fn*0(*x*) = <sup>1</sup>

[*U*′ *n*(*x*)]<sup>2</sup>

*Cn*<sup>1</sup> + *Cn*<sup>2</sup>

*dx* ln(*Cn*<sup>1</sup> <sup>+</sup> *Cn*<sup>2</sup>

level *n* (�= 0) and make the following substitution in Eq.(26)

*<sup>n</sup>* (*x*)+(*λ<sup>n</sup>* <sup>−</sup> <sup>1</sup>)

*Fn*0(*x*)*W*(*x*) = *Cn*2*<sup>e</sup>*

<sup>=</sup> *<sup>d</sup>*

The equation for *Fn*(*x*) can then be written

*U*′′

elementary means, and we deduce that

special case in this work.

<sup>20</sup>(*x*) + [(ln *<sup>W</sup>*(*x*))′ <sup>+</sup> <sup>2</sup>(*F*00(*x*) + *<sup>λ</sup>*1*F*10(*x*))*W*(*x*)]*F*20(*x*) + *<sup>λ</sup>*2*W*(*x*)*F*<sup>2</sup>

*F*′

*F*′

*F*′

. . . . . . *F*′

<sup>00</sup>(*x*)+[ln *<sup>W</sup>*(*x*)]′

The particular solutions for *F*0(*x*) stemming from Eq.(21) can be fed into Eq.(19) (as *F*00(*x*)) and thus expand the space of available concrete deformations. The physical potential *V*ˆ <sup>−</sup>(*x*) generated by *W*ˆ <sup>0</sup>(*x*) can in general thus be written [6]<sup>2</sup>

$$\begin{split} \hat{\mathcal{V}}\_{-}(\mathbf{x}) &= \hat{\mathcal{W}}\_{0}^{2}(\mathbf{x}) - \hat{\mathcal{W}}\_{0}^{\prime}(\mathbf{x}) = \hat{\mathcal{W}}\_{00}^{2}(\mathbf{x}) - \hat{\mathcal{W}}\_{00}^{\prime}(\mathbf{x}) + \\ &+ \frac{4\hat{\mathcal{W}}\_{00}(\mathbf{x})e^{-2\int^{\mathbf{x}}\hat{\mathcal{W}}\_{00}(t)dt}}{\mathbf{C}\_{01} + \int^{\mathbf{x}}e^{-2\int^{\mathbf{x}}\hat{\mathcal{W}}\_{00}(t)dt}dt} + 2\left[\frac{e^{-2\int^{\mathbf{x}}\hat{\mathcal{W}}\_{00}(t)dt}}{\mathbf{C}\_{01} + \int^{\mathbf{x}}e^{-2\int^{\mathbf{x}}\hat{\mathcal{W}}\_{00}(t)dt}dt}\right]^{2} \end{split} \tag{23}$$

with

$$
\hat{\mathcal{W}}\_{00}(\mathbf{x}) = \frac{d}{d\mathbf{x}} \ln \mathcal{U}\_0(\mathbf{x}) \,. \tag{24}
$$

#### **3.2. Recursive linear deformations**

Although the Riccati equation can be transformed into an ordinary second order differential equation the non-linearity of the Riccati equation allows for a solution space which is larger than the one associated with linear differential equations of second order, as became evident in the previous section. It is therefore natural to ask whether the non-linearity of the Riccati equation implies even more isospectral deformations than the ones we already have deduced [6]. We will explore this question in this and the next section.

#### *3.2.1. The sum*

Let us entertain the following idea. Assume that we have derived a particular base deformation *W*ˆ <sup>00</sup>(*x*) from an explicitly given superpotential *W*(*x*). Then assume that we *add* another term *F*1(*x*)*W*(*x*) (possibly multiplied with a constant) to that deformation such that we in principle get a novel deformation on the form *W*ˆ (*x*) = *F*10(*x*)*W*(*x*) + *W*ˆ <sup>00</sup>(*x*). After determining *F*10(*x*) from the isospectrality condition Eq.(14) add yet another term of this kind to the deformation. Let us assume that this process can be repeated indefinitely. Will terms added in this manner give rise to novel deformations? We will in the following show that they do. This represents a recursive deformation scheme.

Following the basic idea, after *m* iterations we thus have the general recursive linear (in *W*(*x*)) deformation

$$\hat{\mathcal{W}}\_{m0}(\mathbf{x}) = (\sum\_{i=0}^{m} \lambda\_i \mathbf{F}\_{i0}(\mathbf{x})) \mathcal{W}(\mathbf{x}) = \lambda\_m \mathbf{F}\_{m0}(\mathbf{x}) \mathcal{W}(\mathbf{x}) + \hat{\mathcal{W}}\_{(m-1)0}(\mathbf{x}) \; , \; \lambda\_0 \equiv 1 \; . \tag{25}$$

<sup>2</sup> Note that the corresponding expression in [6] ((2.14)) is misprinted.

The *λi*'s are assumed to be independent real constants. Starting with a known superpotential *m* consecutive applications of the isospectrality condition yields the following set of equations

$$\begin{cases} F\_{0}'(\mathbf{x}) + [\ln W(\mathbf{x})]'F\_{00}(\mathbf{x}) + W(\mathbf{x})F\_{00}^2(\mathbf{x}) = W(\mathbf{x}) + (\ln W(\mathbf{x}))' \\ F\_{10}'(\mathbf{x}) + [(\ln W(\mathbf{x}))' + 2F\_{00}(\mathbf{x})W(\mathbf{x})]F\_{10}(\mathbf{x}) + \lambda\_1 W(\mathbf{x})F\_{10}^2(\mathbf{x}) = \mathbf{0} \\ F\_{20}'(\mathbf{x}) + [(\ln W(\mathbf{x}))' + 2(F\_{00}(\mathbf{x}) + \lambda\_1 F\_{10}(\mathbf{x}))W(\mathbf{x})]F\_{20}(\mathbf{x}) + \lambda\_2 W(\mathbf{x})F\_{20}^2(\mathbf{x}) = \mathbf{0} \\ \vdots \\ F\_{m0}'(\mathbf{x}) + [(\ln W(\mathbf{x}))' + 2\hat{W}\_{(m-1)0}(\mathbf{x}))]F\_{m0}(\mathbf{x}) + \lambda\_m W(\mathbf{x})F\_{m0}^2(\mathbf{x}) = \mathbf{0} \end{cases} \tag{26}$$

The first equation in Eq.(26) coincides of course per definition with Eq.(16). Note that *Fj*0(*x*) = 1 only solves the first equation in Eq.(26). Let us consider an arbitrary iteration level *n* (�= 0) and make the following substitution in Eq.(26)

$$F\_{n0}(\mathbf{x}) = \frac{1}{W(\mathbf{x})} (\ln \mathcal{U}\_{\boldsymbol{\eta}}(\mathbf{x}))'. \tag{27}$$

The equation for *Fn*(*x*) can then be written

6 Quantum Mechanics

1 is generated by the solution

*V*ˆ

with

*3.2.1. The sum*

deformation

*W*ˆ *<sup>m</sup>*0(*x*)=(

*m* ∑ *i*=0

<sup>2</sup> Note that the corresponding expression in [6] ((2.14)) is misprinted.

<sup>−</sup>(*x*) ≡ *W*ˆ <sup>2</sup>

+

**3.2. Recursive linear deformations**

generated by *W*ˆ <sup>0</sup>(*x*) can in general thus be written [6]<sup>2</sup>

<sup>0</sup> (*x*) <sup>−</sup> *<sup>W</sup>*<sup>ˆ</sup> ′

<sup>4</sup>*W*<sup>ˆ</sup> <sup>00</sup>(*x*)*e*−<sup>2</sup>

[6]. We will explore this question in this and the next section.

show that they do. This represents a recursive deformation scheme.

*<sup>C</sup>*<sup>01</sup> <sup>+</sup> *<sup>x</sup> <sup>e</sup>*−<sup>2</sup>

This equation coincides of course with the zero-energy eigenfunction equation. However, keep in mind that *U*<sup>0</sup> is *not* in general to be identified with the eigenfunction of the system. This is of particular importance to remember in light of Eq.(16). The special solution *F*0(*x*) =

The particular solutions for *F*0(*x*) stemming from Eq.(21) can be fed into Eq.(19) (as *F*00(*x*)) and thus expand the space of available concrete deformations. The physical potential *V*ˆ

<sup>00</sup>(*x*) <sup>−</sup> *<sup>W</sup>*<sup>ˆ</sup> ′

+ 2 

<sup>00</sup>(*x*) +

*e*−<sup>2</sup>

*<sup>C</sup>*<sup>01</sup> <sup>+</sup> *<sup>x</sup> <sup>e</sup>*−<sup>2</sup>

*<sup>x</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dt*

*<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu*

*dx* ln *<sup>U</sup>*0(*x*). (24)

2

*<sup>x</sup> <sup>W</sup>*(*t*)*dt* . (22)

<sup>−</sup>(*x*)

(23)

*<sup>U</sup>*0(*x*) ∼ *<sup>e</sup>*

<sup>0</sup>(*x*) = *<sup>W</sup>*<sup>ˆ</sup> <sup>2</sup>

*<sup>x</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dt*

*<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu*

*<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) = *<sup>d</sup>*

Although the Riccati equation can be transformed into an ordinary second order differential equation the non-linearity of the Riccati equation allows for a solution space which is larger than the one associated with linear differential equations of second order, as became evident in the previous section. It is therefore natural to ask whether the non-linearity of the Riccati equation implies even more isospectral deformations than the ones we already have deduced

Let us entertain the following idea. Assume that we have derived a particular base deformation *W*ˆ <sup>00</sup>(*x*) from an explicitly given superpotential *W*(*x*). Then assume that we *add* another term *F*1(*x*)*W*(*x*) (possibly multiplied with a constant) to that deformation such that we in principle get a novel deformation on the form *W*ˆ (*x*) = *F*10(*x*)*W*(*x*) + *W*ˆ <sup>00</sup>(*x*). After determining *F*10(*x*) from the isospectrality condition Eq.(14) add yet another term of this kind to the deformation. Let us assume that this process can be repeated indefinitely. Will terms added in this manner give rise to novel deformations? We will in the following

Following the basic idea, after *m* iterations we thus have the general recursive linear (in *W*(*x*))

*<sup>λ</sup>iFi*0(*x*))*W*(*x*) = *<sup>λ</sup>mFm*0(*x*)*W*(*x*) + *<sup>W</sup>*<sup>ˆ</sup> (*m*−1)0(*x*) , *<sup>λ</sup>*<sup>0</sup> <sup>≡</sup> 1 . (25)

$$\mathcal{U}\_{\boldsymbol{n}}^{\prime\prime}(\boldsymbol{\chi}) + (\lambda\_{\boldsymbol{n}} - 1) \frac{[\mathcal{U}\_{\boldsymbol{n}}^{\prime}(\boldsymbol{\chi})]^2}{\mathcal{U}\_{\boldsymbol{n}}(\boldsymbol{\chi})} + 2\hat{\mathcal{W}}\_{(\boldsymbol{n}-1)0}(\boldsymbol{\chi})\mathcal{U}\_{\boldsymbol{n}}^{\prime}(\boldsymbol{\chi}) = \boldsymbol{0}. \tag{28}$$

This equation corresponds to Eq.(21) in the case when *n* = 0. It reduces in general to an ordinary linear differential equation only when *λ<sup>n</sup>* = 1 , ∀*n* �= 0. We will focus on this special case in this work.

The general solution of Eq.(28) for arbitrary *n* �= 0, and with *λ<sup>n</sup>* set to unity, can be found by elementary means, and we deduce that

$$F\_{n0}(\mathbf{x})\mathcal{W}(\mathbf{x}) = \frac{\mathsf{C}\_{n2}e^{-2\int^{x}\mathsf{\hat{W}}\_{(n-1)0}(t)dt}}{\mathsf{C}\_{n1} + \mathsf{C}\_{n2}\int^{x}e^{-2\int^{u}\mathsf{\hat{W}}\_{(n-1)0}(t)dt}du} = $$

$$= \frac{d}{dx}\ln(\mathsf{C}\_{n1} + \mathsf{C}\_{n2}\int^{x}e^{-2\int^{u}\mathsf{\hat{W}}\_{(n-1)0}(t)dt}du)\,. \tag{29}$$

*Cn*<sup>1</sup> and *Cn*<sup>2</sup> are integration constants, which we assume to be real. We can reduce the number of integration constants to one at each iteration level, but we will stick to the habit of explicitly writing down the actual number of constants in order to make it easier to compare the various formulas we deduce, which stem from both second and first order differential equations. We also note that the structure of *Fn*0(*x*) implies that previous deformations are not regenerated in general. Of course, this does not exclude this possibility to arise, as we will see in Section 5. Hence, *m* in Eq.(25) has in principle no natural upper bound. From Eq.(25) and Eq.(29) we get the following expression for the superpotential at iteration level *m*

$$
\begin{split}
\hat{\mathsf{W}}\_{\mathsf{m}0}(\mathbf{x}) &= \hat{\mathsf{W}}\_{\mathsf{0}0}(\mathbf{x}) + \sum\_{j=1}^{m} \frac{d}{d\mathbf{x}} \, \ln(\mathsf{C}\_{j1} + \mathsf{C}\_{j2} \int^{\mathsf{x}} e^{-2\int^{u} \hat{\mathsf{W}}\_{(j-1)0}(t) dt} \, du) = \\ &= \hat{\mathsf{W}}\_{\mathsf{0}0}(\mathbf{x}) + \frac{d}{d\mathbf{x}} \ln \prod\_{j=1}^{m} (\mathsf{C}\_{j1} + \mathsf{C}\_{j2} \int^{\mathsf{x}} e^{-2\int^{u} \hat{\mathsf{W}}\_{(j-1)0}(t) dt} \, du) \equiv \\ &\equiv \hat{\mathsf{W}}\_{\mathsf{0}0}(\mathbf{x}) + \frac{d}{d\mathbf{x}} \ln P\_{\mathsf{m}}(\mathbf{x}) \,. \end{split} \tag{30}
$$

 

*<sup>W</sup>* <sup>×</sup>*F*0=*F*<sup>00</sup> −−−−−→ *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>

can be derived as a special case of Eq.(19) with *<sup>X</sup>*−<sup>1</sup>

*d*

−−−−→ *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup> <sup>=</sup> *<sup>W</sup>* <sup>+</sup>*F*10*<sup>W</sup>*

+*F*10*W*

thus in this particular case regenerated by the scheme at the next recursion level, i.e.

From Eq.(26) we find that *Xn*(*x*) satisfies the equation

*dx Xn*(*x*) <sup>−</sup> ( *<sup>d</sup>*

*Fn*0(*x*)*W*(*x*) <sup>→</sup> *Fn*0(*x*)*W*(*x*) + *<sup>d</sup>*

*Qm*(*x*) ≡

general expression for the superpotential in Eq.(35) can be written as

*m* ∏ *i*=0

*<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*(*x*) = *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) + *<sup>d</sup>*

.

*<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*(*x*) = *<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*0(*x*) + *<sup>d</sup>*

(*Ci*<sup>3</sup> + *λ<sup>i</sup>*

� *<sup>x</sup> e* −2

*dx* ln *Pm*(*x*) + *<sup>d</sup>*

*<sup>m</sup>* = 0 in Eq.(38) (*λ*<sup>0</sup> ≡ 1) reproduces Eq.(19). In the special case when *<sup>λ</sup><sup>m</sup>* = 1 , ∀*<sup>m</sup>* in Eq.(25)

When we compare the expressions for (*Pm*(*x*))′ and (*Qm*(*x*))′ we find that they differ by just

−−−−→ *<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup> <sup>=</sup> *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup> + (ln *<sup>P</sup>*1)′ <sup>+</sup>*F*20*<sup>W</sup>*

−−−−→ *<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup> <sup>=</sup> *<sup>W</sup>* + (ln *<sup>P</sup>*1)′ <sup>+</sup>*F*20*<sup>W</sup>*

**Figure 1.** The upper line depicts the solvable deformation chain Eq.(25) to iteration level *m*. There is no upper bound on *m*. The *Fj*0(*x*) functions are given in Eq.(29). The *Pj*(*x*) functions are given in Eq.(30) and Eq.(32). They are functions of a base deformation *W*ˆ <sup>00</sup>(*x*). A base deformation *W*ˆ <sup>00</sup>(*x*) is generated by the zero-energy Schrödinger equation interacting with the partner potential *V*+(*x*), Eq.(21). The second line depicts the important special case when *F*0(*x*) = 1. This particular solution

Eq.(20) and the solution Eq.(22). *W*ˆ <sup>10</sup>(*x*) then coincides with Eq.(19) (when *F*00(*x*) = 1 and *C*<sup>01</sup> is finite in Eq.(19)); Eq.(19) is

This equation is a generalization of Eq.(18). The *n*'th deformation term Eq.(29) thus changes

*Cn*<sup>3</sup> are integrations constants, which we assume to be real. Eq.(37) implies that the more

*dx* ln(*Cn*<sup>3</sup> <sup>+</sup> *<sup>λ</sup><sup>n</sup>*

−−−−→ · · ·

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

−−−−→ · · ·

<sup>0</sup> <sup>=</sup> <sup>0</sup>, which can be achieved by *<sup>C</sup>*<sup>01</sup> <sup>→</sup> <sup>∞</sup>, and *<sup>F</sup>*00(*x*) determined by

*dx* ln *<sup>W</sup>*(*x*) + <sup>2</sup>*W*<sup>ˆ</sup> *<sup>n</sup>*0(*x*))*Xn*(*x*) = *<sup>λ</sup>mW*(*x*). (36)

� *<sup>x</sup> e* −2

+*Fm*0*W* −−−−→ *<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*<sup>0</sup>

http://dx.doi.org/10.5772/53845

715

+*Fm*0*W* −−−−→ *<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*<sup>0</sup>

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> *<sup>n</sup>*0(*t*)*dtdu*). (37)

*dx* ln *Qm*(*x*), (38)

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> *<sup>i</sup>*0(*t*)*dtdu*). (39)

*dx* ln *Qm*(*x*). (40)

into

where

we get

the last term in (*Qm*(*x*))′

*<sup>W</sup>* <sup>×</sup>*F*0=<sup>1</sup>

From Eq.(30) we deduce that

$$e^{-2\int^{x}\hat{W}\_{(j-1)0}(t)dt} = P\_{j-1}^{-2}(\mathbf{x})e^{-2\int^{x}\hat{W}\_{00}(t)dt} \; ; \; P\_0^2(\mathbf{x}) \equiv 1 \; ; \; j \neq 0 \; \tag{31}$$

such that

$$\begin{cases} P\_1(\mathbf{x}) = \mathbb{C}\_{11} + \mathbb{C}\_{12} \int^{\mathbf{x}} e^{-2\int^{\mathbf{u}} \hat{\mathbf{W}}\_{00}(t) dt} d\mathbf{u}\_{\prime} \\ P\_2(\mathbf{x}) = P\_1(\mathbf{x}) (\mathbb{C}\_{21} + \mathbb{C}\_{22} \int^{\mathbf{x}} P\_1^{-2}(\mathbf{u}) e^{-2\int^{\mathbf{u}} \hat{\mathbf{W}}\_{00}(t) dt} d\mathbf{u}) \\ \vdots \\ \vdots \\ P\_m(\mathbf{x}) = P\_{m-1}(\mathbf{x}) (\mathbb{C}\_{m1} + \mathbb{C}\_{m2} \int^{\mathbf{x}} P\_{m-1}^{-2}(\mathbf{u}) e^{-2\int^{\mathbf{u}} \hat{\mathbf{W}}\_{00}(t) dt} d\mathbf{u}) \end{cases} \tag{32}$$

Hence,

$$P\_{\mathbb{H}}(\mathbf{x}) = \prod\_{j=1}^{n} (\mathbb{C}\_{j1} + \mathbb{C}\_{j2} \int^{\infty} P\_{j-1}^{-2}(u) e^{-2 \int^{u} \mathbb{W}\_{00}(t) dt} du) \,. \tag{33}$$

This last form of the *Pn*(*x*) functions neatly exhibits how the base deformation *W*ˆ <sup>00</sup>(*x*) generates the higher order deformations. Some of the details we have deduced so far are presented in Figure 1.

Make the following substitution at each iteration level in Eq.(25)

$$F\_{n0}(\mathbf{x}) \to F\_{n0}(\mathbf{x}) + \frac{1}{X\_{\mathcal{U}}(\mathbf{x})} \,. \tag{34}$$

This implies (with the *λm*'s reinstated in Eq.(25)) a generalized form *W*ˆ *<sup>m</sup>*(*x*) of the superpotential *W*ˆ *<sup>m</sup>*0(*x*)

$$
\hat{\mathcal{W}}\_{m0}(\mathbf{x}) = \sum\_{i=0}^{m} \lambda\_i F\_{i0} \mathcal{W}(\mathbf{x}) \Rightarrow \hat{\mathcal{W}}\_{m}(\mathbf{x}) = \hat{\mathcal{W}}\_{m0}(\mathbf{x}) + \sum\_{i=0}^{m} \frac{\lambda\_i}{X\_i(\mathbf{x})} \mathcal{W}(\mathbf{x}) \,. \tag{35}
$$

$$\begin{cases} \begin{array}{ccccc} \mathcal{W} & \xrightarrow{\times \mathcal{F} = \mathcal{F}\_{00}} & \mathcal{W}\_{00} & \xrightarrow{+\mathcal{F}\_{0} \mathcal{W}} & \hat{\mathcal{W}}\_{10} = \mathcal{W}\_{00} + (\ln P\_{1})' & \xrightarrow{+\mathcal{F}\_{20} \mathcal{W}} & \cdots & \xrightarrow{+\mathcal{F}\_{m0} \mathcal{W}} & \hat{\mathcal{W}}\_{m0} \\\\ \mathcal{W} & \xrightarrow{\times \mathcal{F}\_{0} = \mathcal{I}} & \hat{\mathcal{W}}\_{00} = \mathcal{W} & \xrightarrow{+\mathcal{F}\_{0} \mathcal{W}} & \hat{\mathcal{W}}\_{10} = \mathcal{W} + (\ln P\_{1})' & \xrightarrow{+\mathcal{F}\_{20} \mathcal{W}} & \cdots & \xrightarrow{+\mathcal{F}\_{m0} \mathcal{W}} & \hat{\mathcal{W}}\_{m0} \end{array}$$

**Figure 1.** The upper line depicts the solvable deformation chain Eq.(25) to iteration level *m*. There is no upper bound on *m*. The *Fj*0(*x*) functions are given in Eq.(29). The *Pj*(*x*) functions are given in Eq.(30) and Eq.(32). They are functions of a base deformation *W*ˆ <sup>00</sup>(*x*). A base deformation *W*ˆ <sup>00</sup>(*x*) is generated by the zero-energy Schrödinger equation interacting with the partner potential *V*+(*x*), Eq.(21). The second line depicts the important special case when *F*0(*x*) = 1. This particular solution can be derived as a special case of Eq.(19) with *<sup>X</sup>*−<sup>1</sup> <sup>0</sup> <sup>=</sup> <sup>0</sup>, which can be achieved by *<sup>C</sup>*<sup>01</sup> <sup>→</sup> <sup>∞</sup>, and *<sup>F</sup>*00(*x*) determined by Eq.(20) and the solution Eq.(22). *W*ˆ <sup>10</sup>(*x*) then coincides with Eq.(19) (when *F*00(*x*) = 1 and *C*<sup>01</sup> is finite in Eq.(19)); Eq.(19) is thus in this particular case regenerated by the scheme at the next recursion level, i.e.

From Eq.(26) we find that *Xn*(*x*) satisfies the equation

$$\frac{d}{d\mathbf{x}}\mathbf{X}\_{\mathrm{fl}}(\mathbf{x}) - \left(\frac{d}{d\mathbf{x}}\ln\mathcal{W}(\mathbf{x}) + 2\hat{\mathcal{W}}\_{\mathrm{n0}}(\mathbf{x})\right)\mathbf{X}\_{\mathrm{fl}}(\mathbf{x}) = \lambda\_{\mathrm{fl}}\mathcal{W}(\mathbf{x})\,. \tag{36}$$

This equation is a generalization of Eq.(18). The *n*'th deformation term Eq.(29) thus changes into

$$F\_{n0}(\mathbf{x})\mathcal{W}(\mathbf{x}) \to F\_{n0}(\mathbf{x})\mathcal{W}(\mathbf{x}) + \frac{d}{d\mathbf{x}}\ln(\mathbb{C}\_{n3} + \lambda\_n \int^{\mathbf{x}} e^{-2\int^{u} \hat{\mathcal{W}}\_{n0}(t)dt} du) \,. \tag{37}$$

*Cn*<sup>3</sup> are integrations constants, which we assume to be real. Eq.(37) implies that the more general expression for the superpotential in Eq.(35) can be written as

$$
\hat{\mathcal{W}}\_m(\mathbf{x}) = \hat{\mathcal{W}}\_{m0}(\mathbf{x}) + \frac{d}{d\mathbf{x}} \ln Q\_m(\mathbf{x}) \,. \tag{38}
$$

where

8 Quantum Mechanics

*W*ˆ *<sup>m</sup>*0(*x*) = *W*ˆ <sup>00</sup>(*x*) +

From Eq.(30) we deduce that

such that

Hence,

*e* −2

. . . . . .

presented in Figure 1.

superpotential *W*ˆ *<sup>m</sup>*0(*x*)

*W*ˆ *<sup>m</sup>*0(*x*) =

*m* ∑ *i*=0

<sup>=</sup> *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) + *<sup>d</sup>*

<sup>≡</sup> *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) + *<sup>d</sup>*

� *<sup>x</sup> <sup>W</sup>*<sup>ˆ</sup> (*j*−1)0(*t*)*dt* <sup>=</sup> *<sup>P</sup>*−<sup>2</sup>

*P*1(*x*) = *C*<sup>11</sup> + *C*<sup>12</sup>

*Pn*(*x*) =

*P*2(*x*) = *P*1(*x*)(*C*<sup>21</sup> + *C*<sup>22</sup>

*Pm*(*x*) = *Pm*−1(*x*)(*Cm*<sup>1</sup> + *Cm*<sup>2</sup>

(*Cj*<sup>1</sup> + *Cj*<sup>2</sup>

� *<sup>x</sup> <sup>P</sup>*−<sup>2</sup> *<sup>j</sup>*−1(*u*)*e*

This last form of the *Pn*(*x*) functions neatly exhibits how the base deformation *W*ˆ <sup>00</sup>(*x*) generates the higher order deformations. Some of the details we have deduced so far are

*Fn*0(*x*) <sup>→</sup> *Fn*0(*x*) + <sup>1</sup>

This implies (with the *λm*'s reinstated in Eq.(25)) a generalized form *W*ˆ *<sup>m</sup>*(*x*) of the

*<sup>λ</sup>iFi*0*W*(*x*) ⇒ *<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*(*x*) = *<sup>W</sup>*<sup>ˆ</sup> *<sup>m</sup>*0(*x*) +

*n* ∏ *j*=1

Make the following substitution at each iteration level in Eq.(25)

*m* ∑ *j*=1

*dx* ln

*d*

*m* ∏ *j*=1

*<sup>j</sup>*−1(*x*)*e*

� *<sup>x</sup> <sup>e</sup>*−<sup>2</sup>

−2

*dx* ln(*Cj*<sup>1</sup> <sup>+</sup> *Cj*<sup>2</sup>

(*Cj*<sup>1</sup> + *Cj*<sup>2</sup>

� *<sup>x</sup> e* −2

� *<sup>x</sup> e* −2

� *<sup>x</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dt* ; *<sup>P</sup>*<sup>2</sup>

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu* ,

<sup>1</sup> (*u*)*e*−<sup>2</sup>

� *<sup>x</sup> <sup>P</sup>*−<sup>2</sup>

*m*−1(*u*)*e*−<sup>2</sup>

−2

� *<sup>x</sup> <sup>P</sup>*−<sup>2</sup>

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> (*j*−1)0(*t*)*dtdu*) =

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> (*j*−1)0(*t*)*dtdu*) <sup>≡</sup>

<sup>0</sup> (*x*) <sup>≡</sup> 1 , *<sup>j</sup>* � 0 , (31)

(32)

*dx* ln *Pm*(*x*). (30)

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu*),

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu*).

� *<sup>u</sup> <sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*t*)*dtdu*). (33)

*Xn*(*x*) . (34)

*W*(*x*). (35)

*m* ∑ *i*=0

*λi Xi*(*x*)

$$Q\_{\mathfrak{M}}(\mathbf{x}) \equiv \prod\_{i=0}^{m} (\mathbb{C}\_{i\mathfrak{I}} + \lambda\_i \int^{\mathbf{x}} e^{-2\int^{u} \hat{\mathcal{W}}\_{\mathfrak{I}0}(t) dt} d\mathbf{u}) \,. \tag{39}$$

*<sup>m</sup>* = 0 in Eq.(38) (*λ*<sup>0</sup> ≡ 1) reproduces Eq.(19). In the special case when *<sup>λ</sup><sup>m</sup>* = 1 , ∀*<sup>m</sup>* in Eq.(25) we get

$$
\hat{\mathcal{W}}\_m(\mathbf{x}) = \hat{\mathcal{W}}\_{00}(\mathbf{x}) + \frac{d}{d\mathbf{x}} \ln P\_m(\mathbf{x}) + \frac{d}{d\mathbf{x}} \ln Q\_m(\mathbf{x}) \,. \tag{40}
$$

When we compare the expressions for (*Pm*(*x*))′ and (*Qm*(*x*))′ we find that they differ by just the last term in (*Qm*(*x*))′ .

#### *3.2.2. The product*

What happens if we in Eq.(25) assume a product structure instead of a sum structure ? Let us assume that we have determined a base deformation. Let this be the seed superpotential for the deformation

$$
\hat{\mathcal{W}}\_{00}(\mathbf{x}) \to \hat{\mathcal{W}}\_{10}(\mathbf{x}) = F\_{10}(\mathbf{x}) \hat{\mathcal{W}}\_{00}(\mathbf{x}) = F\_{10}(\mathbf{x}) F\_{00}(\mathbf{x}) \mathcal{W}(\mathbf{x}) \,. \tag{41}
$$

The isospectrality condition then implies

*<sup>i</sup>*0(*x*)+[*k*(ln *<sup>W</sup>*(*x*))′ <sup>+</sup> <sup>2</sup>*W*<sup>ˆ</sup> (*i*−1)0(*x*)]*Fi*0(*x*) + *<sup>W</sup>k*(*x*)*F*<sup>2</sup>

*Fi*0(*x*) = <sup>1</sup>

*<sup>k</sup>* (*x*) + <sup>2</sup>*W*<sup>ˆ</sup> *<sup>i</sup>*−1(*x*)*U*′

Another canonical generalization of our work is to consider deformations on the form

*<sup>H</sup>*0(*x*) + F(*W*)*H*<sup>2</sup>

easily verified that Eq.(50) can be cast into the form Eq.(21) by the substitution

*<sup>H</sup>*0(*x*) = <sup>1</sup>

where *H*00(*x*) is a particular solution of Eq.(50). We then get the equation

where *Uk*(*x*) is some function. This expression inserted into Eq.(46) gives

*U*′′

Clearly, the range of values of *k* can be expanded to the real numbers.

This is a Riccati type equation of the kind we have met earlier in this work. Apparently, different *k*-values give rise to very different equations to solve. However, and rather intriguingly, all the possible *k*-values implies the same deformation. This is seen by making

*Wk*(*x*)

Hence, *W*ˆ *<sup>i</sup>*0(*x*) is independent of *k* and we are essentially left with a linear deformation.

where F is any *functional* of the seed superpotential *W*(*x*). The isospectrality condition then

Note that *<sup>H</sup>*0(*x*) = 1 does *not* solve this equation unless F(*W*) = *<sup>W</sup>*, since Eq.(50) with

choice of functional will fail to satisfy the isospectrality condition. Hence the conclusion. The particular solution *H*0(*x*) = 1 is not forced upon us. We can in principle do without it. It is

F(*W*)

*<sup>H</sup>*0(*x*) = *<sup>H</sup>*00(*x*) + <sup>1</sup>

*U*′ *<sup>k</sup>*(*x*)

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

*<sup>i</sup>*0(*x*) = 0 . (46)

http://dx.doi.org/10.5772/53845

717

(*x*)). (50)

*Uk*(*x*) , (47)

*<sup>k</sup>*(*x*) = 0 . (48)

(ln *<sup>U</sup>*(*x*))′ . (51)

*<sup>Z</sup>*0(*x*) , (52)

*<sup>W</sup>*<sup>ˆ</sup> (*x*) = *<sup>H</sup>*0(*x*)F(*W*), (49)

<sup>0</sup> (*x*) = <sup>F</sup>(*W*)−1(*W*2(*x*) + *<sup>W</sup>*′

*dx* ln <sup>F</sup>(*W*) + <sup>2</sup>*H*00(*x*)F(*W*))*Z*0(*x*) = <sup>F</sup>(*W*), (53)

(*x*) + F2(*x*) = *V*+(*x*). Since *V*+(*x*) is uniquely given in Eq.(2) any other

*F*′

the following substitution

implies

*H*′

*<sup>H</sup>*0(*x*) = 1 implies <sup>F</sup>′

<sup>0</sup>(*x*)+(ln <sup>F</sup>(*W*))′

We can also look for an expanded solution by writing

*d*

*dx <sup>Z</sup>*0(*x*) <sup>−</sup> ( *<sup>d</sup>*

where *F*10(*x*) is some function to be determined by the isospectrality condition. This product scheme can of course in principle be repeated an arbitrary number *m* times

$$
\hat{\mathcal{W}}\_{m0}(\mathbf{x}) = (\prod\_{i=0}^{m} F\_{i0}(\mathbf{x})) \mathcal{W}(\mathbf{x}) = F\_{m0}(\mathbf{x}) \hat{\mathcal{W}}\_{(m-1)0} \,. \tag{42}
$$

This structure gives rise to the following set of equations

$$\begin{cases} F\_{0}'(\mathbf{x}) + (\ln \mathcal{W}(\mathbf{x}))' \mathbf{F}\_{00}(\mathbf{x}) + \mathcal{W}(\mathbf{x}) \mathbf{F}\_{00}^{2}(\mathbf{x}) = \mathcal{W}(\mathbf{x}) + (\ln \mathcal{W}(\mathbf{x}))',\\ F\_{10}'(\mathbf{x}) + (\ln \hat{\mathcal{W}}\_{00}(\mathbf{x}))' \mathbf{F}\_{10}(\mathbf{x}) + \hat{\mathcal{W}}\_{00}(\mathbf{x}) \mathbf{F}\_{10}^{2}(\mathbf{x}) = \frac{1}{\text{Fo}(\mathbf{x})} (\mathcal{W} + (\ln \mathcal{W}(\mathbf{x}))'),\\ \vdots\\ \vdots\\ F\_{m0}'(\mathbf{x}) + (\ln \hat{\mathcal{W}}\_{(m-1)0}(\mathbf{x}))' \mathbf{F}\_{m0}(\mathbf{x}) + \hat{\mathcal{W}}\_{(m-1)0}(\mathbf{x}) \mathbf{F}\_{m0}^{2}(\mathbf{x}) = \frac{\mathcal{W}(\mathbf{x})}{\mathcal{W}\_{(m-1)0}(\mathbf{x})} (\mathcal{W}(\mathbf{x}) + (\ln \mathcal{W}(\mathbf{x}))'). \end{cases} \tag{43}$$

Clearly, each iteration level depends on all the previous ones, and at each level we are dealing with a non-homogenous non-linear differential equation. Interestingly, by making the following substitution at an arbitrary iteration level *n* �= 0

$$F\_{n0}(\mathbf{x}) = \frac{1}{\hat{W}\_{(n-1)0}(\mathbf{x})} (\ln \mathcal{U}\_n(\mathbf{x}))',\tag{44}$$

where *Un*(*x*) is some function, the equations Eq.(43) all reduce to Eq.(21). Hence, attempting to generate novel deformations recursively via a product structure, of the kind above, fails. This conclusion was also reached in [6], but at the level of the second order linear differential equation Eq.(21).

#### **3.3. Recursive non-linear deformations**

We have so far only considered linear (in the superpotential) deformations. In this section we will briefly consider two non-linear deformation schemes. Let us first consider a polynomial kind of deformation. That is, given a superpotential *<sup>W</sup>*<sup>ˆ</sup> (*i*−1)0(*x*) which we will assume is derived, in some way or another, from some seed superpotential *W*(*x*). Consider then the polynomial deformation

$$
\hat{\mathcal{W}}\_{i0}(\mathbf{x}) = F\_{i0}(\mathbf{x})\mathcal{W}^k(\mathbf{x}) + \hat{\mathcal{W}}\_{(i-1)0}(\mathbf{x}); k \in \{1, 2, 3, \dots\}. \tag{45}
$$

The isospectrality condition then implies

10 Quantum Mechanics

*3.2.2. The product*

for the deformation

*F*′

*F*′

. . . . . . *F*′

equation Eq.(21).

polynomial deformation

<sup>00</sup>(*x*)+(ln *<sup>W</sup>*(*x*))′

<sup>10</sup>(*x*)+(ln *<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*))′

*<sup>m</sup>*0(*x*)+(ln *<sup>W</sup>*<sup>ˆ</sup> (*m*−1)0(*x*))′

**3.3. Recursive non-linear deformations**

What happens if we in Eq.(25) assume a product structure instead of a sum structure ? Let us assume that we have determined a base deformation. Let this be the seed superpotential

where *F*10(*x*) is some function to be determined by the isospectrality condition. This product

scheme can of course in principle be repeated an arbitrary number *m* times

*m* ∏ *i*=0

*W*ˆ *<sup>m</sup>*0(*x*)=(

This structure gives rise to the following set of equations

*F*00(*x*) + *W*(*x*)*F*<sup>2</sup>

the following substitution at an arbitrary iteration level *n* �= 0

*F*10(*x*) + *W*ˆ <sup>00</sup>(*x*)*F*<sup>2</sup>

*<sup>W</sup>*<sup>ˆ</sup> <sup>00</sup>(*x*) → *<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup>(*x*) = *<sup>F</sup>*10(*x*)*W*<sup>ˆ</sup> <sup>00</sup>(*x*) = *<sup>F</sup>*10(*x*)*F*00(*x*)*W*(*x*), (41)

<sup>00</sup>(*x*) = *<sup>W</sup>*(*x*)+(ln *<sup>W</sup>*(*x*))′ ,

<sup>10</sup>(*x*) = <sup>1</sup>

Clearly, each iteration level depends on all the previous ones, and at each level we are dealing with a non-homogenous non-linear differential equation. Interestingly, by making

*<sup>W</sup>*<sup>ˆ</sup> (*n*−1)0(*x*)

where *Un*(*x*) is some function, the equations Eq.(43) all reduce to Eq.(21). Hence, attempting to generate novel deformations recursively via a product structure, of the kind above, fails. This conclusion was also reached in [6], but at the level of the second order linear differential

We have so far only considered linear (in the superpotential) deformations. In this section we will briefly consider two non-linear deformation schemes. Let us first consider a polynomial kind of deformation. That is, given a superpotential *<sup>W</sup>*<sup>ˆ</sup> (*i*−1)0(*x*) which we will assume is derived, in some way or another, from some seed superpotential *W*(*x*). Consider then the

*<sup>W</sup>*<sup>ˆ</sup> *<sup>i</sup>*0(*x*) = *Fi*0(*x*)*Wk*(*x*) + *<sup>W</sup>*<sup>ˆ</sup> (*i*−1)0(*x*); *<sup>k</sup>* ∈ {1, 2, 3, . . .} . (45)

*Fm*0(*x*) + *<sup>W</sup>*<sup>ˆ</sup> (*m*−1)0(*x*)*F*<sup>2</sup>

*Fn*0(*x*) = <sup>1</sup>

*Fi*0(*x*))*W*(*x*) = *Fm*0(*x*)*W*<sup>ˆ</sup> (*m*−1)<sup>0</sup> . (42)

),

(ln *Un*(*x*))′ , (44)

(*W*(*x*)+(ln *<sup>W</sup>*(*x*))′

). (43)

*<sup>F</sup>*00(*x*)(*<sup>W</sup>* + (ln *<sup>W</sup>*(*x*))′

*<sup>m</sup>*0(*x*) = *<sup>W</sup>*(*x*)

*W*ˆ (*m*−1)0(*x*)

$$F\_{i0}'(\mathbf{x}) + [k(\ln \mathcal{W}(\mathbf{x}))' + 2\hat{\mathcal{W}}\_{(i-1)0}(\mathbf{x})]F\_{i0}(\mathbf{x}) + \mathcal{W}^k(\mathbf{x})F\_{i0}^2(\mathbf{x}) = \mathbf{0}.\tag{46}$$

This is a Riccati type equation of the kind we have met earlier in this work. Apparently, different *k*-values give rise to very different equations to solve. However, and rather intriguingly, all the possible *k*-values implies the same deformation. This is seen by making the following substitution

$$F\_{l0}(\mathbf{x}) = \frac{1}{W^k(\mathbf{x})} \frac{\mathcal{U}\_k'(\mathbf{x})}{\mathcal{U}\_k(\mathbf{x})},\tag{47}$$

where *Uk*(*x*) is some function. This expression inserted into Eq.(46) gives

$$\mathcal{U}l\_k''(\mathbf{x}) + 2\hat{\mathcal{W}}\_{l-1}(\mathbf{x})\mathcal{U}\_k'(\mathbf{x}) = \mathbf{0}.\tag{48}$$

Hence, *W*ˆ *<sup>i</sup>*0(*x*) is independent of *k* and we are essentially left with a linear deformation. Clearly, the range of values of *k* can be expanded to the real numbers.

Another canonical generalization of our work is to consider deformations on the form

$$
\hat{W}(\mathbf{x}) = H\_0(\mathbf{x}) \mathcal{F}(\mathcal{W}) \, , \tag{49}
$$

where F is any *functional* of the seed superpotential *W*(*x*). The isospectrality condition then implies

$$H\_0'(\mathbf{x}) + (\ln \mathcal{F}(\mathcal{W}))' H\_0(\mathbf{x}) + \mathcal{F}(\mathcal{W}) H\_0^2(\mathbf{x}) = \mathcal{F}(\mathcal{W})^{-1} (\mathcal{W}^2(\mathbf{x}) + \mathcal{W}'(\mathbf{x})) \,. \tag{50}$$

Note that *<sup>H</sup>*0(*x*) = 1 does *not* solve this equation unless F(*W*) = *<sup>W</sup>*, since Eq.(50) with *<sup>H</sup>*0(*x*) = 1 implies <sup>F</sup>′ (*x*) + F2(*x*) = *V*+(*x*). Since *V*+(*x*) is uniquely given in Eq.(2) any other choice of functional will fail to satisfy the isospectrality condition. Hence the conclusion. The particular solution *H*0(*x*) = 1 is not forced upon us. We can in principle do without it. It is easily verified that Eq.(50) can be cast into the form Eq.(21) by the substitution

$$H\_0(\mathbf{x}) = \frac{1}{\mathcal{F}(\mathcal{W})} (\ln \mathcal{U}(\mathbf{x}))'. \tag{51}$$

We can also look for an expanded solution by writing

$$H\_0(\mathbf{x}) = H\_{00}(\mathbf{x}) + \frac{1}{Z\_0(\mathbf{x})},\tag{52}$$

where *H*00(*x*) is a particular solution of Eq.(50). We then get the equation

$$\frac{d}{d\mathbf{x}}Z\_0(\mathbf{x}) - (\frac{d}{d\mathbf{x}}\ln \mathcal{F}(\mathcal{W}) + 2H\_{00}(\mathbf{x})\mathcal{F}(\mathcal{W}))Z\_0(\mathbf{x}) = \mathcal{F}(\mathcal{W}),\tag{53}$$

which is a generalized form of Eq.(18). The reciprocal solution has the general form

$$\frac{1}{Z\_0(\boldsymbol{x})} = \frac{e^{-2\int^{\boldsymbol{x}} H\_{00}(t) \mathcal{F}(\boldsymbol{W}) dt}}{\mathcal{F}(\boldsymbol{W}) (\mathbb{C} + \int^{\boldsymbol{X}} e^{-2\int^{\boldsymbol{x}} H\_{00}(t) \mathcal{F}(\boldsymbol{W}) dt} d\boldsymbol{u})} \,. \tag{54}$$

Let us consider the *s*-state with *l* = 0 in order to get a better grasp on the content buried in

after redefining *C*<sup>11</sup> such that the lower integration limit of the integral in Eq.(60) does not appear explicitly in the expression for the potential. We will automatically do such redefinitions in the following when it is appropriate. The corresponding physical potential

with *W*ˆ <sup>00</sup>(*x*) = *W*(*x*) and *C*<sup>01</sup> finite. This is a consequence of a regeneration of Eq.(19) by the

In the special case when we set *C*<sup>11</sup> = 0 the last term in Eq.(61) becomes independent of the exponentials (and *C*12) and thus reduces to a pure rational function. The physical potential

Let us go to the second iteration level starting from the expression for *W*ˆ <sup>10</sup>(*x*) in Eq.(61) with

Note that when *C*<sup>21</sup> = 0 we get *W*ˆ <sup>20</sup>(*x*) = *W*(*x*). Hence, the deformation scheme allows in general for the possibility that additional iterations in particular cases may regenerate previous potentials in a nontrivial fashion. The expression for the corresponding physical

<sup>−</sup>1(*x*) + *<sup>C</sup>*22*x*2*e<sup>x</sup>*

1 2 *x*2 *x*<sup>2</sup> + 2*x* + 2

(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)<sup>2</sup> <sup>≡</sup> *<sup>V</sup>*−(*x*) + <sup>4</sup>*x*(*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)

*C*21(*x*<sup>2</sup> + 2*x* + 2)<sup>2</sup> + *C*22(*x*<sup>2</sup> + 2*x* + 2)*e<sup>x</sup>*

) + <sup>2</sup>(*C*21(2*<sup>x</sup>* <sup>+</sup> <sup>2</sup>) + *<sup>C</sup>*22*ex*) *C*21(*x*<sup>2</sup> + 2*x* + 2) + *C*22*e<sup>x</sup>*

4*x*(*x* + 2)

*<sup>W</sup>*<sup>ˆ</sup> <sup>20</sup>(*x*) = *<sup>W</sup>*<sup>ˆ</sup> <sup>1</sup>(*x*) + *<sup>C</sup>*22*x*2*e<sup>x</sup>*

recursion scheme which was noted in Figure 1. From the definition it follows that

*<sup>C</sup>*12*x*(2*<sup>x</sup>* <sup>−</sup> <sup>4</sup>)*e*−*<sup>x</sup> <sup>C</sup>*<sup>11</sup> <sup>−</sup> *<sup>C</sup>*12(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)*e*−*<sup>x</sup>* <sup>+</sup>

*<sup>C</sup>*12*x*2*e*−*<sup>x</sup>*

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

<sup>−</sup>1(*x*) ≡ *<sup>W</sup>*<sup>ˆ</sup> <sup>2</sup>

*<sup>C</sup>*<sup>11</sup> <sup>−</sup> *<sup>C</sup>*12(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)*e*−*<sup>x</sup>* (61)

<sup>10</sup>(*x*) <sup>−</sup> *<sup>W</sup>*<sup>ˆ</sup> ′

2*C*<sup>2</sup>

*<sup>C</sup>*21(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)<sup>2</sup> <sup>+</sup> *<sup>C</sup>*22(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)*e<sup>x</sup>* . (64)

12*x*4*e*−2*<sup>x</sup>* (*C*<sup>11</sup> <sup>−</sup> *<sup>C</sup>*12(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)*e*−*x*)<sup>2</sup> . (62)

(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>)<sup>2</sup> . (63)

 ×

. (65)

<sup>10</sup>(*x*) or from Eq.(23)

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719

Eq.(60). We also set *<sup>q</sup>* ≡ 1. The expression for *<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup>(*x*) then reduces to

<sup>2</sup> <sup>−</sup> <sup>1</sup> *x* +

*<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup>(*x*) = <sup>1</sup>

<sup>−</sup>1(*x*) can either be derived from the definition *V*ˆ

*V*ˆ

*V*ˆ

*V*ˆ

<sup>−</sup>1(*x*) = <sup>1</sup>

<sup>4</sup> <sup>−</sup> <sup>1</sup> *x* +

*V*ˆ

potential is given by

*V*ˆ

<sup>−</sup>2(*x*) = *V*ˆ

× −4( 1 *x* +

<sup>−</sup>1(*x*) generated by *W*ˆ <sup>10</sup>(*x*) is then given by

<sup>−</sup>1(*x*) = <sup>1</sup>

*C*<sup>11</sup> = 0, for convenience. It then follows that

<sup>4</sup> <sup>−</sup> <sup>1</sup> *x* +

*C* is an integration constant, which we assume to be real. Utilizing that *H*00(*x*) = F−1(*x*)(ln *<sup>U</sup>*(*x*))′ the resulting deformation coincides with Eq.(19). We thus therefore conclude that non-linear deformations on the form Eq.(49) does not generate additional deformations to the ones already generated by Eq.(13).

#### **3.4. Deforming the Coulomb potential**

As a relatively simple application of the linear deformation scheme let us briefly consider deformations of the Coulomb potential. This potential has, within the framework of SUSYQM, been treated in several previous works [5]. The superpotential and the partner potential for the Coulomb potential are given by [5]

$$W(\mathbf{x}) = \frac{q^2}{2(l+1)} - \frac{(l+1)}{\mathbf{x}}\,'\,\tag{55}$$

$$V\_+(\mathbf{x}) = \frac{1}{4}(\frac{q^2}{l+1})^2 - \frac{q^2}{\mathbf{x}} + \frac{(l+1)(l+2)}{\mathbf{x}^2} \,. \tag{56}$$

*q* and *l* in these expressions are the electric charge and the angular momentum quantum numbers, respectively. These potentials result in the following general solution for *U*0(*x*) in Eq.(21) [6]

$$\mathcal{U}\_0(\mathbf{x}) = \mathbb{C}\_1 \mathcal{M}\_{l+1,l+\frac{3}{2}}(\frac{q^2 \mathbf{x}}{l+1}) + \mathbb{C}\_2 \mathcal{W}\_{l+1,l+\frac{3}{2}}(\frac{q^2 \mathbf{x}}{l+1}) \,. \tag{57}$$

The *M*(*x*)- and *W*(*x*)-functions are the Whittaker functions. The solution Eq.(22) is given by [6]

$$\mathrm{GL}\_0(\mathbf{x}) \sim e^{\frac{q^2 \mathbf{x}}{2(l+1)} - (l+1)\ln(2\mathbf{x})}.\tag{58}$$

We will for simplicity assume this solution in the following. We will let *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> in Eq.(19) such that we deal with the identity deformation *W*ˆ <sup>0</sup>(*x*) = *W*ˆ <sup>00</sup>(*x*) = *W*(*x*). We will also ignore the *Qj*(*x*) contributions in the following. Define *<sup>A</sup>* ≡ *<sup>q</sup>*2/(2(*<sup>l</sup>* + <sup>1</sup>)) and *<sup>B</sup>* ≡ *<sup>l</sup>* + 1. It then follows that

$$P\_1(\mathbf{x}) = \mathbb{C}\_{11} + \mathbb{C}\_{12} \int^{\mathbf{x}} t^{2B} e^{-2At} dt \,\mathrm{d}\mathbf{t} \,\tag{59}$$

such that

$$
\hat{\mathcal{W}}\_{10}(\mathbf{x}) = A - \frac{B}{\mathbf{x}} + \frac{\mathbf{C}\_{12} \mathbf{x}^{2B} e^{-2A\mathbf{x}}}{\mathbf{C}\_{11} + \mathbf{C}\_{12} \int^{\mathbf{x}} t^{2B} e^{-2At} dt}. \tag{60}
$$

Let us consider the *s*-state with *l* = 0 in order to get a better grasp on the content buried in Eq.(60). We also set *<sup>q</sup>* ≡ 1. The expression for *<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup>(*x*) then reduces to

12 Quantum Mechanics

Eq.(21) [6]

then follows that

such that

[6]

which is a generalized form of Eq.(18). The reciprocal solution has the general form

F(*W*)(*<sup>C</sup>* + *<sup>x</sup> <sup>e</sup>*−<sup>2</sup>

*C* is an integration constant, which we assume to be real. Utilizing that *H*00(*x*) = F−1(*x*)(ln *<sup>U</sup>*(*x*))′ the resulting deformation coincides with Eq.(19). We thus therefore conclude that non-linear deformations on the form Eq.(49) does not generate additional

As a relatively simple application of the linear deformation scheme let us briefly consider deformations of the Coulomb potential. This potential has, within the framework of SUSYQM, been treated in several previous works [5]. The superpotential and the partner

<sup>2</sup>(*<sup>l</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> (*<sup>l</sup>* <sup>+</sup> <sup>1</sup>)

)<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup> *x*

*q* and *l* in these expressions are the electric charge and the angular momentum quantum numbers, respectively. These potentials result in the following general solution for *U*0(*x*) in

The *M*(*x*)- and *W*(*x*)-functions are the Whittaker functions. The solution Eq.(22) is given by

*q*2 *x*

We will for simplicity assume this solution in the following. We will let *<sup>C</sup>*<sup>01</sup> → <sup>∞</sup> in Eq.(19) such that we deal with the identity deformation *W*ˆ <sup>0</sup>(*x*) = *W*ˆ <sup>00</sup>(*x*) = *W*(*x*). We will also ignore the *Qj*(*x*) contributions in the following. Define *<sup>A</sup>* ≡ *<sup>q</sup>*2/(2(*<sup>l</sup>* + <sup>1</sup>)) and *<sup>B</sup>* ≡ *<sup>l</sup>* + 1. It

> *<sup>x</sup> t* <sup>2</sup>*Be*

*C*<sup>11</sup> + *C*<sup>12</sup>

*<sup>C</sup>*12*x*<sup>2</sup>*Be*<sup>−</sup>2*Ax*

2 ( *<sup>q</sup>*2*<sup>x</sup> l* + 1

*<sup>U</sup>*0(*x*) ∼ *<sup>e</sup>*

*P*1(*x*) = *C*<sup>11</sup> + *C*<sup>12</sup>

*x* +

*<sup>W</sup>*<sup>ˆ</sup> <sup>10</sup>(*x*) = *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*

*<sup>x</sup> <sup>H</sup>*00(*t*)F(*W*)*dt*

*<sup>H</sup>*00(*t*)F(*W*)*dtdu*)

<sup>+</sup> (*<sup>l</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>l</sup>* <sup>+</sup> <sup>2</sup>)

) + *C*2*Wl*<sup>+</sup>1,*l*<sup>+</sup> <sup>3</sup>

2 ( *<sup>q</sup>*2*<sup>x</sup> l* + 1

<sup>2</sup>(*l*+1) <sup>−</sup>(*l*+1)ln(2*x*) . (58)

<sup>−</sup>2*Atdt* , (59)

*<sup>x</sup> <sup>t</sup>*<sup>2</sup>*Be*−2*Atdt* . (60)

. (54)

*<sup>x</sup>* , (55)

*<sup>x</sup>*<sup>2</sup> . (56)

). (57)

1

deformations to the ones already generated by Eq.(13).

potential for the Coulomb potential are given by [5]

*<sup>W</sup>*(*x*) = *<sup>q</sup>*<sup>2</sup>

4 ( *<sup>q</sup>*<sup>2</sup> *l* + 1

*<sup>V</sup>*+(*x*) = <sup>1</sup>

*U*0(*x*) = *C*1*Ml*<sup>+</sup>1,*l*<sup>+</sup> <sup>3</sup>

**3.4. Deforming the Coulomb potential**

*<sup>Z</sup>*0(*x*) <sup>=</sup> *<sup>e</sup>*−<sup>2</sup>

$$\hat{W}\_{10}(\mathbf{x}) = \frac{1}{2} - \frac{1}{\mathbf{x}} + \frac{\mathbf{C}\_{12}\mathbf{x}^2 e^{-\mathbf{x}}}{\mathbf{C}\_{11} - \mathbf{C}\_{12}(\mathbf{x}^2 + 2\mathbf{x} + 2)e^{-\mathbf{x}}} \tag{61}$$

after redefining *C*<sup>11</sup> such that the lower integration limit of the integral in Eq.(60) does not appear explicitly in the expression for the potential. We will automatically do such redefinitions in the following when it is appropriate. The corresponding physical potential *V*ˆ <sup>−</sup>1(*x*) can either be derived from the definition *V*ˆ <sup>−</sup>1(*x*) ≡ *<sup>W</sup>*<sup>ˆ</sup> <sup>2</sup> <sup>10</sup>(*x*) <sup>−</sup> *<sup>W</sup>*<sup>ˆ</sup> ′ <sup>10</sup>(*x*) or from Eq.(23) with *W*ˆ <sup>00</sup>(*x*) = *W*(*x*) and *C*<sup>01</sup> finite. This is a consequence of a regeneration of Eq.(19) by the recursion scheme which was noted in Figure 1. From the definition it follows that

$$\hat{\mathcal{V}}\_{-1}(\mathbf{x}) = \frac{1}{4} - \frac{1}{\mathbf{x}} + \frac{\mathsf{C}\_{12}\mathbf{x}(2\mathbf{x} - 4)e^{-\mathbf{x}}}{\mathsf{C}\_{11} - \mathsf{C}\_{12}(\mathbf{x}^2 + 2\mathbf{x} + 2)e^{-\mathbf{x}}} + \frac{2\mathsf{C}\_{12}^2\mathbf{x}^4e^{-2\mathbf{x}}}{(\mathsf{C}\_{11} - \mathsf{C}\_{12}(\mathbf{x}^2 + 2\mathbf{x} + 2)e^{-\mathbf{x}})^2} . \tag{62}$$

In the special case when we set *C*<sup>11</sup> = 0 the last term in Eq.(61) becomes independent of the exponentials (and *C*12) and thus reduces to a pure rational function. The physical potential *V*ˆ <sup>−</sup>1(*x*) generated by *W*ˆ <sup>10</sup>(*x*) is then given by

$$\hat{\mathcal{V}}\_{-1}(\mathbf{x}) = \frac{1}{4} - \frac{1}{\mathbf{x}} + \frac{4\mathbf{x}(\mathbf{x} + 2)}{(\mathbf{x}^2 + 2\mathbf{x} + 2)^2} \equiv V\_-(\mathbf{x}) + \frac{4\mathbf{x}(\mathbf{x} + 2)}{(\mathbf{x}^2 + 2\mathbf{x} + 2)^2}.\tag{63}$$

Let us go to the second iteration level starting from the expression for *W*ˆ <sup>10</sup>(*x*) in Eq.(61) with *C*<sup>11</sup> = 0, for convenience. It then follows that

$$
\hat{\mathcal{W}}\_{20}(\mathbf{x}) = \hat{\mathcal{W}}\_1(\mathbf{x}) + \frac{\mathsf{C}\_{22}\mathbf{x}^2 e^{\mathbf{x}}}{\mathsf{C}\_{21}(\mathbf{x}^2 + 2\mathbf{x} + 2)^2 + \mathsf{C}\_{22}(\mathbf{x}^2 + 2\mathbf{x} + 2)e^{\mathbf{x}}} \,. \tag{64}
$$

Note that when *C*<sup>21</sup> = 0 we get *W*ˆ <sup>20</sup>(*x*) = *W*(*x*). Hence, the deformation scheme allows in general for the possibility that additional iterations in particular cases may regenerate previous potentials in a nontrivial fashion. The expression for the corresponding physical potential is given by

$$
\hat{\mathcal{V}}\_{-2}(\mathbf{x}) = \hat{\mathcal{V}}\_{-1}(\mathbf{x}) + \left[ \frac{\mathsf{C}\_{22}\mathbf{x}^2 e^{\mathbf{x}}}{\mathsf{C}\_{21}(\mathbf{x}^2 + 2\mathbf{x} + 2)^2 + \mathsf{C}\_{22}(\mathbf{x}^2 + 2\mathbf{x} + 2)e^{\mathbf{x}}} \right] \times \\
$$

$$
\times \left[ -4(\frac{1}{\mathbf{x}} + \frac{\frac{1}{2}\mathbf{x}^2}{\mathbf{x}^2 + 2\mathbf{x} + 2}) + \frac{2(\mathsf{C}\_{21}(2\mathbf{x} + 2) + \mathsf{C}\_{22}e^{\mathbf{x}})}{\mathsf{C}\_{21}(\mathbf{x}^2 + 2\mathbf{x} + 2) + \mathsf{C}\_{22}e^{\mathbf{x}}} \right]. \tag{65}
$$

**4. Conclusion**

expression Eq.(33).

**Author details**

Bjørn Jensen

the *non-linear* differential equations in Eq.(28).

In a previous paper we showed that isospectral deformations on the form Eq.(15) are contained in the space of deformations generated by isospectral deformations on the form Eq.(13). In this work we have shown that Eq.(13) can be considered as the initial, or base, deformation of a novel infinite recursive isospectral deformation chain. This thus answers to some extend the question by which we ended our previous paper [6]; how does the most general isospectral deformation of the kind considered there (Eq.(13) in this paper) look like. The results in this work do obviously only give a partial answer. We deduced in particular that a class of recursive deformations exists which is generated by the solutions of

A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics

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721

We briefly discussed various ways to construct alternative recursive deformation structures. We considered a linear product structure, polynomial deformations and completely generalized base deformations. They all either failed to provide a recursive structure or

We applied the linear recursive scheme to the Coulomb potential. We derived novel superpotentials which all per construction satisfy the isospectrality condition. It is straightforward, although very tedious, to check that the corresponding physical potentials *V*<sup>+</sup> at the various iteration levels all satisfy the isospectrality condition Eq.(26). This application did also demonstrate how easily novel isospectral deformations can be generated in this approach. It did also demonstrate an increased relative complexity of the generated potentials with the number of iterations, as one also naively would expect from the

The results in this work is obviously only a starting point for further research. One issue which needs clarification is the more general implications which can be drawn from Eq.(28). Another obvious issue is the behaviour of the transmission *T* and reflection *R* coefficients when the deformation chain is applied to some known initial scattering process. It is known that *T* and *R* are invariant under the simple deformation in Eq.(15) [5]. However, it is unclear whether this property is also a property of the general deformation chain. It is also of great interest to study the relation between our chain construction and the conventional operator approaches. One possible strategy one might follow in order to cast some light on this issue is to study the relation between the deformation chain and the concept of intertwining operators. It is known that many exactly solvable potentials are related by intertwining operator transformations including the Darboux transformations which also appear in our context [6]. Clearly, key to our construction in this work is the non-linearity of the Riccati equation. One could contemplate studying the associated JET-space and its deformations. Furthermore, an analysis of the intertwining of the hierarchy of the JET-spaces associated

with the system of equations in Eq.(26) might also cast new light on our subject.

<sup>⋆</sup> Address all correspondence to: bjorn.jensen@hive.no

Vestfold University College, Norway

they turned out to be identical to the deformation scheme developed in this work.

**Figure 2.** Generic plots depicting *V*ˆ <sup>−</sup>(*x*), *V*ˆ <sup>−</sup>1(*x*) and *V*ˆ <sup>−</sup>2(*x*). The plots for *V*ˆ <sup>−</sup>2(*x*) show how drastic the nature of a potential might change as the values of the integration constants change.

The superpotential stemming from the third iteration with *W*ˆ <sup>20</sup>(*x*) in Eq.(64) as the starting point is given by

$$
\hat{\mathcal{W}}\_{30}(\mathbf{x}) = \hat{\mathcal{W}}\_{20}(\mathbf{x}) + \frac{\left(\frac{\mathbf{C}\_{32}\mathbf{x}^2\mathbf{e}^x}{\mathbf{C}\_{21}(\mathbf{x}^2 + 2\mathbf{x} + 2)^2 + \mathbf{C}\_{22}(\mathbf{x}^2 + 2\mathbf{x} + 2)\mathbf{e}^x}\right)}{\left(\mathbf{C}\_{31} + \frac{\mathbf{C}\_{32}}{\mathbf{C}\_{22}}\ln\left|\frac{\mathbf{C}\_{21}(\mathbf{x}^2 + 2\mathbf{x} + 2) + \mathbf{C}\_{22}\mathbf{e}^x}{\mathbf{x}^2 + 2\mathbf{x} + 2}\right|\right)}.\tag{66}
$$

ˆˆ ˆ ˆ

This superpotential introduces the possibility for a logarithmic singularity away from the origin when *C*22/*C*<sup>21</sup> < 0. We note that setting *C*<sup>31</sup> = 0 does not regenerate a previous potential as was possible at the previous iteration level when we correspondingly put *C*<sup>21</sup> = 0. From Eq.(66) we can deduce the physical potential *V*ˆ <sup>−</sup>3(*x*) at the third iteration level. We do not reproduce it here due to its complexity. Due to the complicated integrals appearing we are not able to provide the analytical expression for *W*ˆ <sup>40</sup>(*x*). We leave detailed studies of the Coulomb potential for the future.

### **4. Conclusion**

14 Quantum Mechanics



**Figure 2.** Generic plots depicting *V*ˆ

point is given by


V−(x)

might change as the values of the integration constants change.

*W*ˆ <sup>30</sup>(*x*) = *W*ˆ <sup>20</sup>(*x*) +

the Coulomb potential for the future.

0. From Eq.(66) we can deduce the physical potential *V*ˆ

<sup>−</sup>(*x*), *V*ˆ

<sup>−</sup>1(*x*) and *V*ˆ

 *C*<sup>31</sup> +

Vˆ−2(x) ; C = - C = +1

21 22

Vˆ−1(x)

0.5

1

1.5

123456 7

Vˆ−2(x)

; C = C = +1 21 22

ˆˆ ˆ ˆ

*<sup>C</sup>*21(*x*<sup>2</sup> + <sup>2</sup>*<sup>x</sup>* + <sup>2</sup>)<sup>2</sup> + *<sup>C</sup>*22(*x*<sup>2</sup> + <sup>2</sup>*<sup>x</sup>* + <sup>2</sup>)*ex*

*<sup>C</sup>*21(*x*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>) + *<sup>C</sup>*22*e<sup>x</sup> x*<sup>2</sup> + 2*x* + 2

<sup>−</sup>2(*x*) show how drastic the nature of a potential

. (66)

 

<sup>−</sup>3(*x*) at the third iteration level. We

<sup>−</sup>2(*x*). The plots for *V*ˆ

The superpotential stemming from the third iteration with *W*ˆ <sup>20</sup>(*x*) in Eq.(64) as the starting

*<sup>C</sup>*32*x*2*e<sup>x</sup>*

This superpotential introduces the possibility for a logarithmic singularity away from the origin when *C*22/*C*<sup>21</sup> < 0. We note that setting *C*<sup>31</sup> = 0 does not regenerate a previous potential as was possible at the previous iteration level when we correspondingly put *C*<sup>21</sup> =

do not reproduce it here due to its complexity. Due to the complicated integrals appearing we are not able to provide the analytical expression for *W*ˆ <sup>40</sup>(*x*). We leave detailed studies of

*C*<sup>32</sup> *C*<sup>22</sup> ln  In a previous paper we showed that isospectral deformations on the form Eq.(15) are contained in the space of deformations generated by isospectral deformations on the form Eq.(13). In this work we have shown that Eq.(13) can be considered as the initial, or base, deformation of a novel infinite recursive isospectral deformation chain. This thus answers to some extend the question by which we ended our previous paper [6]; how does the most general isospectral deformation of the kind considered there (Eq.(13) in this paper) look like. The results in this work do obviously only give a partial answer. We deduced in particular that a class of recursive deformations exists which is generated by the solutions of the *non-linear* differential equations in Eq.(28).

We briefly discussed various ways to construct alternative recursive deformation structures. We considered a linear product structure, polynomial deformations and completely generalized base deformations. They all either failed to provide a recursive structure or they turned out to be identical to the deformation scheme developed in this work.

We applied the linear recursive scheme to the Coulomb potential. We derived novel superpotentials which all per construction satisfy the isospectrality condition. It is straightforward, although very tedious, to check that the corresponding physical potentials *V*<sup>+</sup> at the various iteration levels all satisfy the isospectrality condition Eq.(26). This application did also demonstrate how easily novel isospectral deformations can be generated in this approach. It did also demonstrate an increased relative complexity of the generated potentials with the number of iterations, as one also naively would expect from the expression Eq.(33).

The results in this work is obviously only a starting point for further research. One issue which needs clarification is the more general implications which can be drawn from Eq.(28). Another obvious issue is the behaviour of the transmission *T* and reflection *R* coefficients when the deformation chain is applied to some known initial scattering process. It is known that *T* and *R* are invariant under the simple deformation in Eq.(15) [5]. However, it is unclear whether this property is also a property of the general deformation chain. It is also of great interest to study the relation between our chain construction and the conventional operator approaches. One possible strategy one might follow in order to cast some light on this issue is to study the relation between the deformation chain and the concept of intertwining operators. It is known that many exactly solvable potentials are related by intertwining operator transformations including the Darboux transformations which also appear in our context [6]. Clearly, key to our construction in this work is the non-linearity of the Riccati equation. One could contemplate studying the associated JET-space and its deformations. Furthermore, an analysis of the intertwining of the hierarchy of the JET-spaces associated with the system of equations in Eq.(26) might also cast new light on our subject.

### **Author details**

Bjørn Jensen

<sup>⋆</sup> Address all correspondence to: bjorn.jensen@hive.no

Vestfold University College, Norway

#### **References**

[1] E. Schrödinger, Proc. R. Irish Acad. **A46**, 9 (1940); **A46**, 183 (1940); A47, 53 (1941), L.Infeld and T.E. Hull, Rev. Mod. Phys. 23, 21 (1951) (see also the references in this paper).

**Chapter 31**

**Quantum Effects Through a Fractal Theory of Motion**

Scale Relativity Theory (SRT) affirms that the laws of physics apply in all reference systems, whatever its state of motion and its scale. In consequence, SRT imply [1-3] the followings:

**i.** Particle movement on continuous and non-differentiable curve (or almost nowhere

**ii.** Physical quantities will be expressed through fractal functions, namely through

Let *F* (*x*) be a fractal function in the interval *x* ∈ *a*, *b* and let the sequence of values for *x* be:

Let us now consider as a *ε*¯-scale approximation of the same function. Since *F* (*x*) is everywhere almost self-similar, if *ε* and *ε*¯ are sufficiently small, both approximations *F* (*x*, *ε*) and must lead to same results. By comparing the two cases, one notices that scale expansion is related

r

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Agop et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Agop et al.; licensee InTech. This is a paper 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.

eee

(1)

= = (2)

01 0 <sup>0</sup> <sup>0</sup> ,, , *a kn b x xx x x x kx x n x* = =+ =+ =+=

to the increase *dε* of *ε*, according to an increase *dε*¯ of *ε*¯. But, in this case we have:

e

*d d <sup>d</sup>* e e

e

differentiable), that is explicitly scale dependent and its length tends to infinity, when

functions that are dependent both on coordinate field and resolution scale. The invariance of the physical quantities in relation with the resolution scale generates special types of transformations, called resolution scale transformations. In what

M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa

Additional information is available at the end of the chapter

the scale interval tends to zero.

follows we will explain the above statement.

We can now say that *F* (*x*, *ε*) is a –scale approximation.

http://dx.doi.org/10.5772/54172

**1. Introduction**

