**Emergent un-Quantum Mechanics**

**Emergent un-Quantum Mechanics**

John P. Ralston Additional information is available at the end of the chapter

John P. Ralston

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

[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 Mechan‐

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

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

[13] Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kin‐

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

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

ematik und Mechanik. *Zeitschrift für Physik,* , 43(3-4), 172-198.

uary

436 Advances in Quantum Mechanics

ics and Relativity Theory.

*December 11, 1954*

323(6089), 595-598.

Additional information is available at the end of the chapter 10.5772/55954

http://dx.doi.org/10.5772/55954

### **1. Introduction**

There is great interest in "emergent" dynamical systems and the possibility of quantum mechanics as emergent phenomena. We engage the topic by making a sharp distinction between models of microphysics, and the so-called quantum framework. We find the models have all the information. Given that the framework of quantum theory is mathematically self-consistent we propose it should be viewed as an information management tool not derived from physics nor depending on physics. That encourages practical applications of quantum-style information management to near arbitrary data systems. As part of developing the physics, we show there is no intrinsic distinction between quantum dynamics and classical dynamics in its general form, and there is no observable function for the unit converter known as Planck's constant. The main accomplishment of quantum-style theory is a expanding the notion of probability. A map exists going from macroscopic information as "data" to quantum probability. The map allows a hidden variable description for quantum states, and broadens the scope of quantum information theory. Probabilities defined for mutually exclusive objects equal the classical ones, while probabilities of objects in more general equivalence classes yield the quantum values. Quantum probability is a remarkably efficient data processing device; the *Principle of Minimum Entropy* explains how it serves to construct order out of chaos.

### **2. Complexity and symmetry induce dynamics**

The framework of quantum mechanics is intricately structured and thought the perfection of fundamental theory. It predicts an absolute and unvarying law of time evolution. There is a tightly defined space of possible states, upon which strictly prescribed operators act to produce crisp possibilities for observables. There is an unprecedented universal rule for predicting probabilities of observations. The general predictions of the framework are incompatible with hidden variables defined by distributions, and have been confirmed by every experiment so far conducted.

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 Ralston; 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 Ralston; 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 Ralston, licensee InTech. This is an open access chapter distributed under the terms of the Creative

Meanwhile, the *particular realizations* of physical theories are widely believed to be *emergent.* That means they probably do not really represent fundamental physical law, but instead represent generic outputs of complicated systems not driven by the same laws. The Standard Model of particle physics is the most sophisticated prototype. It explains all data from all experiments done so far, except for a few outliers. Yet it is hard to find anyone involved that will argue the Standard Model is more than a generic derivative expansion embodying certain symmetries of some more complicated theory, of which numberless possibilities exist. Like Hooke's Law, Standard Model Laws are no longer imagined to be serious candidates for Laws, because they are so contrived and of the type that had to occur one way or the other. Nor does one really need the machinery of the Standard Model to understand most of the Universe. For most of what matters, the non-relativistic Schroedinger Equation is a "theory of everything"[1].

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"photon" of the 1920's thought to be so fundamental and ultimate are not fundamental objects. They are approximations that never stop interacting with an infinite ocean of quantum fields, if not something more unknown and more interesting. The self-consistency of the *framework* suggests that living in peaceful co-existence with what it must ignore may

Many workers seek to derive quantum mechanics. An active movement suspects or maintains that it is emergent, not fundamental[2–4]. We think it would be a waste to obtain exactly the quantum framework already known. Progress requires new features and new viewpoints. Progress usually involves dropping obsolete views and clearing out deadwood:

We start by considering what can and cannot be given up. Quantum mechanics wiped out the previous vision of a Newtonian universe made of point particles. It's gone forever. There remains some confusion and disagreement about whether the theory is about point particles of some subtle magic kind. While point particles are loosely cited in press releases, and inaccurately associated with theories based on *local* space-time interactions, we cannot find any evidence for them, and give that up. We believe all the rest of early quantum lore also can be given up, especially those parts leading to pedagogical confusion. We can't explain why it is unconventional or even scandalous to admit that the pre-history of quantum mechanics –meaning that period between 1900 and 1926 – was characterized by great theoretical advances that were all wrong or limiting in some way or other. Being

Many professional physicists are still influenced by the cult of the "quantum of action," forgetting it went away when quantum mechanics found action is not quantized. Physicists have been programmed from birth to hold Einstein's relations *<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>ω</sup>*, *<sup>p</sup>* <sup>=</sup> *<sup>h</sup>*¯*<sup>k</sup>* as highly fundamental. They actually know these relations are not universal, but derived facts coming from special cases, yet downgrading them is taboo. Physicists often believe that classical Newtonian mechanics shall be a starting point to be "quantized" to predict quantum systems, forgetting this recipe only predicts textbook problems for training purposes. It would forbid the Standard Model of physics to exist at all. Physicists are also trained that Feynman's path integral creates a quantum theory from a classical one, forgetting that what's integrated over nowadays has no relation to the starting point young Mr. Feynman used. When we un-Quantize this, an integral representation of correlation functions is a math tool, not an

We must honor our forebears and we do. Yet why are historical misdirections kept around with special emphasis in quantum physics? The culture of quantum mechanics almost seems to maintain mistakes of pre-quantum history on purpose. Instead of giving them up, misconceptions are kept around as philosophical quandaries and paradoxes because there is

Quantum mechanics is a misnomer held-over from the pre-quantum era. The Schroedinger equation explained how and when quantization of physical quantities occurs as an outcome

no other way to perpetuate mistakes except as paradoxes. *All that can be given up.*

wrong is normal in physics but covering it for generations is very strange.

independent principle, nor does it "come from physics."

*2.1.1. Quantum-Style Things Not to Give Up*

be its main accomplishments.

**2.1. What would emerge?**

call it *un-Quantization*.

Then it is very reasonable to expect that quantum mechanics itself should be a generic, self-defining "emergent" feature of the Universe. In simple terms, an output, not an input.

#### *2.0.1. Practical goals*

What does that mean? Discussions of emergent quantum mechanics tend to become confused by discord over what we mean by quantum mechanics. Different writers will disagree about what is fundamental and even about what experimental data says. Not far behind is a superstition that quantum objects cannot possibly be understood, so that making them even more obscure and more difficult might be the intellectual high ground. We reject that holdover from the 1920's, but it's not clear whether it has died out, or might be coming back

To skirt the morass we have a new point of view. We want physics to be practical and simple. In this century not many find the mathematics of quantum theory so intimidating. Perhaps physics "quantum-style" is not so profound after all. What is the evidence that the quantum *framework* itself is so meaningful and fundamental?

Most accomplishments of "quantum mechanics" come from the model details. For example, understanding the Hydrogen atom and calculating *g* − 2 of the electron are astounding achievements. However those accomplishments come directly from the model details, and systematic laborious tuning of theory to experimental facts, rather than from the framework of quantum mechanics itself. That is not always noticed while the framework lays claim for every accomplishment. The framework passes every test, especially when tested by thought-experiments set up for validation by pencil and typewriter challenges that recycle the framework. But the *framework* is rather hard to falsify experimentally. When it might have failed, a little ingenuity never fails to bring it back. That is quite unlike models challenging conventional theory, such as non-linear Schroedinger equations and so on, which have the decency to be able to fail.

Following this and other clues, we propose the quantum *framework* amounts to *descriptive* tools and *classification tools* that categorize data beautifully, but predict very little at all. When the "Laws" of quantum mechanics are considered as procedural and classification structures, it's much easier to guess how they would *emerge* as human-made bookkeeping.

Not everything in quantum mechanics is procedural, and some of its general workings contain clues to Nature. It is rarely noticed that quantum theory has infinities of hidden variables. They are not classical hidden variables of the usual kind. The "electron" and

"photon" of the 1920's thought to be so fundamental and ultimate are not fundamental objects. They are approximations that never stop interacting with an infinite ocean of quantum fields, if not something more unknown and more interesting. The self-consistency of the *framework* suggests that living in peaceful co-existence with what it must ignore may be its main accomplishments.

### **2.1. What would emerge?**

2 Advances in Quantum Mechanics

of everything"[1].

*2.0.1. Practical goals*

decency to be able to fail.

*framework* itself is so meaningful and fundamental?

Meanwhile, the *particular realizations* of physical theories are widely believed to be *emergent.* That means they probably do not really represent fundamental physical law, but instead represent generic outputs of complicated systems not driven by the same laws. The Standard Model of particle physics is the most sophisticated prototype. It explains all data from all experiments done so far, except for a few outliers. Yet it is hard to find anyone involved that will argue the Standard Model is more than a generic derivative expansion embodying certain symmetries of some more complicated theory, of which numberless possibilities exist. Like Hooke's Law, Standard Model Laws are no longer imagined to be serious candidates for Laws, because they are so contrived and of the type that had to occur one way or the other. Nor does one really need the machinery of the Standard Model to understand most of the Universe. For most of what matters, the non-relativistic Schroedinger Equation is a "theory

Then it is very reasonable to expect that quantum mechanics itself should be a generic, self-defining "emergent" feature of the Universe. In simple terms, an output, not an input.

What does that mean? Discussions of emergent quantum mechanics tend to become confused by discord over what we mean by quantum mechanics. Different writers will disagree about what is fundamental and even about what experimental data says. Not far behind is a superstition that quantum objects cannot possibly be understood, so that making them even more obscure and more difficult might be the intellectual high ground. We reject that holdover from the 1920's, but it's not clear whether it has died out, or might be coming back To skirt the morass we have a new point of view. We want physics to be practical and simple. In this century not many find the mathematics of quantum theory so intimidating. Perhaps physics "quantum-style" is not so profound after all. What is the evidence that the quantum

Most accomplishments of "quantum mechanics" come from the model details. For example, understanding the Hydrogen atom and calculating *g* − 2 of the electron are astounding achievements. However those accomplishments come directly from the model details, and systematic laborious tuning of theory to experimental facts, rather than from the framework of quantum mechanics itself. That is not always noticed while the framework lays claim for every accomplishment. The framework passes every test, especially when tested by thought-experiments set up for validation by pencil and typewriter challenges that recycle the framework. But the *framework* is rather hard to falsify experimentally. When it might have failed, a little ingenuity never fails to bring it back. That is quite unlike models challenging conventional theory, such as non-linear Schroedinger equations and so on, which have the

Following this and other clues, we propose the quantum *framework* amounts to *descriptive* tools and *classification tools* that categorize data beautifully, but predict very little at all. When the "Laws" of quantum mechanics are considered as procedural and classification structures,

Not everything in quantum mechanics is procedural, and some of its general workings contain clues to Nature. It is rarely noticed that quantum theory has infinities of hidden variables. They are not classical hidden variables of the usual kind. The "electron" and

it's much easier to guess how they would *emerge* as human-made bookkeeping.

Many workers seek to derive quantum mechanics. An active movement suspects or maintains that it is emergent, not fundamental[2–4]. We think it would be a waste to obtain exactly the quantum framework already known. Progress requires new features and new viewpoints. Progress usually involves dropping obsolete views and clearing out deadwood: call it *un-Quantization*.

We start by considering what can and cannot be given up. Quantum mechanics wiped out the previous vision of a Newtonian universe made of point particles. It's gone forever. There remains some confusion and disagreement about whether the theory is about point particles of some subtle magic kind. While point particles are loosely cited in press releases, and inaccurately associated with theories based on *local* space-time interactions, we cannot find any evidence for them, and give that up. We believe all the rest of early quantum lore also can be given up, especially those parts leading to pedagogical confusion. We can't explain why it is unconventional or even scandalous to admit that the pre-history of quantum mechanics –meaning that period between 1900 and 1926 – was characterized by great theoretical advances that were all wrong or limiting in some way or other. Being wrong is normal in physics but covering it for generations is very strange.

Many professional physicists are still influenced by the cult of the "quantum of action," forgetting it went away when quantum mechanics found action is not quantized. Physicists have been programmed from birth to hold Einstein's relations *<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>ω</sup>*, *<sup>p</sup>* <sup>=</sup> *<sup>h</sup>*¯*<sup>k</sup>* as highly fundamental. They actually know these relations are not universal, but derived facts coming from special cases, yet downgrading them is taboo. Physicists often believe that classical Newtonian mechanics shall be a starting point to be "quantized" to predict quantum systems, forgetting this recipe only predicts textbook problems for training purposes. It would forbid the Standard Model of physics to exist at all. Physicists are also trained that Feynman's path integral creates a quantum theory from a classical one, forgetting that what's integrated over nowadays has no relation to the starting point young Mr. Feynman used. When we un-Quantize this, an integral representation of correlation functions is a math tool, not an independent principle, nor does it "come from physics."

We must honor our forebears and we do. Yet why are historical misdirections kept around with special emphasis in quantum physics? The culture of quantum mechanics almost seems to maintain mistakes of pre-quantum history on purpose. Instead of giving them up, misconceptions are kept around as philosophical quandaries and paradoxes because there is no other way to perpetuate mistakes except as paradoxes. *All that can be given up.*

#### *2.1.1. Quantum-Style Things Not to Give Up*

Quantum mechanics is a misnomer held-over from the pre-quantum era. The Schroedinger equation explained how and when quantization of physical quantities occurs as an outcome of *dynamics*, and quantization is not the primary new feature of the theory. We have coined the term "quantum-style" to describe things done in the organizational style of quantum mechanics that we don't want to give up.

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of freedom, equilibrium thermodynamics notwithstanding. Whether or not microphysics had any new and spooky elements, we find that quantum-style methods would need to be invented to handle the complexity. It is very efficient: And then the classification system and information processing power of quantum data organization should be used wherever a

Towards that end, we have developed an approach which largely avoids the historical path. That path and its traditional presentation interweaves a little dynamics, measurement theory, microphysical facts, experimental claims and the prehistory of failed theories in an alternating web. It is designed not to be challengeable, which is a cheat. To make progress we must change something. We first separate the dynamics, meaning rules of time evolution, from the rest, and identify it as being trivial. This is developed in Section 3. For us the ordinary form of quantum dynamics is a "toy model", which at first seems too simple to be realistic. This is not to say that the *models* describing microphysics are trivial or easy to solve. Actually the models are the real discovery, while claims that the dynamical framework is the

The great roadblock to using the organizational scheme of quantum theory more generally is Planck's constant. Once the claim is made that Nature partitions itself into little cells determined by ¯*h*, one cannot do without ¯*h*, nor use quantum theory for anything else but issues of ¯*h*. But the claim is wrong. In Section 4 we discuss a quantum-style Universe without ¯*h*. We claim it is the ordinary Universe, but if that is too provocative, the step of

We also dispense with needlessly obscure definitions of "observables." We define observables (Section 5) as maps from the system coordinates to numbers. This is plain and unpretentious. Satisfying things, including relations of Poisson Brackets usually postulated as independent, can actually be derived using symmetry. In principle the map between system and observables is invertible: the system coordinates (wave function, density matrix) are observable. It is a non-trivial fact that real physical systems seem to have infinite complexity,

In the end the new thing that came out of quantum mechanics are new definitions of "probability", Section 6. The working of quantum probability has always been explained by physicists using a self-validating logic that "it works" because "features of microphysical object make it work". (And this is very mysterious and too profound for humans to grasp,

Instead of buying it, we seek a rational explanation why certain mathematical tools work *sometimes* and other rules work *other times* without depending on circular postulates. In our approach the information management of quantum theory is a topic of mathematical classification, and for that reason mostly devoid of physical content. Since it is mathematics, we can derive the quantum rules of probability as a bookkeeping system that does not need any special features from the objects they describe. And we do this to increase the scope and

making exhaustive measurements a bit out of reach, but this "bug" is a "feature."

etc.) Every time that thinking style is used we find it unprogressive and circular.

*utility* of the rules so we can use them in new applications.

never introducing ¯*h* is a part of *our approach* where it never appears.

useful result might come out.

real discovery have things reversed.

*2.1.2. How to Proceed*

Quantum-style mechanics describes certain data of the Universe, and the ability to describe experimental data absolutely cannot be given up. But that does not mean in some future time we would interpret success the same way. A certain vagueness of description is probably tied to success. For example, it is rarely noticed that the dramatic demonstrations of beginning quantum probability, such as the Bell inequalities and EPR "paradoxes," are realized experimentally only by virtuoso fiddling and selection of systems nice enough to make them work out. When an experiment produces nothing but the mundane predictions of ordinary classical probability, nobody notices that was also a retrospective quantum prediction of certain dirty density matrices, protecting the framework from falsification. But we have noticed, and it suggests that quantum mechanics must be a framework that is so broad and flexible it does not restrict much. For example, if a future civilization discovered a true and ultimate Newtonian particle, contradicting everything now believed, any competent theorist would find no great difficulty describing it with an appropriately constructed density matrix.

All of this suggests what a new vision of "quantum theory" should include. *The restriction of the subject to describing micro-physical objects of fundamental physical character is obsolete.* It seems the same mistake as thinking complex numbers have no practical use because they have an "imaginary" part. We believe the framework of quantum theory is mathematically self-consistent. And when there are self-consistent structures, they ought to have more uses than microphysics, and not depend on microphysics for their uses. Even more directly, asking a self-consistent framework to "emerge" means mainly to start using it without fear it could go wrong.

This idea appears radical because it contradicts a few existing ideas. Physicists are educated in the magical antics of quantum objects and convinced they are impossible to understand before they begin serious coursework. As a result, misidentifying mathematical relations as inexplicable Facts of Nature is very common. That is fatal to physics, because Facts and Behaviors of a special Universe are not supposed to work on other Universes. That is why it is generally considered stupid to apply quantum-style methods to non-quantumy objects.

The degree of stupidity depends strongly one's Bayesian priors. 100,000 years ago humans would know the concepts of integers but probably be unable to separate them from empirical facts of Nature. It would be considered absurd and dangerous to imagine integers disassociated from their experimental realization in the number of rocks or rabbits. Yet those humans were not stupid. They lacked the background about the Universe to decide whether integers came from physics or came from human thinking, with physics probably having the edge because it was real. In much the same way the occult mysteries of ¯*h*, *<sup>i</sup>* <sup>=</sup> √−1, and electrons "in all states at once" conditioned the physics community to think "quantum methods are for quantum objects. "

Anyone thoughtful ought to notice that is circular, and ought to be open to using mathematical structures more liberally. But thoughtful people are given false information by the physicists about what's established, and what can be contradicted, and even about what experiments find. We feel it is significant that the early 20th century was the first time physics needed to seriously deal with the details of experiments involving a large number of degrees

of freedom, equilibrium thermodynamics notwithstanding. Whether or not microphysics had any new and spooky elements, we find that quantum-style methods would need to be invented to handle the complexity. It is very efficient: And then the classification system and information processing power of quantum data organization should be used wherever a useful result might come out.

#### *2.1.2. How to Proceed*

4 Advances in Quantum Mechanics

matrix.

it could go wrong.

methods are for quantum objects. "

mechanics that we don't want to give up.

of *dynamics*, and quantization is not the primary new feature of the theory. We have coined the term "quantum-style" to describe things done in the organizational style of quantum

Quantum-style mechanics describes certain data of the Universe, and the ability to describe experimental data absolutely cannot be given up. But that does not mean in some future time we would interpret success the same way. A certain vagueness of description is probably tied to success. For example, it is rarely noticed that the dramatic demonstrations of beginning quantum probability, such as the Bell inequalities and EPR "paradoxes," are realized experimentally only by virtuoso fiddling and selection of systems nice enough to make them work out. When an experiment produces nothing but the mundane predictions of ordinary classical probability, nobody notices that was also a retrospective quantum prediction of certain dirty density matrices, protecting the framework from falsification. But we have noticed, and it suggests that quantum mechanics must be a framework that is so broad and flexible it does not restrict much. For example, if a future civilization discovered a true and ultimate Newtonian particle, contradicting everything now believed, any competent theorist would find no great difficulty describing it with an appropriately constructed density

All of this suggests what a new vision of "quantum theory" should include. *The restriction of the subject to describing micro-physical objects of fundamental physical character is obsolete.* It seems the same mistake as thinking complex numbers have no practical use because they have an "imaginary" part. We believe the framework of quantum theory is mathematically self-consistent. And when there are self-consistent structures, they ought to have more uses than microphysics, and not depend on microphysics for their uses. Even more directly, asking a self-consistent framework to "emerge" means mainly to start using it without fear

This idea appears radical because it contradicts a few existing ideas. Physicists are educated in the magical antics of quantum objects and convinced they are impossible to understand before they begin serious coursework. As a result, misidentifying mathematical relations as inexplicable Facts of Nature is very common. That is fatal to physics, because Facts and Behaviors of a special Universe are not supposed to work on other Universes. That is why it is generally considered stupid to apply quantum-style methods to non-quantumy objects. The degree of stupidity depends strongly one's Bayesian priors. 100,000 years ago humans would know the concepts of integers but probably be unable to separate them from empirical facts of Nature. It would be considered absurd and dangerous to imagine integers disassociated from their experimental realization in the number of rocks or rabbits. Yet those humans were not stupid. They lacked the background about the Universe to decide whether integers came from physics or came from human thinking, with physics probably having the edge because it was real. In much the same way the occult mysteries of ¯*h*, *<sup>i</sup>* <sup>=</sup> √−1, and electrons "in all states at once" conditioned the physics community to think "quantum

Anyone thoughtful ought to notice that is circular, and ought to be open to using mathematical structures more liberally. But thoughtful people are given false information by the physicists about what's established, and what can be contradicted, and even about what experiments find. We feel it is significant that the early 20th century was the first time physics needed to seriously deal with the details of experiments involving a large number of degrees Towards that end, we have developed an approach which largely avoids the historical path. That path and its traditional presentation interweaves a little dynamics, measurement theory, microphysical facts, experimental claims and the prehistory of failed theories in an alternating web. It is designed not to be challengeable, which is a cheat. To make progress we must change something. We first separate the dynamics, meaning rules of time evolution, from the rest, and identify it as being trivial. This is developed in Section 3. For us the ordinary form of quantum dynamics is a "toy model", which at first seems too simple to be realistic. This is not to say that the *models* describing microphysics are trivial or easy to solve. Actually the models are the real discovery, while claims that the dynamical framework is the real discovery have things reversed.

The great roadblock to using the organizational scheme of quantum theory more generally is Planck's constant. Once the claim is made that Nature partitions itself into little cells determined by ¯*h*, one cannot do without ¯*h*, nor use quantum theory for anything else but issues of ¯*h*. But the claim is wrong. In Section 4 we discuss a quantum-style Universe without ¯*h*. We claim it is the ordinary Universe, but if that is too provocative, the step of never introducing ¯*h* is a part of *our approach* where it never appears.

We also dispense with needlessly obscure definitions of "observables." We define observables (Section 5) as maps from the system coordinates to numbers. This is plain and unpretentious. Satisfying things, including relations of Poisson Brackets usually postulated as independent, can actually be derived using symmetry. In principle the map between system and observables is invertible: the system coordinates (wave function, density matrix) are observable. It is a non-trivial fact that real physical systems seem to have infinite complexity, making exhaustive measurements a bit out of reach, but this "bug" is a "feature."

In the end the new thing that came out of quantum mechanics are new definitions of "probability", Section 6. The working of quantum probability has always been explained by physicists using a self-validating logic that "it works" because "features of microphysical object make it work". (And this is very mysterious and too profound for humans to grasp, etc.) Every time that thinking style is used we find it unprogressive and circular.

Instead of buying it, we seek a rational explanation why certain mathematical tools work *sometimes* and other rules work *other times* without depending on circular postulates. In our approach the information management of quantum theory is a topic of mathematical classification, and for that reason mostly devoid of physical content. Since it is mathematics, we can derive the quantum rules of probability as a bookkeeping system that does not need any special features from the objects they describe. And we do this to increase the scope and *utility* of the rules so we can use them in new applications.

### *2.1.3. Question From the Bottom Up*

It is not always helpful to put the framework of quantum mechanics on a high pedestal. It is sometimes assumed that quantum mechanics might only be "explained" by progress at the far edge of the research frontier involving quantum gravity, foamy space-time, strings, and so on[5]. But if true that would put our topic among those not seeking to deal with what is observable and testable. We are only interested in what is observable and testable.

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We assume the reader knows how to get equations by varying an action *S*, expressed using

Symbols *qi* are generalized coordinates, namely numbers describing a system, labeled by *i* = 1...*N*, and the dot indicates a time derivative. We are not going to suggest that the "action principle" will be our foundation postulate. *When and if* an emergent quantum system has sufficiently nice dynamics that it gets noticed as an experimental regularity, the action is a

By familiar steps, finding the extrema of the action produces Lagrange's equations. Define

*<sup>H</sup>*(*qi*, *pi*) = *piq*˙*<sup>i</sup>* − *<sup>L</sup>*(*qi <sup>q</sup>*˙*i*).

Repeated indices are summed. When these transformations can be done, then Lagrange's

We will pause at this point to repeat that (*qi*, *pi*) are real-valued *numbers*, that everything above was known by (say) 1850, and that our subject is nevertheless quantum dynamics. We will never confuse a classical coordinate with an operator, we will never use the abusive term "quantum particle" except to reject it, and when an operator is intended it will be indicated

The thing for our discussion not known in 1850 lies in the number of dynamical degrees of freedom (*do f*) we intend to use. A Newtonian particle has three *do f* usually taken to be its Cartesian coordinates. The Newtonian particle is not a valid prototype and (unlike the early history) we base nothing on making contact with it. In our approach we have no advance information on the number of *do f* describing a system, because that is an arbitrary defining feature of a system. For *N do f* the phase-space of (*qi*, *pi*) is 2*N* dimensional. We also pretend to no advance information on the Hamiltonian, although some properties will be specified to make contact with existing models. We claim this freedom not to commit is a defining fact

Now proceed: Hamilton's equations are invariant under symplectic (*Sp*) transformations. It is usually developed by combining ( *qi*, *pi* ) into a 2*N* dimensional multiplet Φ =

*∂*H

Φ˙ = *J*

; *<sup>p</sup>*˙*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>H</sup>*

*∂qi*

. (1)

*<sup>∂</sup>*<sup>Φ</sup> . (2)

*<sup>q</sup>*˙*<sup>i</sup>* <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂pi*

of basic quantum mechanics: but if that is not agreed, it is a fact of *our theory.*

( *q*1...*qN*, *p*1...*pN* ). Hamilton's equations become

*dt L*(*qi q*˙*i*).

*δS* = *δ* 

a Lagrangian *L*(*qi q*˙*i*):

the Hamiltonian *H* by

by a ˆ *hat*.

fine invariant notation to express it.

equations are equivalent to Hamilton's:

Progress needs to come from revising the bottom. Successful work at the bottom revises basic notions that are actually harder to challenge than advanced work, because the whole system rests on the base The mathematics of our discussion is not advanced. It is little more than linear algebra, and deliberate choices not to use mathematics that is more advanced than needs to be used.

There is a very elementary point often overlooked. Mathematical subjects can be reduced to a definite minimal number of axioms, which might be swapped around, but not decreased. Early on quantum theory looked ripe for axiomization, and it tends to be accepted today. Yet every effort to make physics into axioms fails because we don't know what the Universe is. However physics can *often* reduce the number of postulates, axioms or guesses by swapping around the order, which actually changes their meaning and power. The everyday assumption that this was optimized long ago is not true. Thus, it is a form of progress to explore how *post-quantum* physics-axioms can be eliminated by re-ordering and re-interpreting the logic. The process will help quantum mechanics "emerge" more clearly from its own tangles.

### **3. Dynamics**

#### *3.0.4. Where to Start?*

By very curious structuring, the usual approach to quantum mechanics starts with the doctrine of measurement postulates. What is out of order in those approaches is failing to first define the system and its dynamics. For example the Stern-Gerlach experiment is traditionally developed as a raw mystery of 1922 involving point-like electrons and two spots, then requiring a new principle[6]. If one were given from the start that a two-component wave function was involved the separation into two spots does not really need any new principles. It was known to Fresnel from calcite crystals and explained without requiring a new principle. And if one knew the particular two-component wave functions of electrons were expected from representations of the rotation group none of it would be a terrible surprise. While ordering things to make physics more mysterious and inexplicable was an early promotional tool, we lack any interest in it. That is why we will start with the dynamics, because it can be explained. We will discuss how there is nothing new contributed by quantum mechanics to its own framework of dynamics: *at least in our approach.*

#### **3.1. Hamilton's equations in Schroedinger's notation**

Physics predicts little more than evolution of systems with time, symbol *t*. By the end of this work we will argue the predictions (above and beyond the empirically-found model details!) originate in symmetry.

We assume the reader knows how to get equations by varying an action *S*, expressed using a Lagrangian *L*(*qi q*˙*i*):

6 Advances in Quantum Mechanics

than needs to be used.

from its own tangles.

*3.0.4. Where to Start?*

originate in symmetry.

**3. Dynamics**

*2.1.3. Question From the Bottom Up*

It is not always helpful to put the framework of quantum mechanics on a high pedestal. It is sometimes assumed that quantum mechanics might only be "explained" by progress at the far edge of the research frontier involving quantum gravity, foamy space-time, strings, and so on[5]. But if true that would put our topic among those not seeking to deal with what is

Progress needs to come from revising the bottom. Successful work at the bottom revises basic notions that are actually harder to challenge than advanced work, because the whole system rests on the base The mathematics of our discussion is not advanced. It is little more than linear algebra, and deliberate choices not to use mathematics that is more advanced

There is a very elementary point often overlooked. Mathematical subjects can be reduced to a definite minimal number of axioms, which might be swapped around, but not decreased. Early on quantum theory looked ripe for axiomization, and it tends to be accepted today. Yet every effort to make physics into axioms fails because we don't know what the Universe is. However physics can *often* reduce the number of postulates, axioms or guesses by swapping around the order, which actually changes their meaning and power. The everyday assumption that this was optimized long ago is not true. Thus, it is a form of progress to explore how *post-quantum* physics-axioms can be eliminated by re-ordering and re-interpreting the logic. The process will help quantum mechanics "emerge" more clearly

By very curious structuring, the usual approach to quantum mechanics starts with the doctrine of measurement postulates. What is out of order in those approaches is failing to first define the system and its dynamics. For example the Stern-Gerlach experiment is traditionally developed as a raw mystery of 1922 involving point-like electrons and two spots, then requiring a new principle[6]. If one were given from the start that a two-component wave function was involved the separation into two spots does not really need any new principles. It was known to Fresnel from calcite crystals and explained without requiring a new principle. And if one knew the particular two-component wave functions of electrons were expected from representations of the rotation group none of it would be a terrible surprise. While ordering things to make physics more mysterious and inexplicable was an early promotional tool, we lack any interest in it. That is why we will start with the dynamics, because it can be explained. We will discuss how there is nothing new contributed

by quantum mechanics to its own framework of dynamics: *at least in our approach.*

Physics predicts little more than evolution of systems with time, symbol *t*. By the end of this work we will argue the predictions (above and beyond the empirically-found model details!)

**3.1. Hamilton's equations in Schroedinger's notation**

observable and testable. We are only interested in what is observable and testable.

$$
\delta S = \delta \int dt \, L(q\_i \, \dot{q}\_i).
$$

Symbols *qi* are generalized coordinates, namely numbers describing a system, labeled by *i* = 1...*N*, and the dot indicates a time derivative. We are not going to suggest that the "action principle" will be our foundation postulate. *When and if* an emergent quantum system has sufficiently nice dynamics that it gets noticed as an experimental regularity, the action is a fine invariant notation to express it.

By familiar steps, finding the extrema of the action produces Lagrange's equations. Define the Hamiltonian *H* by

$$H(q\_{i\nu} \ p\_i) = p\_i \dot{q}\_i - L(q\_i \dot{q}\_i).$$

Repeated indices are summed. When these transformations can be done, then Lagrange's equations are equivalent to Hamilton's:

$$\dot{\eta}\_{i} = \frac{\partial H}{\partial p\_{i}}; \quad \dot{p}\_{i} = -\frac{\partial H}{\partial q\_{i}}.\tag{1}$$

We will pause at this point to repeat that (*qi*, *pi*) are real-valued *numbers*, that everything above was known by (say) 1850, and that our subject is nevertheless quantum dynamics. We will never confuse a classical coordinate with an operator, we will never use the abusive term "quantum particle" except to reject it, and when an operator is intended it will be indicated by a ˆ *hat*.

The thing for our discussion not known in 1850 lies in the number of dynamical degrees of freedom (*do f*) we intend to use. A Newtonian particle has three *do f* usually taken to be its Cartesian coordinates. The Newtonian particle is not a valid prototype and (unlike the early history) we base nothing on making contact with it. In our approach we have no advance information on the number of *do f* describing a system, because that is an arbitrary defining feature of a system. For *N do f* the phase-space of (*qi*, *pi*) is 2*N* dimensional. We also pretend to no advance information on the Hamiltonian, although some properties will be specified to make contact with existing models. We claim this freedom not to commit is a defining fact of basic quantum mechanics: but if that is not agreed, it is a fact of *our theory.*

Now proceed: Hamilton's equations are invariant under symplectic (*Sp*) transformations. It is usually developed by combining ( *qi*, *pi* ) into a 2*N* dimensional multiplet Φ = ( *q*1...*qN*, *p*1...*pN* ). Hamilton's equations become

$$
\dot{\Phi} = J \frac{\partial \mathcal{H}}{\partial \Phi}.\tag{2}
$$

Here *J* is a matrix with block representation

$$J = \begin{pmatrix} 0 & \mathbf{1}\_{N \times N} \\ -\mathbf{1}\_{N \times N} & 0 \end{pmatrix}. \tag{3}$$

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(11)

= <sup>Γ</sup>*ij* − <sup>Γ</sup>*ji*. (10)

The Hamiltonian of a classical, 3-dimensional, non-relativistic particle in an external electromagnetic field is *Hem* = (*<sup>p</sup>* <sup>−</sup> *eA* (*<sup>q</sup>*))2/2*<sup>m</sup>* <sup>+</sup> *<sup>V</sup>*(*<sup>q</sup>*), where *<sup>A</sup>* is the vector potential. Except for allowing a tensor mass, a quantum system with three *do f* (spin-1, say) is dynamically indistinguishable. For more *do f* the symbol A(q) continues to serves as a vector

> <sup>−</sup> *<sup>∂</sup>*A*<sup>j</sup> ∂qi*

These theories have gauge symmetries. From Eq. 10 any symmetric part of Γ drops out of F*ij* and the equations of motion. That is equivalent to gauge transformation A(*q*) → A(*q*) + Σ*q*, where Σ = Σ*T*. As a rule gauge symmetries indicate a system that is being described with more coordinates than are truly dynamical: the redundant coordinates may

Go to coordinates where the symplectic metric *J* is diagonal. Since *J* is antisymmetric, the

<sup>F</sup>*ij* <sup>=</sup> *<sup>∂</sup>*A*<sup>i</sup> ∂qj*

be hard to eradicate, and easier to treat as "symmetries." That will be a clue.

potential with an associated curvature

transformation goes from real to complex numbers:

Φ =

*q*1 *q*<sup>2</sup> ... *p*1 *p*2 ...

→ Ψ =

� 1*N*×*<sup>N</sup> i* 1*N*×*<sup>N</sup>* <sup>1</sup>*N*×*<sup>N</sup>* −*<sup>i</sup>* <sup>1</sup>*N*×*<sup>N</sup>*

� *i* 1*N*×*<sup>N</sup>* 0

<sup>0</sup> −*<sup>i</sup>* <sup>1</sup>*N*×*N*.

*ψ*1 *ψ*2 ... *ψ*∗ 1 *ψ*∗ 2 ...

� ,

UU† = 12*N*×2*N*, (12)

. (13)

�

Ψ = UΦ;

<sup>U</sup> <sup>=</sup> <sup>1</sup> √2

U *J*U† = −

*3.1.2. Diagonal Frame*

An appropriate map is

with

and then

Under a real-valued 2*N* × 2*N* transformation *S*, the equation transforms

$$
\Phi \to \Phi\_S = S\Phi;$$

$$
\Phi\_S = S \cdot J \cdot \mathbf{S}^T \frac{\partial \mathcal{H}}{\partial \Phi}.\tag{4}$$

Super-*T* denotes the *transpose*. The *symplectic group* of 2*N* dimensions is the set of transformations such that

$$\mathbf{S} \cdot \mathbf{J} \cdot \mathbf{S}^T = \mathbf{J}.\tag{5}$$

It can be shown the determinant *det*(*S*) = 1. Transformations satisfying Eq. 5 with determinant -1 will be called *Sp* − *parity changing*.

#### *3.1.1. Simplistic Linear Theories Are Not Our Burden To Defend*

By a linear theory we mean that the Hamiltonian *H* is bilinear in *qi*, *pi*:

$$\mathcal{H}(q,p) = \frac{1}{2} \Phi^T H\_{\Phi} \Phi;$$

$$H\_{\Phi} = \frac{1}{2} \begin{pmatrix} h\_{qq} & h\_{qp} \\ h\_{qp}^T & h\_{pp} \end{pmatrix} \tag{6}$$

Matrix multiplication is implied, and *hqq*, *hqp*...*etc* are *N* × *N* arrays of constant parameters. We are not writing linear terms like *αq* + *βp*, which can be removed by translating coordinates. We have no commitment here to the bilinear form, which is presented to make contact with ordinary quantum mechanics.

The most general such theory has a familiar form, seen by writing1

$$h\_{qq} = \text{K}; \quad h\_{pp} = M^{-1} \quad h\_{qp} = -\Gamma^T M. \tag{7}$$

Complete the square:

$$\mathcal{H}(q, p) = \frac{1}{2}p\boldsymbol{M}^{-1}\boldsymbol{p} + \frac{1}{2}q\boldsymbol{K}\boldsymbol{q} + q\boldsymbol{M}\boldsymbol{\Gamma}\boldsymbol{p} + p\boldsymbol{\Gamma}^T\boldsymbol{M}\boldsymbol{q}.\tag{8}$$

$$=\frac{1}{2}(p-\mathcal{A}(q))M^{-1}(p-\mathcal{A}(q))+\mathcal{V};\tag{9}$$

$$where \quad \mathcal{A}(q) = \Gamma q; \quad \mathcal{V} = \frac{1}{2}q(\mathcal{K} - \Gamma^T M^{-1} \Gamma)q.$$

<sup>1</sup> There's no loss of generality using these symbols, as *M*−<sup>1</sup> is meant to be the inverse on the space *M* does not send to zero, i.e. the pseudoinverse.

The Hamiltonian of a classical, 3-dimensional, non-relativistic particle in an external electromagnetic field is *Hem* = (*<sup>p</sup>* <sup>−</sup> *eA* (*<sup>q</sup>*))2/2*<sup>m</sup>* <sup>+</sup> *<sup>V</sup>*(*<sup>q</sup>*), where *<sup>A</sup>* is the vector potential. Except for allowing a tensor mass, a quantum system with three *do f* (spin-1, say) is dynamically indistinguishable. For more *do f* the symbol A(q) continues to serves as a vector potential with an associated curvature

$$\mathcal{F}\_{ij} = \frac{\partial \mathcal{A}\_i}{\partial q\_j} - \frac{\partial \mathcal{A}\_j}{\partial q\_i} = \Gamma\_{ij} - \Gamma\_{ji}. \tag{10}$$

These theories have gauge symmetries. From Eq. 10 any symmetric part of Γ drops out of F*ij* and the equations of motion. That is equivalent to gauge transformation A(*q*) → A(*q*) + Σ*q*, where Σ = Σ*T*. As a rule gauge symmetries indicate a system that is being described with more coordinates than are truly dynamical: the redundant coordinates may be hard to eradicate, and easier to treat as "symmetries." That will be a clue.

#### *3.1.2. Diagonal Frame*

8 Advances in Quantum Mechanics

transformations such that

Here *J* is a matrix with block representation

determinant -1 will be called *Sp* − *parity changing*.

contact with ordinary quantum mechanics.

Complete the square:

to zero, i.e. the pseudoinverse.

*3.1.1. Simplistic Linear Theories Are Not Our Burden To Defend*

By a linear theory we mean that the Hamiltonian *H* is bilinear in *qi*, *pi*:

The most general such theory has a familiar form, seen by writing1

<sup>2</sup> *pM*−<sup>1</sup> *<sup>p</sup>* <sup>+</sup>

*where* <sup>A</sup>(*q*) = <sup>Γ</sup>*q*; <sup>V</sup> <sup>=</sup> <sup>1</sup>

<sup>=</sup> <sup>1</sup> 2

<sup>H</sup>(*q*, *<sup>p</sup>*) = <sup>1</sup>

<sup>H</sup>(*q*, *<sup>p</sup>*) = <sup>1</sup>

*<sup>H</sup>*<sup>Φ</sup> <sup>=</sup> <sup>1</sup> 2

*J* =

Under a real-valued 2*N* × 2*N* transformation *S*, the equation transforms

 0 1*N*×*<sup>N</sup>* −1*N*×*<sup>N</sup>* <sup>0</sup>

<sup>Φ</sup> → <sup>Φ</sup>*<sup>S</sup>* = *<sup>S</sup>*Φ;

<sup>Φ</sup>˙ *<sup>S</sup>* <sup>=</sup> *<sup>S</sup>* · *<sup>J</sup>* · *<sup>S</sup><sup>T</sup> <sup>∂</sup>*<sup>H</sup>

Super-*T* denotes the *transpose*. The *symplectic group* of 2*N* dimensions is the set of

It can be shown the determinant *det*(*S*) = 1. Transformations satisfying Eq. 5 with

2

Matrix multiplication is implied, and *hqq*, *hqp*...*etc* are *N* × *N* arrays of constant parameters. We are not writing linear terms like *αq* + *βp*, which can be removed by translating coordinates. We have no commitment here to the bilinear form, which is presented to make

> 1 2

<sup>1</sup> There's no loss of generality using these symbols, as *M*−<sup>1</sup> is meant to be the inverse on the space *M* does not send

2

Φ*TH*ΦΦ;

*hqq* <sup>=</sup> *<sup>K</sup>*; *hpp* <sup>=</sup> *<sup>M</sup>*−<sup>1</sup> *hqp* <sup>=</sup> <sup>−</sup>Γ*TM*. (7)

*qKq* + *qM*Γ*p* + *p*Γ*TMq*, (8)

(*<sup>p</sup>* − A(*q*))*M*−1(*<sup>p</sup>* − A(*q*)) + <sup>V</sup>; (9)

*<sup>q</sup>*(*<sup>K</sup>* <sup>−</sup> <sup>Γ</sup>*TM*<sup>−</sup>1Γ)*q*.

 *hqq hqp hT qp hpp* . (3)

*<sup>∂</sup>*<sup>Φ</sup> . (4)

(6)

*S* · *J* · *S<sup>T</sup>* = *J*. (5)

Go to coordinates where the symplectic metric *J* is diagonal. Since *J* is antisymmetric, the transformation goes from real to complex numbers:

$$
\Phi = \begin{pmatrix} q\_1 \\ q\_2 \\ p\_1 \\ p\_2 \\ \dots \end{pmatrix} \to \Psi = \begin{pmatrix} \psi\_1 \\ \psi\_2 \\ \dots \\ \psi\_1^\* \\ \psi\_2^\* \\ \dots \end{pmatrix} \tag{11}
$$

An appropriate map is

$$\begin{aligned} \Psi &= \mathcal{U}\Phi; \\ \mathcal{U} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1\_{N \times N} & i \mathbf{1}\_{N \times N} \\ 1\_{N \times N} & -i \mathbf{1}\_{N \times N} \end{pmatrix} \end{aligned}$$

with

$$\mathcal{U}\mathcal{U}^{\dagger} = \mathbf{1}\_{2N \times 2N\prime} \tag{12}$$

and then

$$
\mathcal{U}\mathcal{U}^{\dagger} = -\begin{pmatrix} i\mathbf{1}\_{N \times N} & \mathbf{0} \\ \mathbf{0} & -i\mathbf{1}\_{N \times N \cdot \mathbf{}} \end{pmatrix}.\tag{13}
$$

The transformation produces a remarkable simplification of linear dynamical systems. Hamilton's first equation (Eq. 2) become

$$i\frac{\partial\Psi}{\partial t} = \frac{\partial H(\Psi, \Psi^\*)}{\partial \Psi^\*}.\tag{14}$$

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, (20)

Ωˆ *ψ<sup>n</sup>* = *ωnψn*. (21)

*3.1.3. Conventionally Assumed Properties*

its eigenvalues will not change.

Schroedinger equation.

frequently asked questions:

*3.1.4. Discussion*

Eq. 19 is an expansion in normal modes,

where *ψn* are solutions to the eigenvalue problem

appears as *<sup>e</sup>*−*iEnt*/¯*<sup>h</sup>* ≡ *<sup>e</sup>*−*iωnt* is a complete waste of time.

Hermitian Ωˆ , which a short exercise shows is equivalent to Eq. 18.

in matrix elements <sup>Ω</sup><sup>ˆ</sup> *ij* =< *<sup>i</sup>*|Ω<sup>ˆ</sup> |*<sup>j</sup>* >, which remains unspecified.

We do not have hermiticity in the form Ωˆ = Ωˆ † automatically. First as Bender and collaborators has emphasized[8], the self-adjoint test does not have a magnificent degree of invariance. If an operator is self-adjoint it will be Hermitian, as defined by having real eigenvalues. It will also remain self-adjoint under unitary transformations. But if a Hermitian operator is subject to arbitrary similarity transformations it may cease to be self-adjoint, while

Second, the class of eigenvalues of Ωˆ are a physics decision. The most general solution to

This happens to eliminate a postulate, as we'll explain, and just to re-iterate, we're discussing a system of generalized classical coordinate where the assumption of units ¯*h* = 1 has definitely *not* been imposed. We have no need for ¯*h* as explained in Section 4. Eq. 21 is self explanatory. Frequencies are the eigenvalues of the *frequency operator* Ωˆ . As Feynman must have said, the textbook business of multiplying *ω<sup>n</sup>* → *h*¯ *ω<sup>n</sup>* ≡ *En* so the time evolution

If *ωn* has a complex part the time evolution contains exponentially growing or damped solutions, which were frowned upon by the authorities in charge of setting up quantum mechanics. That eliminates another postulate (the postulate of Hermiticity), replacing it by the Decision of Hermiticity. To conform with this reasonable decision we specify real *ωn*,

To reiterate, in our approach we have available every freedom to consider non-linear or non-Hermitian systems, at least up to here. We have taken a *more general* framework and reduced it to the *less general* dynamical rules of quantum theory by identifying the restrictions assumed in standard lore. Classical mechanics is a vast general framework not at all the same as Newtonian physics. Understanding that, there is nothing but classical physics in the

Eq. 19 comes from mere algebraic manipulations. While developed a bit in Ref.[7] it is surprisingly unknown to most physicists. Discussions with many physicists find several

• *Every interesting quantum theory is non-linear. Right? Why is the linearization H* →< *ψ*|Ω|*ψ* > *relevant?* No, quantum dynamics is *always* linear. The mixup about what is linear comes from habitual sloppiness in physics discussions to mix operators with numbers, and then constructing Hamiltonians as non-linear combinations of operators. An operator appears

*ψn*(0)*e*

−*iωnt*

<sup>Ψ</sup>(*t*) = ∑

*n*

Since it is an important point we show the algebra for one *do f* . We are given

$$
\dot{q} = \frac{\partial H}{\partial p}; \qquad \dot{p} = -\frac{\partial H}{\partial q}.
$$

Combine two real numbers into one complex one:

$$
\psi(q+ip)/\sqrt{2}.\tag{15}
$$

We call this the "quantum map". It explains how complex numbers came to be essential in quantum theory. Compute

$$
\psi = (\dot{q} + \dot{i}\dot{p}) / \sqrt{2} = (\frac{\partial H}{\partial p} - \dot{i}\frac{\partial H}{\partial q}) / \sqrt{2}. \tag{16}
$$

The chain rule gives

$$\frac{\partial}{\partial p} - i \frac{\partial}{\partial q} = \sqrt{2} \frac{\partial}{\partial \psi^\*} \gamma$$

and then Hamilton's equations are

$$i\psi = \frac{\partial H}{\partial \psi^\*}.\tag{17}$$

*Continuing:* When *<sup>H</sup>*(*q*, *<sup>p</sup>*) is bilinear then *<sup>H</sup>*(Ψ, <sup>Ψ</sup>∗) is bilinear. In quantum mechanics one always chooses parameters so that

$$H(\Psi, \Psi^\*) = \Psi^\* \Omega \Psi,\tag{18}$$

where Ω now contains the parameters. Eq. 14 and 18 give

$$i\frac{\partial \Psi}{\partial t} = \hat{\Omega} \cdot \Psi. \tag{19}$$

This is Schroedinger's equation, which is nothing more than Hamilton's equation in complex *notation*. We prefer symbol Ωˆ to *H*ˆ for reasons to be explained soon.

#### *3.1.3. Conventionally Assumed Properties*

10 Advances in Quantum Mechanics

quantum theory. Compute

and then Hamilton's equations are

always chooses parameters so that

where Ω now contains the parameters. Eq. 14 and 18 give

The chain rule gives

Hamilton's first equation (Eq. 2) become

Combine two real numbers into one complex one:

The transformation produces a remarkable simplification of linear dynamical systems.

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>∂</sup>H*(Ψ, <sup>Ψ</sup>∗)

*ψ*(*q* + *ip*)/

√

We call this the "quantum map". It explains how complex numbers came to be essential in

<sup>2</sup> = ( *<sup>∂</sup><sup>H</sup>*

*<sup>∂</sup><sup>q</sup>* <sup>=</sup> <sup>√</sup>

*<sup>i</sup>ψ*˙ <sup>=</sup> *<sup>∂</sup><sup>H</sup>*

*Continuing:* When *<sup>H</sup>*(*q*, *<sup>p</sup>*) is bilinear then *<sup>H</sup>*(Ψ, <sup>Ψ</sup>∗) is bilinear. In quantum mechanics one

This is Schroedinger's equation, which is nothing more than Hamilton's equation in complex

*i ∂*Ψ

*notation*. We prefer symbol Ωˆ to *H*ˆ for reasons to be explained soon.

*<sup>∂</sup><sup>p</sup>* <sup>−</sup> *<sup>i</sup>*

<sup>2</sup> *<sup>∂</sup> ∂ψ*<sup>∗</sup> ,

*∂H <sup>∂</sup><sup>q</sup>* )/

√

; *<sup>p</sup>*˙ <sup>=</sup> <sup>−</sup>*∂<sup>H</sup>*

√

*∂q* .

*<sup>∂</sup>*Ψ<sup>∗</sup> . (14)

2. (15)

*∂ψ*<sup>∗</sup> . (17)

*<sup>H</sup>*(Ψ, <sup>Ψ</sup>∗) = <sup>Ψ</sup>∗ΩΨ, (18)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>Ω</sup><sup>ˆ</sup> · <sup>Ψ</sup>. (19)

2. (16)

*i ∂*Ψ

Since it is an important point we show the algebra for one *do f* . We are given

*<sup>q</sup>*˙ <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂p*

*ψ*˙ = (*q*˙ + *ip*˙)/

*∂ <sup>∂</sup><sup>p</sup>* <sup>−</sup> *<sup>i</sup> <sup>∂</sup>* We do not have hermiticity in the form Ωˆ = Ωˆ † automatically. First as Bender and collaborators has emphasized[8], the self-adjoint test does not have a magnificent degree of invariance. If an operator is self-adjoint it will be Hermitian, as defined by having real eigenvalues. It will also remain self-adjoint under unitary transformations. But if a Hermitian operator is subject to arbitrary similarity transformations it may cease to be self-adjoint, while its eigenvalues will not change.

Second, the class of eigenvalues of Ωˆ are a physics decision. The most general solution to Eq. 19 is an expansion in normal modes,

$$\Psi(t) = \sum\_{n} \psi\_{n}(0)e^{-i\omega\_{n}t},\tag{20}$$

where *ψn* are solutions to the eigenvalue problem

$$
\hat{\Omega}\Psi\_n = \omega\_n \psi\_n. \tag{21}
$$

This happens to eliminate a postulate, as we'll explain, and just to re-iterate, we're discussing a system of generalized classical coordinate where the assumption of units ¯*h* = 1 has definitely *not* been imposed. We have no need for ¯*h* as explained in Section 4. Eq. 21 is self explanatory. Frequencies are the eigenvalues of the *frequency operator* Ωˆ . As Feynman must have said, the textbook business of multiplying *ω<sup>n</sup>* → *h*¯ *ω<sup>n</sup>* ≡ *En* so the time evolution appears as *<sup>e</sup>*−*iEnt*/¯*<sup>h</sup>* ≡ *<sup>e</sup>*−*iωnt* is a complete waste of time.

If *ωn* has a complex part the time evolution contains exponentially growing or damped solutions, which were frowned upon by the authorities in charge of setting up quantum mechanics. That eliminates another postulate (the postulate of Hermiticity), replacing it by the Decision of Hermiticity. To conform with this reasonable decision we specify real *ωn*, Hermitian Ωˆ , which a short exercise shows is equivalent to Eq. 18.

To reiterate, in our approach we have available every freedom to consider non-linear or non-Hermitian systems, at least up to here. We have taken a *more general* framework and reduced it to the *less general* dynamical rules of quantum theory by identifying the restrictions assumed in standard lore. Classical mechanics is a vast general framework not at all the same as Newtonian physics. Understanding that, there is nothing but classical physics in the Schroedinger equation.

#### *3.1.4. Discussion*

Eq. 19 comes from mere algebraic manipulations. While developed a bit in Ref.[7] it is surprisingly unknown to most physicists. Discussions with many physicists find several frequently asked questions:

• *Every interesting quantum theory is non-linear. Right? Why is the linearization H* →< *ψ*|Ω|*ψ* > *relevant?* No, quantum dynamics is *always* linear. The mixup about what is linear comes from habitual sloppiness in physics discussions to mix operators with numbers, and then constructing Hamiltonians as non-linear combinations of operators. An operator appears in matrix elements <sup>Ω</sup><sup>ˆ</sup> *ij* =< *<sup>i</sup>*|Ω<sup>ˆ</sup> |*<sup>j</sup>* >, which remains unspecified.

• *Does this method assume a finite-dimensional system, and why would that be relevant?* No. The whole point of applying linear algebra and Hilbert space methods to quantum mechanics is one unified notation. The usual infinite dimensional expression for the Hamiltonian is

$$H(\psi, \psi^\*) = \int d\mathbf{x} \,\psi^\*(\mathbf{x}) \hat{H} \psi(\mathbf{x}).\tag{22}$$

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**4. A World without Planck's constant**

differently.

waste of time.

**4.1. The culprit is mass**

parameters from the theory where they appear.

Under changes of scale *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*′ <sup>=</sup> *<sup>λ</sup><sup>x</sup> <sup>x</sup>*, *<sup>t</sup>* <sup>→</sup> *<sup>t</sup>*

automatically. Consider the constant *c* in an ordinary wave equation:

*∂*2*φ*

Hamilton's equations in its three equivalent forms (Eq. 1, 2, 19) lack Planck's constant. Most physicists believe that Planck's constant is a fundamental feature of our Universe, cannot imagine a world without it, and also have no idea how ¯*h* could possible "emerge" from the (possible noise and chaos) of theory more fundamental. But a Universe without Planck's constant is not hard to imagine[12]. It is a Universe where human history would have gone

**Figure 1.** A record of physics from a Universe without Planck's constant. Rydberg's original fits to frequency (wavenumber) data for the Hydrogen spectrum did not need to be converted into Newtonian units and back again to frequency fix the parameters of quantum theory. As Feynman must have said, converting units and the associated conversion constant is a total

Human history defined a notion of *mass* as a quantity of matter such as silver or butter long before physics defined mass. Imagine a history where Hamiltonian methods were developed first. Then *mass* would be more neutral, a particular "coupling constant" appearing in the Hamiltonian. We might find ourselves lacking the Newtonian intuitive picture of "mass," which might be a good thing. We would need to teach ourselves how to get the meaning of

Transformation properties are generally a key. Just as *q*'s and *p*'s transform under a change of variables, the parameters of a theory transform. However parameters do not transform

unless *c* is transformed by hand. The equation is form-unchanged (has a symmetry) under

*<sup>∂</sup>t*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*2<sup>∇</sup> <sup>2</sup>*<sup>φ</sup>* <sup>=</sup> 0. (23)

′ <sup>=</sup> *<sup>λ</sup>tt* the equation changes, and becomes false,

Apply Hamilton's equations (Eq. 14) using a functional derivative and you are done.

• *Current physics of quantum field theory uses highly non-linear Hamiltonians, for which Eq. 22 fails. Why bother with beginning quantum mechanics?* Once again the question is about the matrix elements and dimension of the Hamiltonian operator, which we've left unspecified, and which is generally a non-linear function of the fields. That does not change (repeat) the linearity of the dynamics. It is most directly seen in the functional Schroedinger equation

$$i\dot{\Psi}(\Phi) = \hat{H}(\Phi, -i\delta/\delta\Phi)\Psi(\Phi).$$

This differential equation is equation is *linear* in the dynamical *do f* Ψ, while exactly equivalent to the non-linear operator relations of the usual approach. Feynman himself was very fond of the Schroedinger picture for the practical reason that wave function equations are easier to solve and approximate than operator equations. For us (up to here) quantum field theory is a very large classical dynamical system.


### **4. A World without Planck's constant**

12 Advances in Quantum Mechanics

equation

• *Does this method assume a finite-dimensional system, and why would that be relevant?* No. The whole point of applying linear algebra and Hilbert space methods to quantum mechanics is one unified notation. The usual infinite dimensional expression for the Hamiltonian is

Apply Hamilton's equations (Eq. 14) using a functional derivative and you are done. • *Current physics of quantum field theory uses highly non-linear Hamiltonians, for which Eq. 22 fails. Why bother with beginning quantum mechanics?* Once again the question is about the matrix elements and dimension of the Hamiltonian operator, which we've left unspecified, and which is generally a non-linear function of the fields. That does not change (repeat) the linearity of the dynamics. It is most directly seen in the functional Schroedinger

*i*Ψ˙ (Φ) = *H*ˆ (Φ, −*iδ*/*δ*Φ)Ψ(Φ).

This differential equation is equation is *linear* in the dynamical *do f* Ψ, while exactly equivalent to the non-linear operator relations of the usual approach. Feynman himself was very fond of the Schroedinger picture for the practical reason that wave function equations are easier to solve and approximate than operator equations. For us (up to

• *How can this be the same as the path integral formulation? Moreover, the field-theoretic path-integral is different from the one of beginning quantum mechanics.* We say that basic quantum mechanics is more fundamental than the path integral. *Given* the Schroedinger equation, the path integral comes to be derived as an integral representation of certain correlations. So we also have path integrals as (up to here) a representation of certain

• *Where are the operator equations of motion? What role exists for operators?* It is interesting that the dis-ordering of material in the education of physicists is such that questions like these come up, while everyone knows the answer. *Given* the Schroedinger time evolution, and any arbitrary operator, the Heisenberg picture is developed as *a definition* of time-dependent operators. We must use the Schroedinger picture because it's not really true that Heisenberg operator equations of motion makes an "equivalent theory". The operators lack a wave function to encode a system's initial conditions and state, and

• *Where is Planck's constant? With classical mechanics and without Planck's constant how are you going to quantize the Hydrogen atom ?* One of the advantages of our approach is the ability to discard deadwood. "Deriving the Hamiltonian" of the Hydrogen atom is schoolbook bunk: at least in our approach! Planck's constant deserves a separate discussion: the next

*dx <sup>ψ</sup>*∗(*x*)*H*<sup>ˆ</sup> *<sup>ψ</sup>*(*x*). (22)

*<sup>H</sup>*(*ψ*, *<sup>ψ</sup>*∗) =

here) quantum field theory is a very large classical dynamical system.

quantities evolving by generalized classical mechanics.

which develops proper observables.

topic.

Hamilton's equations in its three equivalent forms (Eq. 1, 2, 19) lack Planck's constant. Most physicists believe that Planck's constant is a fundamental feature of our Universe, cannot imagine a world without it, and also have no idea how ¯*h* could possible "emerge" from the (possible noise and chaos) of theory more fundamental. But a Universe without Planck's constant is not hard to imagine[12]. It is a Universe where human history would have gone differently.

**Figure 1.** A record of physics from a Universe without Planck's constant. Rydberg's original fits to frequency (wavenumber) data for the Hydrogen spectrum did not need to be converted into Newtonian units and back again to frequency fix the parameters of quantum theory. As Feynman must have said, converting units and the associated conversion constant is a total waste of time.

#### **4.1. The culprit is mass**

Human history defined a notion of *mass* as a quantity of matter such as silver or butter long before physics defined mass. Imagine a history where Hamiltonian methods were developed first. Then *mass* would be more neutral, a particular "coupling constant" appearing in the Hamiltonian. We might find ourselves lacking the Newtonian intuitive picture of "mass," which might be a good thing. We would need to teach ourselves how to get the meaning of parameters from the theory where they appear.

Transformation properties are generally a key. Just as *q*'s and *p*'s transform under a change of variables, the parameters of a theory transform. However parameters do not transform automatically. Consider the constant *c* in an ordinary wave equation:

$$\frac{\partial^2 \phi}{\partial t^2} - c^2 \vec{\nabla}^2 \phi = 0. \tag{23}$$

Under changes of scale *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*′ <sup>=</sup> *<sup>λ</sup><sup>x</sup> <sup>x</sup>*, *<sup>t</sup>* <sup>→</sup> *<sup>t</sup>* ′ <sup>=</sup> *<sup>λ</sup>tt* the equation changes, and becomes false, unless *c* is transformed by hand. The equation is form-unchanged (has a symmetry) under

*<sup>c</sup>* <sup>→</sup> *<sup>c</sup>*′ <sup>=</sup> *<sup>λ</sup>xc*/*λt*. That is read "*<sup>c</sup>* has units of *length* over *time*." Transforming constants is so familiar the need for derivation escapes notice.

Notice that transforming parameters is not treated as an ordinary symmetry. Under an ordinary symmetry, including space-time symmetries such as Lorentz transformations, the Hamiltonian is unchanged, including the parameters. The reason *c* must be changed by hand is that the value *c* = 3 × 108*m*/*s* refers to a particular Universe where units of *length* and *time* measured it. Once *c* is measured and fixed in our particular universe, then changing its value (with fixed scales of space and time) is *definitely not* a physical symmetry. Educating the math about a passive scale change of coordinates requires we transform parameters measured in old units into new numbers so the Universe described remains the same.

Review such scale changes more generally. We noted that the group of canonical transformations, which preserve the action of our theory, is the symplectic group *Sp*(2*N*). Every element *<sup>S</sup>* ∈ *Sp*(2*N*) can be written locally as a product of three factors *<sup>S</sup>* = *<sup>K</sup>*<sup>1</sup> · <sup>Λ</sup> · *<sup>K</sup>*2, where *K*1, *K*<sup>2</sup> are "rotations" from the maximal compact subgroup of *S*. The elements Λ are scaling transformations of the form

$$q\_i \rightarrow q'\_i = \lambda\_i q\_i \mathbf{\dot{\iota}}$$

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*p* = *mq*˙; *qi* <sup>→</sup> *<sup>q</sup>*′

The transformation is precisely the inverse of Eq. 24. That is a good paradox.

of time *t*, a spatial coordinate *q*, its conjugate momentum *p*, and mass *m* are:

′ <sup>=</sup> *<sup>λ</sup>tt*; *<sup>q</sup>* <sup>→</sup> *<sup>q</sup>*′ <sup>=</sup> *<sup>λ</sup>qq*;

*dt* <sup>→</sup> *<sup>p</sup>*′ <sup>=</sup> *<sup>m</sup>*′ *<sup>λ</sup><sup>q</sup>*

*<sup>m</sup>* <sup>→</sup> *<sup>m</sup>*′ <sup>=</sup> *<sup>λ</sup><sup>t</sup>*

The last relation tells us that Newtonian mass has the scaling properties of *time* over *length*2,

Now just as we are accustomed to saying that the number for the speed of light is meaningless until it is expressed as a number of meters per second, or miles per hour, we

**Example** Consider a Newtonian object with mass *m* = 3(*seconds*/*meter*2) moving at 2*meter*/*second*, carrying momentum *p* = *mv* = 6/*meter* at position *<sup>q</sup>* = <sup>5</sup>*meter*. Transform to *<sup>q</sup>*′ = <sup>5</sup>*meter*(100*centimeter*/*meter*) = <sup>500</sup> *centimeter* and *<sup>p</sup>*′ = 6/*meter*(1*meter*/100*centimeter*) = 0.06/*centimeter*. The area of the initial phase space between the origin and the canonical coordinates is ∆*A* = ∆*p*∆*q* = 5 × 6 = 30 and equals

<sup>∆</sup>*q*′ = <sup>500</sup> × 0.06 = 30.

**Example** Under the force of gravity on Earth, an object falls with acceleration *g* = 9.8 *meter*/*second*2. The gravitational force *F* on a given mass *m*<sup>1</sup> = 1 *second*/*meter*<sup>2</sup> is

*meter*

*seconds*<sup>2</sup> <sup>=</sup> 9.8

*meter* · *second* .

*λt p <sup>m</sup>* <sup>=</sup> *<sup>p</sup> λq*

*λ*2 *q*

*t* → *t*

*<sup>p</sup>* <sup>=</sup> *<sup>m</sup> dq*

need to get accustomed to mass as a number of seconds per square meter.

*<sup>F</sup>* <sup>=</sup> *<sup>m</sup>*1*<sup>g</sup>* <sup>=</sup> 9.8 *seconds*

*meter*<sup>2</sup>

Try it with the speed of light, *x* = *ct*. Under *t* → *t*

or *seconds*/*meter*2.

the area of the final phase space <sup>∆</sup>*p*′

*<sup>i</sup>* = *λiqi*;

The paradox comes because Eq. 26 has re-scaled coordinates without re-scaling parameters.

Changing units of one second to one hour changes the unit of a kilometer by a factor of 3600. The correct parameter transformation properties can be found from the Hamiltonian. The Newtonian mass symbol *m* is defined by *HN* = *p*2/2*m*. The scaling transformation properties

*<sup>p</sup>* <sup>→</sup> *<sup>p</sup>*′ <sup>=</sup> *<sup>m</sup>λq*˙ <sup>=</sup> *<sup>λ</sup>p*. (26)

′ <sup>=</sup> *<sup>λ</sup>tt*, then *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*′ <sup>=</sup> *<sup>c</sup>λtt* <sup>=</sup> *<sup>λ</sup>tx*.

*m*. (27)

$$p\_{\mathbf{i}} \to p\_{\mathbf{i}}' = p\_{\mathbf{i}}'/\lambda\_{\mathbf{i}};\tag{24}$$

$$dq\_i dp\_i = dq'\_i dp'\_i. \tag{25}$$

The phase-space volume on every pair (not just the entire space) is preserved by re-scaling and rotations. There are *N* parameters in Λ, and 2*N*<sup>2</sup> in *K*1, *K*2, accounting for all *N*(2*N* + 1) parameters of *Sp*(2*N*). The decomposition is unique up to discrete row-swapping transformations maintaining *det*(*S*) = 1.

The action-preserving transformations predict momenta *pi* scale like 1/*qi*. Unless a different definition is made, that requires *pi* to have units of 1/*qi*. This also follows from the momentum being the generator of translations. Similarly the energy *E* as the value of the Hamiltonian, and the time *t* are conjugate. Under scaling transformations preserving the action, they transform with *dtdE* = *dt*′ *dE*′ , as seen from *Ldt* = *pidqi* − *Hdt*. The intrinsic units of energy are inverse time, as also seen using the action of a solved system directly: *H* = −*∂S*/*∂t*.

There is more than one way to apply this. It applies to our complexified dynamical wave functions, *ψ<sup>i</sup>* = (*qi* + *ipi*)/ √2. But it also applies at the most beginning level known as "high school physics." It is astonishing that mixups at the level of high school physics might affect deep questions of quantum mechanics. But this is not as unlikely as it seems. High school students and their teachers are seldom given freedom to challenge what they are taught. Later on it is difficult to give up what we were taught as children.

#### *4.1.1. Mass Paradox*

Knowing the scaling information of Eq. 25, consider changing the units of a translational coordinate *q*, for example changing the units of length from meters to centimeters. There are 100 *cm*/*m*, hence *qcm* = *λcm*/*mqm* with *λcm*/*<sup>m</sup>* = 100. Ordinary usage of Newtonian momentum predicts

$$\begin{aligned} p &= m\dot{q};\\ q\_{\dot{l}} &\rightarrow q'\_{\dot{l}} = \lambda\_i q\_{\dot{l}};\\ p &\rightarrow p' = m\lambda \dot{q} = \lambda p. \end{aligned} \tag{26}$$

The transformation is precisely the inverse of Eq. 24. That is a good paradox.

14 Advances in Quantum Mechanics

familiar the need for derivation escapes notice.

scaling transformations of the form

transformations maintaining *det*(*S*) = 1.

action, they transform with *dtdE* = *dt*′

√

Later on it is difficult to give up what we were taught as children.

*H* = −*∂S*/*∂t*.

functions, *ψ<sup>i</sup>* = (*qi* + *ipi*)/

*4.1.1. Mass Paradox*

momentum predicts

*<sup>c</sup>* <sup>→</sup> *<sup>c</sup>*′ <sup>=</sup> *<sup>λ</sup>xc*/*λt*. That is read "*<sup>c</sup>* has units of *length* over *time*." Transforming constants is so

Notice that transforming parameters is not treated as an ordinary symmetry. Under an ordinary symmetry, including space-time symmetries such as Lorentz transformations, the Hamiltonian is unchanged, including the parameters. The reason *c* must be changed by hand is that the value *c* = 3 × 108*m*/*s* refers to a particular Universe where units of *length* and *time* measured it. Once *c* is measured and fixed in our particular universe, then changing its value (with fixed scales of space and time) is *definitely not* a physical symmetry. Educating the math about a passive scale change of coordinates requires we transform parameters measured in

Review such scale changes more generally. We noted that the group of canonical transformations, which preserve the action of our theory, is the symplectic group *Sp*(2*N*). Every element *<sup>S</sup>* ∈ *Sp*(2*N*) can be written locally as a product of three factors *<sup>S</sup>* = *<sup>K</sup>*<sup>1</sup> · <sup>Λ</sup> · *<sup>K</sup>*2, where *K*1, *K*<sup>2</sup> are "rotations" from the maximal compact subgroup of *S*. The elements Λ are

*qi* <sup>→</sup> *<sup>q</sup>*′

*dqidpi* <sup>=</sup> *dq*′

The phase-space volume on every pair (not just the entire space) is preserved by re-scaling and rotations. There are *N* parameters in Λ, and 2*N*<sup>2</sup> in *K*1, *K*2, accounting for all *N*(2*N* + 1) parameters of *Sp*(2*N*). The decomposition is unique up to discrete row-swapping

The action-preserving transformations predict momenta *pi* scale like 1/*qi*. Unless a different definition is made, that requires *pi* to have units of 1/*qi*. This also follows from the momentum being the generator of translations. Similarly the energy *E* as the value of the Hamiltonian, and the time *t* are conjugate. Under scaling transformations preserving the

units of energy are inverse time, as also seen using the action of a solved system directly:

There is more than one way to apply this. It applies to our complexified dynamical wave

school physics." It is astonishing that mixups at the level of high school physics might affect deep questions of quantum mechanics. But this is not as unlikely as it seems. High school students and their teachers are seldom given freedom to challenge what they are taught.

Knowing the scaling information of Eq. 25, consider changing the units of a translational coordinate *q*, for example changing the units of length from meters to centimeters. There are 100 *cm*/*m*, hence *qcm* = *λcm*/*mqm* with *λcm*/*<sup>m</sup>* = 100. Ordinary usage of Newtonian

*dE*′

*pi* <sup>→</sup> *<sup>p</sup>*′

*<sup>i</sup>* = *λiqi*;

*idp*′ *i*

*<sup>i</sup>*/*λi*; (24)

, as seen from *Ldt* = *pidqi* − *Hdt*. The intrinsic

2. But it also applies at the most beginning level known as "high

. (25)

*<sup>i</sup>* <sup>=</sup> *<sup>p</sup>*′

old units into new numbers so the Universe described remains the same.

The paradox comes because Eq. 26 has re-scaled coordinates without re-scaling parameters. Try it with the speed of light, *x* = *ct*. Under *t* → *t* ′ <sup>=</sup> *<sup>λ</sup>tt*, then *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*′ <sup>=</sup> *<sup>c</sup>λtt* <sup>=</sup> *<sup>λ</sup>tx*. Changing units of one second to one hour changes the unit of a kilometer by a factor of 3600.

The correct parameter transformation properties can be found from the Hamiltonian. The Newtonian mass symbol *m* is defined by *HN* = *p*2/2*m*. The scaling transformation properties of time *t*, a spatial coordinate *q*, its conjugate momentum *p*, and mass *m* are:

$$t \to t' = \lambda\_t t;$$

$$q \to q' = \lambda\_q q;$$

$$p = m\frac{dq}{dt} \to p' = m'\frac{\lambda\_q}{\lambda\_t}\frac{p}{m} = \frac{p}{\lambda\_q}$$

$$m \to m' = \frac{\lambda\_t}{\lambda\_q^2}m. \tag{27}$$

The last relation tells us that Newtonian mass has the scaling properties of *time* over *length*2, or *seconds*/*meter*2.

Now just as we are accustomed to saying that the number for the speed of light is meaningless until it is expressed as a number of meters per second, or miles per hour, we need to get accustomed to mass as a number of seconds per square meter.

**Example** Consider a Newtonian object with mass *m* = 3(*seconds*/*meter*2) moving at 2*meter*/*second*, carrying momentum *p* = *mv* = 6/*meter* at position *<sup>q</sup>* = <sup>5</sup>*meter*. Transform to *<sup>q</sup>*′ = <sup>5</sup>*meter*(100*centimeter*/*meter*) = <sup>500</sup> *centimeter* and *<sup>p</sup>*′ = 6/*meter*(1*meter*/100*centimeter*) = 0.06/*centimeter*. The area of the initial phase space between the origin and the canonical coordinates is ∆*A* = ∆*p*∆*q* = 5 × 6 = 30 and equals the area of the final phase space <sup>∆</sup>*p*′ <sup>∆</sup>*q*′ = <sup>500</sup> × 0.06 = 30.

**Example** Under the force of gravity on Earth, an object falls with acceleration *g* = 9.8 *meter*/*second*2. The gravitational force *F* on a given mass *m*<sup>1</sup> = 1 *second*/*meter*<sup>2</sup> is

$$F = m\_1 \\ g = 9.8 \frac{seconds}{meter^2} \\ \frac{meter}{seconds^2} = \frac{9.8}{meter \cdot second} .$$

The work lifting the mass one meter is

$$work = m\_1gh = 9.8 \, meter \frac{1}{meter \cdot second} = 9.8 \, \frac{1}{second}.$$

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0.002 0.004 0.006 0.008 0.010

κ

Here we have a model *frequency operator* <sup>Ω</sup><sup>ˆ</sup> <sup>=</sup> −∇ 2/2*mc* familiar from Schroedinger theory.

Essentially the same analysis is done in Ref[12]. A more complicated frequency operator Ωˆ = −∇ 2/2*<sup>m</sup>* <sup>+</sup> *<sup>U</sup>*(*x*) represents an ansatz for interacting waves. We would not pretend to know the interaction function *U*(*x*) from first principles. (The old predictive recipes, we noted, are just mnemonics and pedagogy.) We find *U* from data for electrons. Basic scattering theory allows one to invert the Born-level differential cross section of electron-atom scattering into *U*(*x*). The same *U*(*x*) predicts the observed Hydrogen frequency spectrum, which is quite non-trivial. Figure 1 shows an example of the frequency data of Rydberg[10]. Taken before 1900, the data was of surprisingly high quality. Several other data comparisons are consistent. The entire theory has only two parameters *m* and *κ*. The constant *κ* is dimensionless, as

*<sup>U</sup>*(*x*) = *<sup>κ</sup><sup>c</sup>*

We said we are only concerned with what is observable and testable. When using experimental data to fit the parameters of quantum theory ¯*h* is unobservable, and given up, *in our theory*. Figure 2 shows that we have done the work to fit parameters [12]. With basic information on the *frequencies* observed in the Rydberg spectrum, and the scattering *lengths* observed by Geiger and Marsden, etc. one derives *κ* and *λ<sup>e</sup>* = *c*/*me* directly. The numerical value of *κ* is about 1/137. By a natural coincidence Sommerfeld discovered the dimensionless constant *κ* and called it *α*, the fine structure constant. Dimensionless constants do not depend on the units used to compute them, so that the unobservable unit converter *h*¯ cancelled out for Sommerfeld. In none of this is the introduction of a conversion to archaic *MKS* units necessary, nor is it helpful. As Feynman must have said, "bothering to convert

units with the meaningless constant known as ¯*h* is a total waste of time".

0.002 0.004 0.006 0.008 0.010

κ

show modern values *<sup>c</sup>* = <sup>3</sup> × <sup>10</sup>10*cm*/*s*, *<sup>λ</sup><sup>e</sup>* = 3.87 × <sup>10</sup><sup>−</sup>11, *<sup>κ</sup>* = 1/137 all lie inside the range of *<sup>χ</sup>*<sup>2</sup> <sup>1</sup>.

**Figure 2.** Contours of *χ*2, the summed-squared differences of data versus fit obtained from the data analysis of Ref. [12]. *Left panel*: As a function of parameters *κ* and *λ<sup>e</sup>* with *ce* → *c*. *Right panel*: As a function of parameters *κ* and *ce* with *λ<sup>e</sup>* given by Compton's 1922 experiment. Dots shows the points of minimum *χ*<sup>2</sup> ∼ 0.24 in both cases. Contours are *χ*<sup>2</sup> = 1, 2, 3... Lines

When one redundant unit is dropped, something is gained. The errors in the definition and inconsistent uses of the *kilogram* drop out. Continuing up the ladder, the system where mass


c e (cm/s)

We are still lacking ¯*h*, and in our approach, we will never find it in quantum theory.

consistent with the results of data-fitting

and *U* correctly has dimensions of frequency.

2 x 10-11

4 x 10-11

6 x 10-11

8 x 10-11

λ e

(cm)

1 x 10-10

The power delivered by the force is

$$\begin{split} power &= \vec{F} \cdot \vec{v} = \frac{9.8}{meter \cdot second \, second} \frac{meter}{second} (\frac{v}{meter/second}) \\ &= \frac{9.8}{second^2} v\_{MSA} \end{split}$$

where *vMS* is the dimensionless velocity measured in *meter*/*second*. Using a pulley or spring balance to apply the same force to a second object with mass *m*<sup>2</sup> = 2 *second*/*meter*<sup>2</sup> produces an acceleration

$$a\_2 = F/m\_2 = \frac{9.8}{meter \cdot second} \frac{1}{2} \frac{meter^2}{second} = 4.9 \,\frac{meter}{second^2}.$$

#### *4.1.2. How This is Related to Planck's Constant*

To see how the discussion is related to quantum theory, use a theory that is relevant. Consider a standard wave equation:

$$\frac{\partial^2 \phi}{\partial t^2} - c^2 \vec{\nabla}^2 \phi + m^2 c^4 \phi = 0.$$

This equation comes from a ubiquitous linear Hamiltonian model. It is a trap to "derive" this equation using substitution rules of beginning quantum theory: they are circular. It is better to find the equation generic, as indeed it appears in the vibrations of any collection of oscillators that has an "optical" branch. By inspection the dimensions of the combination *<sup>m</sup>*2*c*<sup>4</sup> are *seconds*<sup>−</sup>2. Then *<sup>m</sup>* scales like a Newtonian mass and we are entitled to call *<sup>m</sup>* <sup>a</sup> "mass parameter."

Make the definition

$$\begin{aligned} \phi &= e^{-imc^2t} \psi; \\ \frac{\partial \phi}{\partial t} &= e^{-imc^2t} (-imc^2\psi + \frac{\partial \psi}{\partial t}). \end{aligned}$$

This transformation removes the *m*2*c*<sup>4</sup> term. Continue to obtain the time evolution equation for *ψ*. Impose a low frequency approximation that drops the term proportional to *ψ*¨. The result is

$$i\dot{\psi} = -\frac{\vec{\nabla}^2}{2mc}\psi.\tag{28}$$

Here we have a model *frequency operator* <sup>Ω</sup><sup>ˆ</sup> <sup>=</sup> −∇ 2/2*mc* familiar from Schroedinger theory. We are still lacking ¯*h*, and in our approach, we will never find it in quantum theory.

Essentially the same analysis is done in Ref[12]. A more complicated frequency operator Ωˆ = −∇ 2/2*<sup>m</sup>* <sup>+</sup> *<sup>U</sup>*(*x*) represents an ansatz for interacting waves. We would not pretend to know the interaction function *U*(*x*) from first principles. (The old predictive recipes, we noted, are just mnemonics and pedagogy.) We find *U* from data for electrons. Basic scattering theory allows one to invert the Born-level differential cross section of electron-atom scattering into *U*(*x*). The same *U*(*x*) predicts the observed Hydrogen frequency spectrum, which is quite non-trivial. Figure 1 shows an example of the frequency data of Rydberg[10]. Taken before 1900, the data was of surprisingly high quality. Several other data comparisons are consistent. The entire theory has only two parameters *m* and *κ*. The constant *κ* is dimensionless, as consistent with the results of data-fitting

$$U(x) = \frac{\kappa c}{|\vec{x}|'},$$

and *U* correctly has dimensions of frequency.

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an acceleration

The work lifting the mass one meter is

The power delivered by the force is

*power* <sup>=</sup>

*4.1.2. How This is Related to Planck's Constant*

a standard wave equation:

"mass parameter." Make the definition

result is

*work* = *m*1*gh* = 9.8 *meter*

*<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>F</sup>*/*m*<sup>2</sup> <sup>=</sup> 9.8

*∂*2*φ*

*φ* = *e*

*∂φ <sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>e</sup>*

−*imc*2*t ψ*;

−*imc*2*t*

This transformation removes the *m*2*c*<sup>4</sup> term. Continue to obtain the time evolution equation for *ψ*. Impose a low frequency approximation that drops the term proportional to *ψ*¨. The

*<sup>i</sup>ψ*˙ <sup>=</sup> <sup>−</sup> <sup>∇</sup> <sup>2</sup>

(−*imc*2*<sup>ψ</sup>* <sup>+</sup> *∂ψ*

*∂t* ).

<sup>2</sup>*mcψ*. (28)

*<sup>F</sup>* ·*<sup>v</sup>* <sup>=</sup> 9.8

*meter* · *second*

*meter* · *second*

where *vMS* is the dimensionless velocity measured in *meter*/*second*. Using a pulley or spring balance to apply the same force to a second object with mass *m*<sup>2</sup> = 2 *second*/*meter*<sup>2</sup> produces

To see how the discussion is related to quantum theory, use a theory that is relevant. Consider

*<sup>∂</sup>t*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*2<sup>∇</sup> <sup>2</sup>*<sup>φ</sup>* <sup>+</sup> *<sup>m</sup>*2*c*4*<sup>φ</sup>* <sup>=</sup> 0.

This equation comes from a ubiquitous linear Hamiltonian model. It is a trap to "derive" this equation using substitution rules of beginning quantum theory: they are circular. It is better to find the equation generic, as indeed it appears in the vibrations of any collection of oscillators that has an "optical" branch. By inspection the dimensions of the combination *<sup>m</sup>*2*c*<sup>4</sup> are *seconds*<sup>−</sup>2. Then *<sup>m</sup>* scales like a Newtonian mass and we are entitled to call *<sup>m</sup>* <sup>a</sup>

1 2 *meter*<sup>2</sup>

1

*meter second* ( *<sup>v</sup>*

*meter* · *second* <sup>=</sup> 9.8 <sup>1</sup>

*second* .

*meter*/*second* )

*second*<sup>2</sup> *vMS*,

*second*<sup>2</sup> .

<sup>=</sup> 9.8

*second* <sup>=</sup> 4.9 *meter*

We said we are only concerned with what is observable and testable. When using experimental data to fit the parameters of quantum theory ¯*h* is unobservable, and given up, *in our theory*. Figure 2 shows that we have done the work to fit parameters [12]. With basic information on the *frequencies* observed in the Rydberg spectrum, and the scattering *lengths* observed by Geiger and Marsden, etc. one derives *κ* and *λ<sup>e</sup>* = *c*/*me* directly. The numerical value of *κ* is about 1/137. By a natural coincidence Sommerfeld discovered the dimensionless constant *κ* and called it *α*, the fine structure constant. Dimensionless constants do not depend on the units used to compute them, so that the unobservable unit converter *h*¯ cancelled out for Sommerfeld. In none of this is the introduction of a conversion to archaic *MKS* units necessary, nor is it helpful. As Feynman must have said, "bothering to convert units with the meaningless constant known as ¯*h* is a total waste of time".

**Figure 2.** Contours of *χ*2, the summed-squared differences of data versus fit obtained from the data analysis of Ref. [12]. *Left panel*: As a function of parameters *κ* and *λ<sup>e</sup>* with *ce* → *c*. *Right panel*: As a function of parameters *κ* and *ce* with *λ<sup>e</sup>* given by Compton's 1922 experiment. Dots shows the points of minimum *χ*<sup>2</sup> ∼ 0.24 in both cases. Contours are *χ*<sup>2</sup> = 1, 2, 3... Lines show modern values *<sup>c</sup>* = <sup>3</sup> × <sup>10</sup>10*cm*/*s*, *<sup>λ</sup><sup>e</sup>* = 3.87 × <sup>10</sup><sup>−</sup>11, *<sup>κ</sup>* = 1/137 all lie inside the range of *<sup>χ</sup>*<sup>2</sup> <sup>1</sup>.

When one redundant unit is dropped, something is gained. The errors in the definition and inconsistent uses of the *kilogram* drop out. Continuing up the ladder, the system where mass is measured in *seconds*/*meter*<sup>2</sup> is such that the accuracy of the best determinations of the electric charge and electron mass were improved[12] by a factor of order 100 compared to the official CODATA determinations[13].

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used the classical adiabatic invariance of the action to relate the electromagnetic energy to its frequency. It was already known that the value of the Hamiltonian *H* = −*∂S*/*∂t*. Hence the clue of a redundant unit existed, and if the connection had been made, it might have led to

**Example:** Due to pre-quantum nonsense everyone is obliged to learn, the quantization of angular momentum is blamed on the value of Planck's constant. One cannot possibly do without ¯*h*, it is claimed, due to the fundamental commutation relations of angular

Strictly deductive algebraic relations developed from those commutators produce the possible representations and the spectrum of observables *Jz* = *nh*¯, where *n* must be an integer or half-integer. And this reproduces data. And in the limit of ¯*h* → 0, the commutators go to

We say: that kind of argumentation fails the quality-control standards of the current millennium. Groups and representations are fine organizing devices that don't originate in claims about physical existence. The representations of *SU*(2) had been worked out before quantum mechanics wanted them. To get Eq. 29 one takes a set of dimensionless *SU*(2)

Then *Ji* = *h*¯ ˜*Ji* come to obey Eq.29. But by remarkable rules of algebra, multiplying both sides of any equation by the same constant does not change the equation. In Eq. 30 one sees that *h*¯ <sup>2</sup> cancels out: including that magic limit that ¯*h* → 0 or ¯*h* → −17.3 or anything else, so it

The quantization of angular momentum eigenvalues is a fact that has nothing to do with *h*¯. The "classical limit" has nothing to do with ¯*h*, assuming one can count and distinguish low quantum numbers from huge ones. The reason that ¯*h* is artificially spliced into Eq. 29, and other algebras, is so that 21st century measurements of angular momentum will be cast into an *MKS* unit system designed for 17th century Newtonian physics. Which continues to assess and obstruct quantum mechanical data by interposing the universal *kilogram* which is

We seek to get as much from the theory as possible without making unnecessary postulates:

The postulate that operators are "physical observables" is redundant: *in our approach*. We find using it slyly abuses language that first defined observables as numbers, in order to slip in the operator as a philosophically transcendent realization of physics. That's too cheap. Very simply, the wave function is observable, and using operators to probe a wave function

zero, which is the classical limit of operators becoming *c*-numbers, etc.

generators ˜*Ji* and arbitrarily multiplies them by ¯*h*:

**5. Observables, quantization and bracket relations**

5.0.0.1. Operators as Physical Observables

may be convenient, but that is not independent.

also cancels out in Eq. 29.

not a feature of *Nature*.

[*Ji*, *Ji*] = *ih*¯ *ǫijk Jk*. (29)

[*h*¯ ˜*Ji*, ¯*h* ˜*Ji*] = *ih*¯ *ǫijk*(*h*¯ ˜*Jk*). (30)

getting rid of the *kilogram* before 1900.

momentum

### **4.2. A redundant convention**

We come to see where Planck's constant entered human history. The space-time scaling properties of *mass* were overlooked early, which continues today, due to a Newtonian prejudice that *mass* is intuitively self-defined. For a long time *mass* was even thought to be a "constant quantity of matter, unchanging by the principle of conservation of mass." That led to a unit of mass unrelated to *meters* and *seconds*, and declared independent by defining a totally arbitrary unit known as the *kilogram*. Introducing an artificial reference standard was found acceptable for Newtonian physics: using artificial standards to weigh silver and butter was quite ancient and obvious.

Notice that introducing artificial unit conventions cannot be detected as faulty by math or logic. Intelligent technicians and business people use a huge array of unnecessary units daily, in many cases imagining that relations between them (such as 1 pascal =0.000145 pounds-per square-inch) must be "laws of physics." Once a redundant unit and its arithmetic enters the scaling laws it can stay around forever.

But as a price for these mistakes, a unit-conversion constant was needed in history to change black-body frequency in terms of temperature, which is energy, which is the frequency of the action, to black body frequency observed in the spectrum, which is frequency. *If and only if* one insists on measuring *mass* in kilograms, one needs a new symbol *mkg* = *ηm*, where *eta* has units of *kgm*2/*s*. The value of the conversion constant *η* is arbitrary, just as the kilogram is arbitrary, and as Ref. [12] shows, fixing one predicts the other. (And that explains[7] the peculiar phenomenon of international unit standardization from global fits finding 100% correlation of the "experimental errors in the kilogram" with the "experimental errors in Planck's constant.")

There is a fast way to reach the same conclusions. The action principle is *δS* = 0. The right hand side "0" has no scale, and no units of *S* can be physically observable. By *S* we don't mean the action of some subsystem which can be compared to another to define an arbitrary fiducial unit, just like the kilogram. We mean the action of the Universe. That causes one overall constant that was defined in quantum pre-history "with the dimensions of action" to drop out.

At least in our approach, every quantum mechanical relation that involves ¯*h* is an ordinary relation not involving ¯*h* that has been multiplied by some power of ¯*h* on both sides: so that ¯*h* cancels out of everything observable.

**Example:** Although Planck is reported to have gotten his constant from black-body data, his original work shows otherwise. From his derivation and data fits [11] the most Planck could get was the ratio of ¯*h*/*kB*, where *kB* is Boltzman's constant. Planck thought *kB* was extremely fundamental, although we now know it is nothing from Nature. It is a conversion constant of energy in temperature units to energy in energy units. The Newtonian convention for *mass* entered in *kB* and ¯*h* both. The *kilogram* cancels out. When measuring quantum data with quantum data ¯*h* cannot be obtained [12], and it is nothing but convention to insert the *kilogram* so as to force a relation. The information was available in 1900. Indeed Wien's Law

used the classical adiabatic invariance of the action to relate the electromagnetic energy to its frequency. It was already known that the value of the Hamiltonian *H* = −*∂S*/*∂t*. Hence the clue of a redundant unit existed, and if the connection had been made, it might have led to getting rid of the *kilogram* before 1900.

**Example:** Due to pre-quantum nonsense everyone is obliged to learn, the quantization of angular momentum is blamed on the value of Planck's constant. One cannot possibly do without ¯*h*, it is claimed, due to the fundamental commutation relations of angular momentum

$$[I\_{i\prime}, I\_i] = i\hbar \epsilon\_{ijk} I\_k. \tag{29}$$

Strictly deductive algebraic relations developed from those commutators produce the possible representations and the spectrum of observables *Jz* = *nh*¯, where *n* must be an integer or half-integer. And this reproduces data. And in the limit of ¯*h* → 0, the commutators go to zero, which is the classical limit of operators becoming *c*-numbers, etc.

We say: that kind of argumentation fails the quality-control standards of the current millennium. Groups and representations are fine organizing devices that don't originate in claims about physical existence. The representations of *SU*(2) had been worked out before quantum mechanics wanted them. To get Eq. 29 one takes a set of dimensionless *SU*(2) generators ˜*Ji* and arbitrarily multiplies them by ¯*h*:

$$[\hbar \tilde{l}\_{\rm i}, \hbar \tilde{l}\_{\rm i}] = i\hbar \epsilon\_{\rm ijk}(\hbar \tilde{l}\_{\rm k}).\tag{30}$$

Then *Ji* = *h*¯ ˜*Ji* come to obey Eq.29. But by remarkable rules of algebra, multiplying both sides of any equation by the same constant does not change the equation. In Eq. 30 one sees that *h*¯ <sup>2</sup> cancels out: including that magic limit that ¯*h* → 0 or ¯*h* → −17.3 or anything else, so it also cancels out in Eq. 29.

The quantization of angular momentum eigenvalues is a fact that has nothing to do with *h*¯. The "classical limit" has nothing to do with ¯*h*, assuming one can count and distinguish low quantum numbers from huge ones. The reason that ¯*h* is artificially spliced into Eq. 29, and other algebras, is so that 21st century measurements of angular momentum will be cast into an *MKS* unit system designed for 17th century Newtonian physics. Which continues to assess and obstruct quantum mechanical data by interposing the universal *kilogram* which is not a feature of *Nature*.

### **5. Observables, quantization and bracket relations**

We seek to get as much from the theory as possible without making unnecessary postulates:

#### 5.0.0.1. Operators as Physical Observables

18 Advances in Quantum Mechanics

the official CODATA determinations[13].

silver and butter was quite ancient and obvious.

scaling laws it can stay around forever.

cancels out of everything observable.

Planck's constant.")

drop out.

**4.2. A redundant convention**

is measured in *seconds*/*meter*<sup>2</sup> is such that the accuracy of the best determinations of the electric charge and electron mass were improved[12] by a factor of order 100 compared to

We come to see where Planck's constant entered human history. The space-time scaling properties of *mass* were overlooked early, which continues today, due to a Newtonian prejudice that *mass* is intuitively self-defined. For a long time *mass* was even thought to be a "constant quantity of matter, unchanging by the principle of conservation of mass." That led to a unit of mass unrelated to *meters* and *seconds*, and declared independent by defining a totally arbitrary unit known as the *kilogram*. Introducing an artificial reference standard was found acceptable for Newtonian physics: using artificial standards to weigh

Notice that introducing artificial unit conventions cannot be detected as faulty by math or logic. Intelligent technicians and business people use a huge array of unnecessary units daily, in many cases imagining that relations between them (such as 1 pascal =0.000145 pounds-per square-inch) must be "laws of physics." Once a redundant unit and its arithmetic enters the

But as a price for these mistakes, a unit-conversion constant was needed in history to change black-body frequency in terms of temperature, which is energy, which is the frequency of the action, to black body frequency observed in the spectrum, which is frequency. *If and only if* one insists on measuring *mass* in kilograms, one needs a new symbol *mkg* = *ηm*, where *eta* has units of *kgm*2/*s*. The value of the conversion constant *η* is arbitrary, just as the kilogram is arbitrary, and as Ref. [12] shows, fixing one predicts the other. (And that explains[7] the peculiar phenomenon of international unit standardization from global fits finding 100% correlation of the "experimental errors in the kilogram" with the "experimental errors in

There is a fast way to reach the same conclusions. The action principle is *δS* = 0. The right hand side "0" has no scale, and no units of *S* can be physically observable. By *S* we don't mean the action of some subsystem which can be compared to another to define an arbitrary fiducial unit, just like the kilogram. We mean the action of the Universe. That causes one overall constant that was defined in quantum pre-history "with the dimensions of action" to

At least in our approach, every quantum mechanical relation that involves ¯*h* is an ordinary relation not involving ¯*h* that has been multiplied by some power of ¯*h* on both sides: so that ¯*h*

**Example:** Although Planck is reported to have gotten his constant from black-body data, his original work shows otherwise. From his derivation and data fits [11] the most Planck could get was the ratio of ¯*h*/*kB*, where *kB* is Boltzman's constant. Planck thought *kB* was extremely fundamental, although we now know it is nothing from Nature. It is a conversion constant of energy in temperature units to energy in energy units. The Newtonian convention for *mass* entered in *kB* and ¯*h* both. The *kilogram* cancels out. When measuring quantum data with quantum data ¯*h* cannot be obtained [12], and it is nothing but convention to insert the *kilogram* so as to force a relation. The information was available in 1900. Indeed Wien's Law

The postulate that operators are "physical observables" is redundant: *in our approach*. We find using it slyly abuses language that first defined observables as numbers, in order to slip in the operator as a philosophically transcendent realization of physics. That's too cheap. Very simply, the wave function is observable, and using operators to probe a wave function may be convenient, but that is not independent.

#### **5.1. Observables as numbers**

An *observable* is a number extracted from the wave function as a projective representation, meaning that *ψ* ≡ *zψ* for any complex *z*. This symmetry exists in the equation of motion. We wish to call attention to it as a feature of "quantum homogeneity." It makes the information in |*<sup>ψ</sup>* > equivalent to the information in the density matrix *ρψ* = |*<sup>ψ</sup>* >< *<sup>ψ</sup>*|/ < *<sup>ψ</sup>*|*<sup>ψ</sup>* >. (The bracket notation, suppressed before to reduce clutter, is useful here.) Equivalence exists because if *ρψ* is given then its unit eigenvector predicts |*<sup>ψ</sup>* > up to a constant *<sup>z</sup>*. (Later we discover that density matrices of rank-1 are too special to base the full theory upon, but that comes with development of quantum probability, which is not yet under discussion.)

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*P*, mistakenly thought to

operator. The interpolation of that rule seems designed to create conflict. Checking a bit of physical data, measuring the eigenvalue of an operator intended is rare. For example experimentalists have been measuring neutrinos for decades while the theorists argued about their operators. We think that measuring a number and calling it an eigenvalue of "some operator " is meaningless. If and when an eigenvalues of a known operator appear in a data set, one obtains that fact that |*ψ* > will then be the corresponding eigenvector of *A*ˆ automatically. We also don't need to discuss "compatible and incompatible" observers in terms of operators that commute. In this century everyone knows how that math works.

The quantum map shows that complex *ψ<sup>i</sup>* are canonical coordinates, up to *i*:

*<sup>i</sup>ψ*˙ <sup>=</sup> *<sup>∂</sup><sup>H</sup> ∂ψ*<sup>∗</sup> *i*

invariant, so that transcribing them to *<sup>ψ</sup>*, *<sup>ψ</sup>*<sup>∗</sup> involves only a factor of *<sup>i</sup>*.

project the wave function into a few particle-like observables *<sup>Q</sup>*� , �

{*Qi*, *Pj*}*PB* <sup>=</sup> <sup>−</sup>*i*∑

{*Qi*, *Pj*}*PB* = −*<sup>i</sup>*

*then <sup>Q</sup>*� <sup>=</sup> *<sup>d</sup>*3*<sup>x</sup> <sup>ψ</sup>*<sup>∗</sup> <sup>ˆ</sup>

*ψ<sup>i</sup>* = (*qi* + *ipi*)/

A factor of *i* is tolerable, in that Hamilton's equations are recognizable including it:

√ 2.

, *<sup>i</sup>ψ*˙ <sup>∗</sup> <sup>=</sup> <sup>−</sup> *<sup>∂</sup><sup>H</sup>*

*<sup>i</sup>* is the canonical momentum conjugate to *ψi*. Poisson brackets (*PB*) are canonically

In fact the *PB* relations among our observable are rather simple due to the decision to make observables bilinear in *<sup>ψ</sup>*, *<sup>ψ</sup>*∗. The most famous application was the early desire to

be important from Newtonian bias. For *<sup>Q</sup>*� to represent a translational coordinate it must

*given <sup>ψ</sup>*(*x*) <sup>→</sup> *<sup>ψ</sup>*�*a*(�*x*) = *<sup>ψ</sup>*(�*<sup>x</sup>* <sup>−</sup>�*a*),

There are few choices but *<sup>Q</sup>*� <sup>=</sup> �*x*. The test that a candidate variable *Pi* is conjugate to these

 *δQi δψx*

> *dx <sup>ψ</sup>*<sup>∗</sup>

*δPj δψ*<sup>∗</sup> *x* <sup>−</sup> *<sup>δ</sup>Pj δψx*

*<sup>x</sup>* [*Q*ˆ*i*, *P*ˆ

*δQi δψ*<sup>∗</sup> *x* .

*<sup>j</sup>*]*ψx*.

*x*

*<sup>Q</sup>*� *<sup>ψ</sup>* <sup>→</sup> *<sup>Q</sup>*� <sup>→</sup> *<sup>Q</sup>*� <sup>+</sup>�*a*.

*∂ψi* .

*5.1.2. Bracket Relations Are Kinematic*

Thus *<sup>i</sup>ψ*<sup>∗</sup>

transform properly:

*Qi* needs the bracket {*Qi*, *Pj*}*PB* = *<sup>δ</sup>ij*.

Computing the derivatives gives

Write this out, assuming operator sandwiches:

It is convenient to extract a number using an "operator sandwich". We define an observable < *A*ˆ > as the number from the map

$$|\psi> \to <\hat{A}> = \frac{<\psi|\hat{A}|\psi>}{<\psi|\psi>} = \frac{tr(\hat{A}\rho\_{\Psi})}{tr(\rho\_{\Psi})}.$$

The last relation is general for cases where *ρ* is not so simple as *rank* − 1. Note we need nothing from quantum probability to make the map. Instead of prescribing < *ψ*|*ψ* > with a normalization postulate, we maintain it is simply unobservable, and drops out. We may then set < *ψ*|*ψ* >→ 1 to simplify expressions.

The trace (symbol *tr*) acts as an inner product between operators. Let *A*ˆ *<sup>i</sup>* be a normalized complete set of operators, which is defined by *tr*(*A*ˆ † *<sup>i</sup> <sup>A</sup>*ˆ*j*) = *<sup>δ</sup>ij*. Since the set is complete,

$$\rho = \sum\_{j} A\_{j} tr(\hat{A}\_{i}^{\dagger}\rho) = \sum\_{j} < \hat{A}\_{j} > \hat{A}\_{j}.$$

Thus *ρ* is equivalent to a number of observables, and |*ψ* > is observable to the exact extent it is defined. At some point this simple relation seems to have been re-packaged as "quantum holography." The very late date of realizing the wave function is observable (to the extent it is defined) supports our case that quantum mechanics is still "emerging" from its history.

#### *5.1.1. Eliminating More Postulates*

There is no particular reason for us to postulate that *A*ˆ must be Hermitian. If it is not Hermitian the operator sandwich gives a complex number, equivalent to two real numbers and two observables, because any operator is the sum of a Hermitian operator and *i* times a Hermitian operator. As for complex numbers being observable in the lab, mathematics tells us that complex numbers are real pairs with 2-vector addition and multiplication rules. It is not unusual to observe such number pairs that have phase relations like *qi*, *pi* which map directly into *ψi*, a complex number. So there is no reason for the 18th century trick of scaring people with complex numbers. And yet: given that *ρ* is Hermitian no harm is done by restricting the operators for observables to being Hermitian. One more grand postulate turns to clay.

*At least in our approach*, those matters of definition need no *foundation postulates.* Neither is there a good reason to insist that an observable be an eigenvalue of some especially known

operator. The interpolation of that rule seems designed to create conflict. Checking a bit of physical data, measuring the eigenvalue of an operator intended is rare. For example experimentalists have been measuring neutrinos for decades while the theorists argued about their operators. We think that measuring a number and calling it an eigenvalue of "some operator " is meaningless. If and when an eigenvalues of a known operator appear in a data set, one obtains that fact that |*ψ* > will then be the corresponding eigenvector of *A*ˆ automatically. We also don't need to discuss "compatible and incompatible" observers in terms of operators that commute. In this century everyone knows how that math works.

#### *5.1.2. Bracket Relations Are Kinematic*

20 Advances in Quantum Mechanics

**5.1. Observables as numbers**

< *A*ˆ > as the number from the map

then set < *ψ*|*ψ* >→ 1 to simplify expressions.

*5.1.1. Eliminating More Postulates*

turns to clay.

complete set of operators, which is defined by *tr*(*A*ˆ †

*<sup>ρ</sup>* <sup>=</sup> ∑ *j*

An *observable* is a number extracted from the wave function as a projective representation, meaning that *ψ* ≡ *zψ* for any complex *z*. This symmetry exists in the equation of motion. We wish to call attention to it as a feature of "quantum homogeneity." It makes the information in |*<sup>ψ</sup>* > equivalent to the information in the density matrix *ρψ* = |*<sup>ψ</sup>* >< *<sup>ψ</sup>*|/ < *<sup>ψ</sup>*|*<sup>ψ</sup>* >. (The bracket notation, suppressed before to reduce clutter, is useful here.) Equivalence exists because if *ρψ* is given then its unit eigenvector predicts |*<sup>ψ</sup>* > up to a constant *<sup>z</sup>*. (Later we discover that density matrices of rank-1 are too special to base the full theory upon, but that comes with development of quantum probability, which is not yet under discussion.)

It is convenient to extract a number using an "operator sandwich". We define an observable

The last relation is general for cases where *ρ* is not so simple as *rank* − 1. Note we need nothing from quantum probability to make the map. Instead of prescribing < *ψ*|*ψ* > with a normalization postulate, we maintain it is simply unobservable, and drops out. We may

> *<sup>i</sup> <sup>ρ</sup>*) = ∑ *j*

Thus *ρ* is equivalent to a number of observables, and |*ψ* > is observable to the exact extent it is defined. At some point this simple relation seems to have been re-packaged as "quantum holography." The very late date of realizing the wave function is observable (to the extent it is defined) supports our case that quantum mechanics is still "emerging" from its history.

There is no particular reason for us to postulate that *A*ˆ must be Hermitian. If it is not Hermitian the operator sandwich gives a complex number, equivalent to two real numbers and two observables, because any operator is the sum of a Hermitian operator and *i* times a Hermitian operator. As for complex numbers being observable in the lab, mathematics tells us that complex numbers are real pairs with 2-vector addition and multiplication rules. It is not unusual to observe such number pairs that have phase relations like *qi*, *pi* which map directly into *ψi*, a complex number. So there is no reason for the 18th century trick of scaring people with complex numbers. And yet: given that *ρ* is Hermitian no harm is done by restricting the operators for observables to being Hermitian. One more grand postulate

*At least in our approach*, those matters of definition need no *foundation postulates.* Neither is there a good reason to insist that an observable be an eigenvalue of some especially known

<sup>&</sup>lt; *<sup>ψ</sup>*|*<sup>ψ</sup>* <sup>&</sup>gt; <sup>=</sup> *tr*(*A*ˆ*ρψ*)

< *A*ˆ*<sup>j</sup>* > *A*ˆ*j*.

*tr*(*ρψ*) .

*<sup>i</sup> <sup>A</sup>*ˆ*j*) = *<sup>δ</sup>ij*. Since the set is complete,

*<sup>i</sup>* be a normalized

<sup>|</sup>*<sup>ψ</sup>* <sup>&</sup>gt;→<sup>&</sup>lt; *<sup>A</sup>*<sup>ˆ</sup> <sup>&</sup>gt;<sup>=</sup> <sup>&</sup>lt; *<sup>ψ</sup>*|*A*ˆ|*<sup>ψ</sup>* <sup>&</sup>gt;

The trace (symbol *tr*) acts as an inner product between operators. Let *A*ˆ

*Ajtr*(*A*ˆ †

The quantum map shows that complex *ψ<sup>i</sup>* are canonical coordinates, up to *i*:

$$\psi\_i = (q\_i + ip\_i) / \sqrt{2}.$$

A factor of *i* is tolerable, in that Hamilton's equations are recognizable including it:

$$i\dot{\psi} = \frac{\partial H}{\partial \psi\_i^\*}, \quad i\dot{\psi}^\* = -\frac{\partial H}{\partial \psi\_i}.$$

Thus *<sup>i</sup>ψ*<sup>∗</sup> *<sup>i</sup>* is the canonical momentum conjugate to *ψi*. Poisson brackets (*PB*) are canonically invariant, so that transcribing them to *<sup>ψ</sup>*, *<sup>ψ</sup>*<sup>∗</sup> involves only a factor of *<sup>i</sup>*.

In fact the *PB* relations among our observable are rather simple due to the decision to make observables bilinear in *<sup>ψ</sup>*, *<sup>ψ</sup>*∗. The most famous application was the early desire to project the wave function into a few particle-like observables *<sup>Q</sup>*� , � *P*, mistakenly thought to be important from Newtonian bias. For *<sup>Q</sup>*� to represent a translational coordinate it must transform properly:

$$\begin{aligned} \text{given} \quad \psi(\mathbf{x}) \to \psi\_{\vec{a}}(\vec{x}) = \psi(\vec{x} - \vec{a}), \\ \text{then} \quad \vec{Q} = d^3 \mathbf{x} \, \psi^\* \hat{Q} \boldsymbol{\psi} \to \vec{Q} \to \vec{Q} + \vec{a}. \end{aligned}$$

There are few choices but *<sup>Q</sup>*� <sup>=</sup> �*x*. The test that a candidate variable *Pi* is conjugate to these *Qi* needs the bracket {*Qi*, *Pj*}*PB* = *<sup>δ</sup>ij*.

Write this out, assuming operator sandwiches:

$$\{Q\_{i\prime}, P\_{\dot{j}}\}\_{PB} = -i \sum\_{\chi} \left( \frac{\delta Q\_{\dot{i}}}{\delta \psi\_{\chi}} \frac{\delta P\_{\dot{j}}}{\delta \psi\_{\chi}^\*} - \frac{\delta P\_{\dot{j}}}{\delta \psi\_{\chi}} \frac{\delta Q\_{\dot{i}}}{\delta \psi\_{\chi}^\*} \right) .$$

Computing the derivatives gives

$$\{Q\_{i\nu} \: P\_{\bar{j}}\}\_{PB} = -i \int d\mathbf{x} \, \psi\_{\mathbf{x}}^\* [\hat{Q}\_{i\nu} \: \hat{P}\_{\bar{j}}] \psi\_{\mathbf{x}}.$$

This must be true for all *ψ*. Quantum homogeneity (the irrelevance of < *ψ*|*ψ* >) then obtains the map between the operator algebra and Poisson bracket

$$\{Q\_{\dot{\nu}}, P\_{\dot{\jmath}}\}\_{PB} = \delta\_{\dot{\imath}\dot{\jmath}} \rightarrow [\hat{Q}\_{\dot{\imath}}, \hat{P}\_{\dot{\jmath}}] = \mathrm{i}\delta\_{\dot{\imath}\dot{\jmath}}\tag{31}$$

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information exists in the algebra than in the direct and simple model of waves. For one thing, the algebra is kinematic, and will work for any Hamiltonian, including non-local ones that do not seem to be observed. In comparison the wave model predicts the algebra, because

It is sometimes thought that "field quantization" proves that quantization principles are a golden road. But what's kinematic on one space is kinematic on a larger space. If one believes there should be wave functions for classical fields, one defines quantum field theory straightforwardly. It also happens to be equivalent to the space made from products of an arbitrary number of beginning quantum systems, which is neat, but which again shows that

Finally, we find there is a perception that abstract operator methods are superior just because they are difficult. It is seldom noticed that an unlimited amount of tortuous and clever operator manipulation can never have more information than just solving the differential equation, which predicts everything, and (in fact) all the differential equations of quantum

These are reason we wrote in the Introduction that the viable models are a higher

In our approach the state space is not going to be predicted by a simplistic algebraic transcription. Finite dimensional quantum models are known and hardly useless: they are models of spin, and molecular rotational dynamics. Finite dimensional models of quantum field theory are known. They are called "lattice theories." The dimensionality of quantum models has no restriction. Leaving the dimension free to grow without limit is

In this Section we explore the origins of probability in our approach. Quantum probability is an old subject with many contributions we cannot possibly review. There is some agreement that the Born rule should be "a Theorem, not a Principle"[14], although our approach is not quite the same. In the first place we must dismiss a common misconception that quantum probability contradicts classical probability, or is inconsistent with it, by defining each.

Probability itself is a subtle topic. It is not well-defined until "objects" are categorized for the purposes of probability. Classical probability (*CP*) of frequentist kind is about classifying objects into mutually exclusive (*me*) equivalence classes3, and assigning numbers to the information by counting. Distributions are a useful tool of *CP*. Quantum probability (*QP*) allows such classifications but does not insist on them. Instead *QP* is a projective map from a system's state, represented by density matrix, into a number. Distributions sometimes exist

<sup>3</sup> Although "equivalence classes" are often mutually exclusive sets by definition, we use the term more broadly, and

what transpired, and one of the reasons the subject is so flexible it cannot fail.

it contains everything, so it is superior.

invoking the quantization principle was redundant.

mechanics are already "solved" by Eq. 20.

accomplishment than framework.

**6. Quantum probability**

in *QP*, but are not always compatible.

add *me* when the term is intended.

**6.1. Define terms**

The other consistency relations of "quantization" are similar. A *PB* algebra predicts a commutator algebra, as follows: If {*A*, *<sup>B</sup>*}*PB* = *<sup>C</sup>* is true for general |*<sup>ψ</sup>* >, and all quantities are operator sandwiches, then [*A*ˆ, *B*ˆ] = *iC*ˆ follows by identity. An interesting application comes from the lack of any non-zero commutator with the unit operator "1." It tells us that < *ψ*|*ψ* > does not transform, time evolve, nor give a non-trivial result, so it is a conserved "momentum" of the theory that drops out as unobservable.2

The early history of quantum theory found the map between Poisson brackets and commutators profound, and it tends to still be viewed that way. In our approach it emerges on its own as useful, but automatic. If one chooses operators that satisfy the bracket-commutator rules, then their observables transform as they are intended, and vice-versa. It is not really necessary to cast around and discover operators by trial and error. Noether's Theorem will manufacture any number of generalized conjugate *Pi* from point transformations on *Qi* as the "charges" of conserved (or un-conserved) currents[7].

It is interesting that in retrospect the original Heisenberg program, based on Poisson brackets, guaranteed such an outcome. The virtue of bracket relations lies in generating Lie algebra and related relations that are inherently coordinate-free. Once a given algebra is transcribed to a different *notation*, it is not going to produce new results. Thus when Heisenberg transcribed the Poisson bracket algebra of Hamiltonian time evolution to a different notation he was building up a classical Hamiltonian dynamics of ordinary kind, if it was not recognized at the time.

The big advance, as mentioned before, comes with the physical model of electrons having an infinite number of *do f* , as Eq. 31 requires, and as found in a *wave theory*. That fact was supposed to be evident in the spectrum of atoms showing a (practically) infinite number of normal mode frequencies. That in turn could have been done by 19th century classical physicists, who knew about spectra and normal modes. And indeed Stokes, Kelvin and Lorentz[9] all deduced the facts that atoms are vibrating jello from such clues before 1900. Lacking any technology to test the speculation, they made little of it which is a pity. Immediately *after* 1900, the cult of the quantum of action went the way of postulating mistakes that could not be expressed without Planck's constant. *All of that can be dropped.*

#### 5.1.2.1. Quantization

The *PB*-commutator relation of Eq. 31 is commonly called the "quantization" principle according to the recipe of Heisenberg or Dirac, which (being a postulate) cannot be explained. While that is what those gentlemen believed, it is not our approach.

Once the physicists have committed to a linear, Hamiltonian theory, there is very little left to determine except its dimension. As already mentioned Eq. 31 realized with *xi*, *<sup>∂</sup><sup>j</sup>* = −*i∂*/*∂xi* requires a space of a continuously infinite number of degrees of freedom: waves. Less

<sup>2</sup> Similarly, the center of mass momentum of the Universe in Newtonian physics is unobservable.

information exists in the algebra than in the direct and simple model of waves. For one thing, the algebra is kinematic, and will work for any Hamiltonian, including non-local ones that do not seem to be observed. In comparison the wave model predicts the algebra, because it contains everything, so it is superior.

It is sometimes thought that "field quantization" proves that quantization principles are a golden road. But what's kinematic on one space is kinematic on a larger space. If one believes there should be wave functions for classical fields, one defines quantum field theory straightforwardly. It also happens to be equivalent to the space made from products of an arbitrary number of beginning quantum systems, which is neat, but which again shows that invoking the quantization principle was redundant.

Finally, we find there is a perception that abstract operator methods are superior just because they are difficult. It is seldom noticed that an unlimited amount of tortuous and clever operator manipulation can never have more information than just solving the differential equation, which predicts everything, and (in fact) all the differential equations of quantum mechanics are already "solved" by Eq. 20.

These are reason we wrote in the Introduction that the viable models are a higher accomplishment than framework.

In our approach the state space is not going to be predicted by a simplistic algebraic transcription. Finite dimensional quantum models are known and hardly useless: they are models of spin, and molecular rotational dynamics. Finite dimensional models of quantum field theory are known. They are called "lattice theories." The dimensionality of quantum models has no restriction. Leaving the dimension free to grow without limit is what transpired, and one of the reasons the subject is so flexible it cannot fail.

### **6. Quantum probability**

In this Section we explore the origins of probability in our approach. Quantum probability is an old subject with many contributions we cannot possibly review. There is some agreement that the Born rule should be "a Theorem, not a Principle"[14], although our approach is not quite the same. In the first place we must dismiss a common misconception that quantum probability contradicts classical probability, or is inconsistent with it, by defining each.

### **6.1. Define terms**

22 Advances in Quantum Mechanics

recognized at the time.

5.1.2.1. Quantization

This must be true for all *ψ*. Quantum homogeneity (the irrelevance of < *ψ*|*ψ* >) then obtains

The other consistency relations of "quantization" are similar. A *PB* algebra predicts a commutator algebra, as follows: If {*A*, *<sup>B</sup>*}*PB* = *<sup>C</sup>* is true for general |*<sup>ψ</sup>* >, and all quantities are operator sandwiches, then [*A*ˆ, *B*ˆ] = *iC*ˆ follows by identity. An interesting application comes from the lack of any non-zero commutator with the unit operator "1." It tells us that < *ψ*|*ψ* > does not transform, time evolve, nor give a non-trivial result, so it is a conserved

The early history of quantum theory found the map between Poisson brackets and commutators profound, and it tends to still be viewed that way. In our approach it emerges on its own as useful, but automatic. If one chooses operators that satisfy the bracket-commutator rules, then their observables transform as they are intended, and vice-versa. It is not really necessary to cast around and discover operators by trial and error. Noether's Theorem will manufacture any number of generalized conjugate *Pi* from point transformations on *Qi* as the "charges" of conserved (or un-conserved) currents[7]. It is interesting that in retrospect the original Heisenberg program, based on Poisson brackets, guaranteed such an outcome. The virtue of bracket relations lies in generating Lie algebra and related relations that are inherently coordinate-free. Once a given algebra is transcribed to a different *notation*, it is not going to produce new results. Thus when Heisenberg transcribed the Poisson bracket algebra of Hamiltonian time evolution to a different notation he was building up a classical Hamiltonian dynamics of ordinary kind, if it was not

The big advance, as mentioned before, comes with the physical model of electrons having an infinite number of *do f* , as Eq. 31 requires, and as found in a *wave theory*. That fact was supposed to be evident in the spectrum of atoms showing a (practically) infinite number of normal mode frequencies. That in turn could have been done by 19th century classical physicists, who knew about spectra and normal modes. And indeed Stokes, Kelvin and Lorentz[9] all deduced the facts that atoms are vibrating jello from such clues before 1900. Lacking any technology to test the speculation, they made little of it which is a pity. Immediately *after* 1900, the cult of the quantum of action went the way of postulating mistakes that could not be expressed without Planck's constant. *All of that can be dropped.*

The *PB*-commutator relation of Eq. 31 is commonly called the "quantization" principle according to the recipe of Heisenberg or Dirac, which (being a postulate) cannot be explained.

Once the physicists have committed to a linear, Hamiltonian theory, there is very little left to determine except its dimension. As already mentioned Eq. 31 realized with *xi*, *<sup>∂</sup><sup>j</sup>* = −*i∂*/*∂xi* requires a space of a continuously infinite number of degrees of freedom: waves. Less

While that is what those gentlemen believed, it is not our approach.

<sup>2</sup> Similarly, the center of mass momentum of the Universe in Newtonian physics is unobservable.

*<sup>j</sup>*] = *iδij* (31)

{*Qi*, *Pj*}*PB* = *<sup>δ</sup>ij* → [*Q*ˆ*i*, *<sup>P</sup>*<sup>ˆ</sup>

the map between the operator algebra and Poisson bracket

"momentum" of the theory that drops out as unobservable.2

Probability itself is a subtle topic. It is not well-defined until "objects" are categorized for the purposes of probability. Classical probability (*CP*) of frequentist kind is about classifying objects into mutually exclusive (*me*) equivalence classes3, and assigning numbers to the information by counting. Distributions are a useful tool of *CP*. Quantum probability (*QP*) allows such classifications but does not insist on them. Instead *QP* is a projective map from a system's state, represented by density matrix, into a number. Distributions sometimes exist in *QP*, but are not always compatible.

<sup>3</sup> Although "equivalence classes" are often mutually exclusive sets by definition, we use the term more broadly, and add *me* when the term is intended.

The breakthrough of quantum probability, we believe, lies in generalizing the notion of probability so as not to *insist* on pre-ordained equivalence classes. Vectors are categorized so they have a great chance of being nearly equivalent. Physicists seem to believe that "quantum objects" from Nature are needed to make sense of quantum probability, and vice-versa. But nothing from physics is involved in developing an accounting system where different vectors are not automatically treated as mutually exclusive. While we will extensively use physics and its examples, the ultimate goal of this Section is not to depend on physics.

#### **6.2. Discovering quantum probability with a hidden-variable map**

Begin with a remarkably simple map from classical to quantum probability which illustrates the necessary class ideas. Let |*D* > be a big vector we call "data". By means which are quite arbitrary it is partitioned in a collection of smaller vectors *<sup>D</sup><sup>J</sup> <sup>i</sup>* =< *i J*|*<sup>D</sup>* > with names *J* = 1...*Jmax* and components *i* = 1...*imax*. We tentatively interpret *J* as labeling the sample number taken from some (deterministic or random) process. The other index is interpreted as describing "objects". The sample space and object space are tentative because certain operations will mix them, as we will see. Formally *<sup>D</sup><sup>J</sup> <sup>i</sup>* exists on the direct product of spaces of dimension *imax* ⊗ *Jmax*, upon which there are certain transformation groups and invariants. For convenience the record is normalized < *D*|*D* >= 1 in the usual way, removing one number set aside.

Now we seek a notion of orderliness or physical regularity. We will expand the vectors in an orthonormal basis set {|*e<sup>α</sup>* >}, where *e<sup>α</sup> <sup>i</sup>* =< *i*|*α* >, and seek some form of statistical repetition. The basis matters, so which basis is used? Every data record actually has two preferred bases, in which the expansion is diagonal:

$$\left| D \right> = \left| \mathbf{e}^{\mathfrak{a}} \right> \Lambda^{\mathfrak{a}} \left| \mathbf{s}^{\mathfrak{a}} \right>. \tag{32}$$

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singular values are positive real numbers, Λ*<sup>α</sup>* > 0. The singular values are invariant with the data is transformed by arbitrary and different unitary transformations on the object and

The interpretation of each term |*s<sup>α</sup>* > |*e<sup>α</sup>* > summed in Eq. 32 is a *strict correlation* of a unique

**Example** Suppose the data consists of integer numbers of objects that are |*apple* > or |*orange* >. This is classical *me* data: by existing in different spaces, < *apple*|*orange* >= 0. Sufficiently fine sampling will produce samples which are either 1 or 0. Typical data is then

> |*D* >= (|*apple* >, 0, |*orange* >, |*apple* >, |*orange* >, ... |*apple* >).

The expression only makes sense if these *me* objects are normalized, < *apple*|*apple* >= 1, and so on, else the normalization would conflict with the number of apples. Expand in the


its sample vector, which is automatically orthogonal to all the other sample vectors:

The diagonal form of *svd* has appeared, up to a normalization. Whenever data consists of disjoint *me* objects, one can show those same objects are *automatically* the *svd* factors. The fact of strict correlation comes with projecting onto one object such as |*apple* > and producing

< *apple*|*D* >= (1, 0, 0, 1, 0, . . . 1).

Conversely, selecting one of the *me* sampling histories automatically selects a unique *object*.

the total number of apples observed. Let *Ntot* be the total of apples and oranges. Remember

*Napple*/*Ntot*|*sapple* > |*apple* >

*Norange*/*Ntot*|*s*

*Ntot*/*Napple*

*Napple*/*Ntot*|*sapple* > . (35)

*orange* > |*orange* > .

+ |*orange* > (0, 0, 1, 0, 1, . . . 0). (34)

*s<sup>α</sup>* | *s<sup>β</sup>* 

= *δαβ*. Let *Napple* be

object vector labeled *α* with a unique sample vector labeled *α*.

sample spaces.

natural basis we started with,

These are features of classical "events."

that we normalized our data. Then

= 

(1, 0, 0, 1, 0, . . . 1) →

Once normalized we can read off the singular values:


> +

To reach the *svd* form implies samples that are normalized:

*Napple*/*Ntot*(1, 0, 0, 1, 0, . . . 1)/

This is the singular value decomposition (*svd*), which is unique. It is proven by diagonalizing two correlations (matrices) that automatically have positive real eigenvalues:

$$DD^\dagger = \sum\_{\mathfrak{a}} |e^{\mathfrak{a}} > (\Lambda^{\mathfrak{a}})^2 < e^{\mathfrak{a}}|;\tag{33}$$

$$D^\dagger D = \sum\_{\mathfrak{a}} |s^{\mathfrak{a}} > (\Lambda^{\mathfrak{a}})^2 < s^{\mathfrak{a}}|.$$

Notice that the decomposition yields vectors which are orthonormal on their respective spaces. Notice that the vectors are defined up to a symmetry:

$$\begin{aligned} \vert e\_{\mathfrak{a}} > \to z\_{\mathfrak{e}} \vert e\_{\mathfrak{a}} >; \\ \vert \mathbf{s}^{\mathfrak{a}} > \to z\_{\mathfrak{s}} \vert \mathbf{s}^{\mathfrak{a}} >; \\ \vert D > \to \vert D > . \end{aligned}$$

Here *ze*, *zs* are arbitrary complex numbers. The factor vectors are eigenvectors which have no normalization except the normalization given by convention. By phase convention the

singular values are positive real numbers, Λ*<sup>α</sup>* > 0. The singular values are invariant with the data is transformed by arbitrary and different unitary transformations on the object and sample spaces.

24 Advances in Quantum Mechanics

number set aside.

The breakthrough of quantum probability, we believe, lies in generalizing the notion of probability so as not to *insist* on pre-ordained equivalence classes. Vectors are categorized so they have a great chance of being nearly equivalent. Physicists seem to believe that "quantum objects" from Nature are needed to make sense of quantum probability, and vice-versa. But nothing from physics is involved in developing an accounting system where different vectors are not automatically treated as mutually exclusive. While we will extensively use physics

Begin with a remarkably simple map from classical to quantum probability which illustrates the necessary class ideas. Let |*D* > be a big vector we call "data". By means which are

*J* = 1...*Jmax* and components *i* = 1...*imax*. We tentatively interpret *J* as labeling the sample number taken from some (deterministic or random) process. The other index is interpreted as describing "objects". The sample space and object space are tentative because certain

of dimension *imax* ⊗ *Jmax*, upon which there are certain transformation groups and invariants. For convenience the record is normalized < *D*|*D* >= 1 in the usual way, removing one

Now we seek a notion of orderliness or physical regularity. We will expand the vectors in

repetition. The basis matters, so which basis is used? Every data record actually has two

This is the singular value decomposition (*svd*), which is unique. It is proven by diagonalizing

Notice that the decomposition yields vectors which are orthonormal on their respective



Here *ze*, *zs* are arbitrary complex numbers. The factor vectors are eigenvectors which have no normalization except the normalization given by convention. By phase convention the

*<sup>α</sup>* >→ *zs*|*s*

*α*� Λ*α* |*s*

*<sup>α</sup>* > (Λ*α*)<sup>2</sup> < *e*

*<sup>α</sup>* > (Λ*α*)<sup>2</sup> < *s*

*<sup>α</sup>* >;

*α*|.


*α* |*e*

*α* |*s*


two correlations (matrices) that automatically have positive real eigenvalues:

*DD*† = ∑

*<sup>D</sup>*†*<sup>D</sup>* = ∑

spaces. Notice that the vectors are defined up to a symmetry:

*<sup>i</sup>* =< *i J*|*<sup>D</sup>* > with names

*<sup>i</sup>* exists on the direct product of spaces

*<sup>α</sup>*�. (32)

*<sup>α</sup>*|; (33)

*<sup>i</sup>* =< *i*|*α* >, and seek some form of statistical

and its examples, the ultimate goal of this Section is not to depend on physics.

**6.2. Discovering quantum probability with a hidden-variable map**

quite arbitrary it is partitioned in a collection of smaller vectors *<sup>D</sup><sup>J</sup>*

operations will mix them, as we will see. Formally *<sup>D</sup><sup>J</sup>*

an orthonormal basis set {|*e<sup>α</sup>* >}, where *e<sup>α</sup>*

preferred bases, in which the expansion is diagonal:

The interpretation of each term |*s<sup>α</sup>* > |*e<sup>α</sup>* > summed in Eq. 32 is a *strict correlation* of a unique object vector labeled *α* with a unique sample vector labeled *α*.

**Example** Suppose the data consists of integer numbers of objects that are |*apple* > or |*orange* >. This is classical *me* data: by existing in different spaces, < *apple*|*orange* >= 0. Sufficiently fine sampling will produce samples which are either 1 or 0. Typical data is then

$$\begin{aligned} \left| D > = (|apple>\,, 0\,\vert \, or \,\text{angle}>\,, \, |apple>\,\vert \, or \,\text{angle}>\,\prime) \right| \\ \ldots \, & \vert \, & \vert \, 1 \, (apple>\,\text{--}\, \,) \end{aligned}$$

The expression only makes sense if these *me* objects are normalized, < *apple*|*apple* >= 1, and so on, else the normalization would conflict with the number of apples. Expand in the natural basis we started with,

$$\begin{split} |D| &>= |
alpha| > (1, 0, 0, 1, 0, \dots 1) \\ &+ |
alpha \text{age} > (0, 0, 1, 0, 1, \dots 0). \end{split} \tag{34}$$

The diagonal form of *svd* has appeared, up to a normalization. Whenever data consists of disjoint *me* objects, one can show those same objects are *automatically* the *svd* factors. The fact of strict correlation comes with projecting onto one object such as |*apple* > and producing its sample vector, which is automatically orthogonal to all the other sample vectors:

$$ = (1,0,0,1,0,\dots1).$$

Conversely, selecting one of the *me* sampling histories automatically selects a unique *object*. These are features of classical "events."

To reach the *svd* form implies samples that are normalized: *s<sup>α</sup>* | *s<sup>β</sup>* = *δαβ*. Let *Napple* be the total number of apples observed. Let *Ntot* be the total of apples and oranges. Remember that we normalized our data. Then

$$\begin{aligned} & (1, \, 0, 0, \, 1, \, 0, \, \dots \, 1) \to \\ & \sqrt{\mathrm{Napple}/N\_{\mathrm{tot}}} (1, \, 0, \, 0, \, 1, \, 0, \, \dots \, 1) / \sqrt{\mathrm{N}\_{\mathrm{tot}}/N\_{\mathrm{apple}}} \\ &= \sqrt{\mathrm{Napple}/N\_{\mathrm{tot}}} |s^{\mathrm{apple}}> \,. \end{aligned}$$

Once normalized we can read off the singular values:

$$\begin{split} |D| &> = \sqrt{N\_{approx}/N\_{tot}} |s^{approx}|s^{approx}> |
alpha> \\ &\quad + \sqrt{N\_{orange}/N\_{tot}} |s^{orange}>|
alpha &> \dots \\ \end{split}$$

Now suppose there is a unitary transformation of our data on either object of sample or both spaces - but not mixing them. The result will involve linear combinations of the form *α*|*apple* > +*β*|*orange* >, which is classically "taboo." For better or worse, we cannot stop linear transformations from being used or being useful. Whatever the coordinate system, we can construct the *svd* factors and singular values as invariants.

In many cases (and always in physics) we are actually forced to suppress some detail by summing over unwanted or unrecorded details of the sampling history. That was already done in Eq. 33. We use that observation as the first example in constructing the *density matrix ρobject* of the object system:

$$\rho\_{object} = tr\_{\mathfrak{s}}(|D| > < D|) = \sum\_{\mathfrak{a}} |\mathfrak{e}^{\mathfrak{a}} > (\Lambda^{\mathfrak{a}})^2 < \mathfrak{e}^{\mathfrak{a}}|.$$

The form *ρobject* we traced-out the sample history. This identity now *defines* the *me* |*object<sup>α</sup>* >= |*e<sup>α</sup>* >. By construction, whenever *me* data is used, we have a convenient invariant formula for the probability of finding such an object:

$$P(|e\_{\mathfrak{A}} > |\rho\_{\text{object}}) = <\text{ e}\_{\mathfrak{A}}|\rho\_{\text{object}}|e\_{\mathfrak{A}} > = \text{tr}(\rho\_{\text{object}}|e\_{\mathfrak{A}} > $$

The symbol *<sup>P</sup>*(|*e<sup>α</sup>* > |*ρobject*) is read as the probability of |*e<sup>α</sup>* > given *<sup>ρ</sup>object*, and exactly coincides with counting numbers: thus

$$P(|apple>\ | \rho) = (\sqrt{\mathcal{N}\_{apple}/\mathcal{N}\_{tot}})^2 = \mathcal{N}\_{apple}/\mathcal{N}\_{tot}.$$

In general form the probability *P* to get an observable < *A*ˆ > is

$$P = \operatorname{tr}(\rho \hat{A}) / \operatorname{tr}(\rho). \tag{36}$$

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In the form above we have both the events and the sample history. The samples are not mutually exclusive, but naturally fall into the corresponding projections. That can be called the "underlying reality." Meanwhile there exists a unitary transformation on the objects and samples where this arbitrary data will be a sum of strictly correlated, *me* elements which are indistinguishable from classical events. The difference between that interpretation and the quantum-style one is a coordinate transformation not available from the density matrix, so it

Normalize the sample history and take the trace over the sample space to form the density matrix. Upon reaching that level, one cannot distinguish the system from one where every event actually was a product of mutually exclusive object and sample vectors. This information is deliberately lost in forming the data categories. That makes it consistent and unique to define probability using the Born rule. No subsequent experiment can make it false. Probabilities defined by naive counting of integer-valued data will agree exactly.

"Outcome dependence" is the name given to statistics that depend on the order of measurement. Our procedure has outcome dependence. Suppose one is selecting channels by simple filters. A "measurement" *β* uses a state |*β* > and an associated projector

*<sup>P</sup>*(*β*, *<sup>γ</sup>*, ...*η*|*ρ*) = *tr*(*πη*...*πγπβρπβπγ*...*πη*).

The projective and non-commutative nature of this kind of probability is self-evident. It follows immediately that no classical distributions can reproduce this kind of probability in

The famous Bell inequalities[15] dramatize this fact, yet there was nothing new in finding that distributions fail in general. It would have been extraordinary for probability based on density matrix projections to be equivalent to distributions in the first place. Years after Bell, Werner[16] formulated the criteria called "separability" of density matrices, which when true allows *QP* to coincide with *CP* as formulated with distributions. Thus the classical probability rules exist inside of *QP*, when and if a special case happens to occur. Conversely *QP* in our approach is a more general extension of the concepts and rules of *CP* that does not contradict any of it. We will argue that *QP* is so general there are no restrictions on how

**About Disturbing Measurements:** It is a geometrical fact that any vector can be considered to be any other vector, plus the difference. It takes one step to make the difference orthogonal,


The term in braces is orthogonal to |*b* >, given normalized vectors. Up to an overall scale, the vector |*b* > that is literally pre-existing in vector |*a* > is |*b* >< *b*|*a* >. This decomposition, of course, does not come from physics. We see that agreeing vectors are" equivalent" up to

a scale is a prelude to counting them as equivalent for probability purposes.

*πβ* <sup>=</sup> <sup>|</sup>*<sup>β</sup>* >< *<sup>β</sup>*|. A series of measurements *<sup>β</sup>*, *<sup>γ</sup>*, *<sup>η</sup>*... in that order, yields

becomes meaningless.

*6.3.1. Outcome Dependence*

general.

it might be used.

by writing

#### **6.3. The quantum-style agreement**

We propose an agreement on how data will be managed: *we agree to describe a system using its density matrix*. We give up the possibility of keeping more information, because it is efficient not to have it.

Review the *apples* − *oranges* discussion with a physical example where transformations are natural. Suppose observations consist of events with 3-vector polarizations |E >. Moreover, only two orthogonal components |*e*<sup>1</sup> >, |*e*<sup>2</sup> > are measured. A generic data set including the sampling history will not generally consist of *me* events, but combinations of the form

$$|D> = (|\mathcal{E}\_1>, 0|\mathcal{E}\_2>, \dots|\mathcal{E}\_n>)$$

which is expanded in the basis

$$\begin{aligned} |D> = |\mathfrak{e}\_1> & (<\mathfrak{e}\_1|\mathfrak{E}\_1>, <\mathfrak{e}\_1|\mathfrak{E}\_{12}> \dots <\mathfrak{e}\_1|\mathfrak{E}\_n>) \\ &+ |\mathfrak{e}\_2> (<\mathfrak{e}\_2|\mathfrak{E}\_1>, <\mathfrak{e}\_2|\mathfrak{E}\_2>, \dots <\mathfrak{e}\_2|\mathfrak{E}\_n>). \end{aligned}$$

In the form above we have both the events and the sample history. The samples are not mutually exclusive, but naturally fall into the corresponding projections. That can be called the "underlying reality." Meanwhile there exists a unitary transformation on the objects and samples where this arbitrary data will be a sum of strictly correlated, *me* elements which are indistinguishable from classical events. The difference between that interpretation and the quantum-style one is a coordinate transformation not available from the density matrix, so it becomes meaningless.

Normalize the sample history and take the trace over the sample space to form the density matrix. Upon reaching that level, one cannot distinguish the system from one where every event actually was a product of mutually exclusive object and sample vectors. This information is deliberately lost in forming the data categories. That makes it consistent and unique to define probability using the Born rule. No subsequent experiment can make it false. Probabilities defined by naive counting of integer-valued data will agree exactly.

#### *6.3.1. Outcome Dependence*

26 Advances in Quantum Mechanics

*ρobject* of the object system:

Now suppose there is a unitary transformation of our data on either object of sample or both spaces - but not mixing them. The result will involve linear combinations of the form *α*|*apple* > +*β*|*orange* >, which is classically "taboo." For better or worse, we cannot stop linear transformations from being used or being useful. Whatever the coordinate system, we

In many cases (and always in physics) we are actually forced to suppress some detail by summing over unwanted or unrecorded details of the sampling history. That was already done in Eq. 33. We use that observation as the first example in constructing the *density matrix*

The form *ρobject* we traced-out the sample history. This identity now *defines* the *me* |*object<sup>α</sup>* >= |*e<sup>α</sup>* >. By construction, whenever *me* data is used, we have a convenient invariant

*<sup>P</sup>*(|*e<sup>α</sup>* > |*ρobject*) =< <sup>e</sup>*α*|*ρobject*|*e<sup>α</sup>* >= *tr*(*ρobject*|*e<sup>α</sup>* >< *<sup>e</sup>α*|).

The symbol *<sup>P</sup>*(|*e<sup>α</sup>* > |*ρobject*) is read as the probability of |*e<sup>α</sup>* > given *<sup>ρ</sup>object*, and exactly

We propose an agreement on how data will be managed: *we agree to describe a system using its density matrix*. We give up the possibility of keeping more information, because it is efficient

Review the *apples* − *oranges* discussion with a physical example where transformations are natural. Suppose observations consist of events with 3-vector polarizations |E >. Moreover, only two orthogonal components |*e*<sup>1</sup> >, |*e*<sup>2</sup> > are measured. A generic data set including the sampling history will not generally consist of *me* events, but combinations of the form



+ |*e*<sup>2</sup> > (< *<sup>e</sup>*2|E<sup>1</sup> >, < *<sup>e</sup>*2|E<sup>2</sup> >, ... < *<sup>e</sup>*2|E*<sup>n</sup>* >).

*α* |*e*

*<sup>α</sup>* > (Λ*α*)<sup>2</sup> < *e*

*Napple*/*Ntot*)<sup>2</sup> = *Napple*/*Ntot*.

*P* = *tr*(*ρA*ˆ)/*tr*(*ρ*). (36)

*α*|.

can construct the *svd* factors and singular values as invariants.

formula for the probability of finding such an object:

*P*(|*apple* > |*ρ*)=(

In general form the probability *P* to get an observable < *A*ˆ > is

coincides with counting numbers: thus

**6.3. The quantum-style agreement**

which is expanded in the basis

not to have it.

*<sup>ρ</sup>object* <sup>=</sup> *trs*(|*<sup>D</sup>* >< *<sup>D</sup>*|) = ∑

"Outcome dependence" is the name given to statistics that depend on the order of measurement. Our procedure has outcome dependence. Suppose one is selecting channels by simple filters. A "measurement" *β* uses a state |*β* > and an associated projector *πβ* <sup>=</sup> <sup>|</sup>*<sup>β</sup>* >< *<sup>β</sup>*|. A series of measurements *<sup>β</sup>*, *<sup>γ</sup>*, *<sup>η</sup>*... in that order, yields

$$P(\beta, \gamma, \ldots \eta | \rho) = \text{tr}(\pi\_{\eta} \ldots \pi\_{\gamma} \pi\_{\beta} \rho \pi\_{\beta} \pi\_{\gamma} \ldots \pi\_{\eta})\,.$$

The projective and non-commutative nature of this kind of probability is self-evident. It follows immediately that no classical distributions can reproduce this kind of probability in general.

The famous Bell inequalities[15] dramatize this fact, yet there was nothing new in finding that distributions fail in general. It would have been extraordinary for probability based on density matrix projections to be equivalent to distributions in the first place. Years after Bell, Werner[16] formulated the criteria called "separability" of density matrices, which when true allows *QP* to coincide with *CP* as formulated with distributions. Thus the classical probability rules exist inside of *QP*, when and if a special case happens to occur. Conversely *QP* in our approach is a more general extension of the concepts and rules of *CP* that does not contradict any of it. We will argue that *QP* is so general there are no restrictions on how it might be used.

**About Disturbing Measurements:** It is a geometrical fact that any vector can be considered to be any other vector, plus the difference. It takes one step to make the difference orthogonal, by writing

$$|a> = |b> < b|a> + (|a> - |b> < b|a>).$$

The term in braces is orthogonal to |*b* >, given normalized vectors. Up to an overall scale, the vector |*b* > that is literally pre-existing in vector |*a* > is |*b* >< *b*|*a* >. This decomposition, of course, does not come from physics. We see that agreeing vectors are" equivalent" up to a scale is a prelude to counting them as equivalent for probability purposes.

It is quite important (we repeat) that the coefficient < *b*|*a* > happens to measure the amount of vector |*a* > that is pre-existing and pre-aligned with |*b* >. It is very reasonable that a quiet, non-disturbing "physical measurement" of a system will filter out a pre-existing component of a vector variable, and pass it through undisturbed. Undisturbed filtering is precisely what happens in textbook discussions of polarizers, Stern-Gerlach, and diffraction gratings. We wish it would be mentioned it has nothing to do with ¯*h* (to repeat). Using the undisturbed, pre-existing projection also totally contradicts the old line of thinking that 'measurement" involves an uncontrollable disturbance of the system due to the finite quantum of action. Indeed if one believed that, the overlaps would change by the process of measurement, and the Born rule would fail.

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The rank of a density matrix obtained from reduction depends on how the reduction was done. Obtaining a *rank* − 1 reduced matrix is exceptional. Finding such systems in the laboratory requires great ingenuity. That is why we treat systems that can be described with wave functions as special cases. That may seem to put our dynamical framework discussing wave functions somewhat askew relative to the probabilistic one. Section 6.3.4 explains why

It is remarkable that Bell's artful introduction of distribution theory[15], which is inappropriate for quantum-style systems, led to a false perception that "hidden variables" had been excluded. The hidden variables of ordinary quantum mechanics are the physical degrees of freedom (wave function or density matrix projections) on those spaces the physicist ignored in setting up his oversimplified model. There are always such spaces in

There is no limit to the number of products or their dimensionality that can be used as "data" or "sample vectors." Partitioning vectors on any given space into a number of factors is a highly coordinate-dependent business. It is essentially a map from one sort of index to a number of composite indices, which can always be done linearly: *Dk* <sup>→</sup> *<sup>D</sup>abc*... <sup>=</sup> <sup>Γ</sup>*abc*... *<sup>k</sup> Dk*, where Γ is an array of constants. In a sufficiently large data vector the "sample" and "object" spaces can be re-configured in practically infinite variations. There is very little that is invariant about entanglement when we allow such freedom. Nature cannot possibly care

In early days the Hilbert spaces of single electrons or single photons were considered utterly fundamental. They were building blocks for lofty postulates that could not be explained. Yet quantum mechanics was long ago enlarged to develop quantum field theory (*QFT*). For some purposes *QFT* is considered to have no new information on quantum mechanics itself, while defining very complicated quantum models. Yet Nature has subtleties. Basic non-relativistic quantum mechanics is incapable of dealing with the very questions of causality and non-locality that cloud interpretation of measurements. Relativistic *QFT* deals with issues of causality directly in terms of correlation functions with very well-defined properties. On that basis *QFT* is the more fundamental topic: it is big enough to support

The relation of ordinary quantum mechanics to field theory is then developed by reduction, where unobserved *do f* are integrated out. As a result all of the phenomenology of ordinary quantum mechanics is subject to the hidden variables known to have been integrated over in developing density matrices that actually occur. This is ignored in ordinary quantum mechanics seeking by itself to be "fundamental." It is not logically consistent to ignore what is known. Now we have shown how naive probabilities of counting emerge from density matrix constructions integrating over quite arbitrary sample spaces. It is hard to escape the inference that the probability interpretation – which beginning quantum mechanics could not explain about its own framework – must certainly originate in reduction of interacting systems of *QFT* down to the experimentally crude probes developed in beginning quantum

**Infrared Example:** There exists certain pure states of the *QFT* called bare electrons: the quanta of a free field theory. If such a state actually participated in an experiment we

about these coordinate conventions: Pause to consider how it affects physics.

and how Hamiltonian time evolution remains relevant.

*6.3.3. What Are Those Hidden Variables?*

Nature.

realistic models.

mechanics.

#### *6.3.2. Division and Reduction*

The conventional approach to "quantum interacting systems" holds that system *A* "exists on" space *A*, system *B* "exists on" space *B*, and when they interact the joint system exists on the direct product of spaces *C* = *A* ⊗ *B*. This is physically puzzling, and we think backwards. Instead we use our idea of equivalence classes developed by partitioning information, or "division".

Given any vector on a space *C*, we may partition it into factor vectors on procedurally-defined spaces *A* and *B*, as used above. For example a vector of 40 dimensions can be written as the product of vectors on 4 dimensions and vectors on 10 dimensions, or products of 5 × 8, etc.

Division is particularly well-developed with Clebsch-series done in the inverse direction: discovering what smaller group representations can be composed to make a given bigger one. A more straightforward division groups a data vector's components into adjacent bins of sub-dimension *imax* and names *J*:

$$D\_a \to D\_{i\parallel} \qquad \mathbf{J} = \text{int}(a/i\_{\text{max}}); \qquad \mathbf{i} = \text{mod}(a, \mathbf{J})\_{\text{rev}}$$

where *int* takes the integer part, and *mod*(*a*, *J*) returns the remainder of *a*/*J*. Arnold's famous "cat map" is an example. The freedom to choose the bins and dimensions is very important.

Division is quite coordinate-dependent. Division can be repeated to divide the factors, and make subdivisions. The process of vector "division" is not profound mathematics, but the arbitrariness is important for physics and data manipulation. Given a particular division, the physicist (knowingly or not) inspects the decomposition searching for simplicity and regularity to "emerge". The factor-states that turn out to make physics easy become well known under many terms...electrons, photons, quarks, etc. The interesting question of the ultimate meaning of such entities is discussed in Section 7.2.

*Given* a density matrix *ρAB* defined on *C*, the decision not to study an observable with a non-trivial operator on space *B* allows us to prepare the density matrix

$$
\rho\_A = tr\_B(\rho\_{AB}).
$$

Here *trB* sums the diagonal elements of the labels on space *B*. This defines reduction in the conventional way. Reduction is inevitable in physics because physics measures very little.

The rank of a density matrix obtained from reduction depends on how the reduction was done. Obtaining a *rank* − 1 reduced matrix is exceptional. Finding such systems in the laboratory requires great ingenuity. That is why we treat systems that can be described with wave functions as special cases. That may seem to put our dynamical framework discussing wave functions somewhat askew relative to the probabilistic one. Section 6.3.4 explains why and how Hamiltonian time evolution remains relevant.

#### *6.3.3. What Are Those Hidden Variables?*

28 Advances in Quantum Mechanics

the Born rule would fail.

"division".

*6.3.2. Division and Reduction*

of sub-dimension *imax* and names *J*:

ultimate meaning of such entities is discussed in Section 7.2.

non-trivial operator on space *B* allows us to prepare the density matrix

It is quite important (we repeat) that the coefficient < *b*|*a* > happens to measure the amount of vector |*a* > that is pre-existing and pre-aligned with |*b* >. It is very reasonable that a quiet, non-disturbing "physical measurement" of a system will filter out a pre-existing component of a vector variable, and pass it through undisturbed. Undisturbed filtering is precisely what happens in textbook discussions of polarizers, Stern-Gerlach, and diffraction gratings. We wish it would be mentioned it has nothing to do with ¯*h* (to repeat). Using the undisturbed, pre-existing projection also totally contradicts the old line of thinking that 'measurement" involves an uncontrollable disturbance of the system due to the finite quantum of action. Indeed if one believed that, the overlaps would change by the process of measurement, and

The conventional approach to "quantum interacting systems" holds that system *A* "exists on" space *A*, system *B* "exists on" space *B*, and when they interact the joint system exists on the direct product of spaces *C* = *A* ⊗ *B*. This is physically puzzling, and we think backwards. Instead we use our idea of equivalence classes developed by partitioning information, or

Given any vector on a space *C*, we may partition it into factor vectors on procedurally-defined spaces *A* and *B*, as used above. For example a vector of 40 dimensions can be written as the product of vectors on 4 dimensions and vectors on 10 dimensions, or products of 5 × 8, etc. Division is particularly well-developed with Clebsch-series done in the inverse direction: discovering what smaller group representations can be composed to make a given bigger one. A more straightforward division groups a data vector's components into adjacent bins

*Da* → *Di J J* = *int*(*a*/*imax*); *i* = *mod*(*a*, *J*),

where *int* takes the integer part, and *mod*(*a*, *J*) returns the remainder of *a*/*J*. Arnold's famous "cat map" is an example. The freedom to choose the bins and dimensions is very important. Division is quite coordinate-dependent. Division can be repeated to divide the factors, and make subdivisions. The process of vector "division" is not profound mathematics, but the arbitrariness is important for physics and data manipulation. Given a particular division, the physicist (knowingly or not) inspects the decomposition searching for simplicity and regularity to "emerge". The factor-states that turn out to make physics easy become well known under many terms...electrons, photons, quarks, etc. The interesting question of the

*Given* a density matrix *ρAB* defined on *C*, the decision not to study an observable with a

*ρ<sup>A</sup>* = *trB*(*ρAB*).

Here *trB* sums the diagonal elements of the labels on space *B*. This defines reduction in the conventional way. Reduction is inevitable in physics because physics measures very little.

It is remarkable that Bell's artful introduction of distribution theory[15], which is inappropriate for quantum-style systems, led to a false perception that "hidden variables" had been excluded. The hidden variables of ordinary quantum mechanics are the physical degrees of freedom (wave function or density matrix projections) on those spaces the physicist ignored in setting up his oversimplified model. There are always such spaces in Nature.

There is no limit to the number of products or their dimensionality that can be used as "data" or "sample vectors." Partitioning vectors on any given space into a number of factors is a highly coordinate-dependent business. It is essentially a map from one sort of index to a number of composite indices, which can always be done linearly: *Dk* <sup>→</sup> *<sup>D</sup>abc*... <sup>=</sup> <sup>Γ</sup>*abc*... *<sup>k</sup> Dk*, where Γ is an array of constants. In a sufficiently large data vector the "sample" and "object" spaces can be re-configured in practically infinite variations. There is very little that is invariant about entanglement when we allow such freedom. Nature cannot possibly care about these coordinate conventions: Pause to consider how it affects physics.

In early days the Hilbert spaces of single electrons or single photons were considered utterly fundamental. They were building blocks for lofty postulates that could not be explained. Yet quantum mechanics was long ago enlarged to develop quantum field theory (*QFT*). For some purposes *QFT* is considered to have no new information on quantum mechanics itself, while defining very complicated quantum models. Yet Nature has subtleties. Basic non-relativistic quantum mechanics is incapable of dealing with the very questions of causality and non-locality that cloud interpretation of measurements. Relativistic *QFT* deals with issues of causality directly in terms of correlation functions with very well-defined properties. On that basis *QFT* is the more fundamental topic: it is big enough to support realistic models.

The relation of ordinary quantum mechanics to field theory is then developed by reduction, where unobserved *do f* are integrated out. As a result all of the phenomenology of ordinary quantum mechanics is subject to the hidden variables known to have been integrated over in developing density matrices that actually occur. This is ignored in ordinary quantum mechanics seeking by itself to be "fundamental." It is not logically consistent to ignore what is known. Now we have shown how naive probabilities of counting emerge from density matrix constructions integrating over quite arbitrary sample spaces. It is hard to escape the inference that the probability interpretation – which beginning quantum mechanics could not explain about its own framework – must certainly originate in reduction of interacting systems of *QFT* down to the experimentally crude probes developed in beginning quantum mechanics.

**Infrared Example:** There exists certain pure states of the *QFT* called bare electrons: the quanta of a free field theory. If such a state actually participated in an experiment we doubt we'd have a statistical explanation for its behavior in the same free field theory. But that dynamics is too trivial to describe anything, or even permit participation, because free electrons are free. The so-called bare electron of free field theory has never been observed and cannot in principle be observed. All electrons and all observable states are "dressed." Calculations addressing infrared divergences find that electrons with zero photons are unobservable, or have zero probability to participate in reactions, as known from the ancient time of the Bloch-Nordseik analysis[21]. When the experimenter finally specifies his experimental resolution adequately, the probabilities of events emerge from density matrix steps integrating over unobserved quanta, exactly as we have discussed, *yet in such a technical fashion that its relation to beginning questions of quantum probability is never recognized.* The same facts also occur for all degrees of freedom not directly pinned down by experimental probes, which is most of them, that are *much more difficult* to categorize. Every single physical experiment involves so many uncontrolled variables that a statistical description via density matrices cannot be avoided. It is a rare experiment that even finds a single wave function will model the data: and a rare experimenter who can tune his instruments to make that expectation come out.

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*<sup>j</sup>* )/2. That is *O*(2*N*),

symmetries of (our approach) to quantum dynamics are *Sp*(2*N*). There is no precedent to define quantum probability with that symmetry. By adopting a linear dynamical model with Hermitian Ωˆ the dynamical symmetry group is no worse than *U*(*N*). This is easier to work

The relation of the groups in our approach is very intimate. Consider the largest set of

a relatively large group overlooked in ordinary quantum discussions. The intersection

complexification. Thus invariance of a probability notion under *U*(*N*) is enough for

Once more our motivation differs from the traditional one. Tradition asserts - blindly and falsely - that since *U*(*N*) is the symmetry group of Schroedinger's equation, the notion of total probability must be preserved under *U*(*N*). We can't buy that. We don't have a reason to preserve the precious *notation* of the Schroedinger equation. (It is *less general* than Hamilton's equations.) What we buy is the fact that *U*(*N*) time evolution will not destroy a *U*(*N*) invariant probability, if and when the time evolution is that simple. It seems unwise to

Supposing the system is so orderly on a chosen space, one cannot be sure of its effective dynamics for a density matrix on a reduced space. This contradicts lore of the Von Neumann (*vN*) equation, which is "derived" by methods hoping it might be correct[18]. The equation is equivalent to predicting the time evolution *ρ*(*t*) = *U*(*t*)*ρ*(0)*U*†(*t*), where *U*(*t*) = *exp*(−*i*Ωˆ *t*). Notice the traditional context assumes that symbol "*ρ*" refers to a unique object space, while we recognize that concept is procedural. Once a particular division and reduction has been done, it is enough for a single eigenvalue of *ρ* to be time dependent for the *vN* equation to fail. While the Von Neumann equation is true by definition in textbooks, it is seldom true in experimental practice. That is because almost all physical systems which are "dirty" enough to need a density matrix are also dirty enough to interact with the environment and spoil the assumptions. There are schemes ("Lindblad theory")[19] to cover the gap. If sometimes a good phenomenology, the cannot be considered general. When energy and interactions leave a subsystem they go into the larger system to return on any number of different time scales. It is not possible in principle for a first order dynamical system to contain enough

Once every system is a subsystem of a larger system, we should never expect to *always* predict

**How the Framework Never Fails:** Nevertheless physicists put great faith in the fundamental existence of a wave function on the largest space they are thinking about. That makes a puzzle of why their faith persists. There is a question of whether that framework can be falsified. We do not believe it is possible to falsify the framework. Any system that fails the test of Hamiltonian evolution can be embedded in a larger system. On the larger system it's always possible to "unitarize" any transformation. One method is the "unitary dilation" found by Sz.-Nagy. Figure 3 illustrates the more painful process that physicists follow. By now particle physicists have added numerous quantum fields to the early quantum theory of electrons and photons following the process the figure illustrates. The infinite capacity of theory to expand practically terminates questions of whether such a theory could fail.

*<sup>j</sup>* (*p*<sup>2</sup> *<sup>j</sup>* <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

*SP*(2*N*) ∼ *U*(*N*), where ∼ means isomorphic after

transformations preserving the unobservable < Ψ|Ψ >= ∑*<sup>N</sup>*

with the actual symmetries *O*(2*N*)

expect more than that from physics.

initial conditions to parameterize all possible cases.

with.

consistency.

dynamics.

#### *6.3.4. Invariants Under Time Evolution*

At the level of *QFT* one can assert certain wave functions exist, and model them with the Hamiltonian time evolution we cited earlier. That is the state of the art, and returns to how the Hamiltonian dynamics is relevant once again.

**Figure 3.** *How to test Quantum Dynamics:* Tests begin at point 0 with maximal optimism, and assuming the Schroedinger equation applies. At stages 1, 2 tests of dynamics and statistics against observations are made. Dynamical freedoms will be added by the user when a test fails: The system dimension is increased. The procedure flows around a closed loop. The framework cannot be falsified; All outputs verify quantum dynamics.

When probabilities refer to physical measurements, it is important that they be invariants of the system being measured. Invariants refer to some definite transformation group. The

symmetries of (our approach) to quantum dynamics are *Sp*(2*N*). There is no precedent to define quantum probability with that symmetry. By adopting a linear dynamical model with Hermitian Ωˆ the dynamical symmetry group is no worse than *U*(*N*). This is easier to work with.

30 Advances in Quantum Mechanics

instruments to make that expectation come out.

the Hamiltonian dynamics is relevant once again.

framework cannot be falsified; All outputs verify quantum dynamics.

add

*6.3.4. Invariants Under Time Evolution*

doubt we'd have a statistical explanation for its behavior in the same free field theory. But that dynamics is too trivial to describe anything, or even permit participation, because free electrons are free. The so-called bare electron of free field theory has never been observed and cannot in principle be observed. All electrons and all observable states are "dressed." Calculations addressing infrared divergences find that electrons with zero photons are unobservable, or have zero probability to participate in reactions, as known from the ancient time of the Bloch-Nordseik analysis[21]. When the experimenter finally specifies his experimental resolution adequately, the probabilities of events emerge from density matrix steps integrating over unobserved quanta, exactly as we have discussed, *yet in such a technical fashion that its relation to beginning questions of quantum probability is never recognized.* The same facts also occur for all degrees of freedom not directly pinned down by experimental probes, which is most of them, that are *much more difficult* to categorize. Every single physical experiment involves so many uncontrolled variables that a statistical description via density matrices cannot be avoided. It is a rare experiment that even finds a single wave function will model the data: and a rare experimenter who can tune his

At the level of *QFT* one can assert certain wave functions exist, and model them with the Hamiltonian time evolution we cited earlier. That is the state of the art, and returns to how

<sup>i</sup> = . ?

1

2

0

no yes

freedoms verified !

i -

 = [ , -] . ?

no yes

**Figure 3.** *How to test Quantum Dynamics:* Tests begin at point 0 with maximal optimism, and assuming the Schroedinger equation applies. At stages 1, 2 tests of dynamics and statistics against observations are made. Dynamical freedoms will be added by the user when a test fails: The system dimension is increased. The procedure flows around a closed loop. The

When probabilities refer to physical measurements, it is important that they be invariants of the system being measured. Invariants refer to some definite transformation group. The

verified !

The relation of the groups in our approach is very intimate. Consider the largest set of transformations preserving the unobservable < Ψ|Ψ >= ∑*<sup>N</sup> <sup>j</sup>* (*p*<sup>2</sup> *<sup>j</sup>* <sup>+</sup> *<sup>q</sup>*<sup>2</sup> *<sup>j</sup>* )/2. That is *O*(2*N*), a relatively large group overlooked in ordinary quantum discussions. The intersection with the actual symmetries *O*(2*N*) *SP*(2*N*) ∼ *U*(*N*), where ∼ means isomorphic after complexification. Thus invariance of a probability notion under *U*(*N*) is enough for consistency.

Once more our motivation differs from the traditional one. Tradition asserts - blindly and falsely - that since *U*(*N*) is the symmetry group of Schroedinger's equation, the notion of total probability must be preserved under *U*(*N*). We can't buy that. We don't have a reason to preserve the precious *notation* of the Schroedinger equation. (It is *less general* than Hamilton's equations.) What we buy is the fact that *U*(*N*) time evolution will not destroy a *U*(*N*) invariant probability, if and when the time evolution is that simple. It seems unwise to expect more than that from physics.

Supposing the system is so orderly on a chosen space, one cannot be sure of its effective dynamics for a density matrix on a reduced space. This contradicts lore of the Von Neumann (*vN*) equation, which is "derived" by methods hoping it might be correct[18]. The equation is equivalent to predicting the time evolution *ρ*(*t*) = *U*(*t*)*ρ*(0)*U*†(*t*), where *U*(*t*) = *exp*(−*i*Ωˆ *t*). Notice the traditional context assumes that symbol "*ρ*" refers to a unique object space, while we recognize that concept is procedural. Once a particular division and reduction has been done, it is enough for a single eigenvalue of *ρ* to be time dependent for the *vN* equation to fail. While the Von Neumann equation is true by definition in textbooks, it is seldom true in experimental practice. That is because almost all physical systems which are "dirty" enough to need a density matrix are also dirty enough to interact with the environment and spoil the assumptions. There are schemes ("Lindblad theory")[19] to cover the gap. If sometimes a good phenomenology, the cannot be considered general. When energy and interactions leave a subsystem they go into the larger system to return on any number of different time scales. It is not possible in principle for a first order dynamical system to contain enough initial conditions to parameterize all possible cases.

Once every system is a subsystem of a larger system, we should never expect to *always* predict dynamics.

**How the Framework Never Fails:** Nevertheless physicists put great faith in the fundamental existence of a wave function on the largest space they are thinking about. That makes a puzzle of why their faith persists. There is a question of whether that framework can be falsified. We do not believe it is possible to falsify the framework. Any system that fails the test of Hamiltonian evolution can be embedded in a larger system. On the larger system it's always possible to "unitarize" any transformation. One method is the "unitary dilation" found by Sz.-Nagy. Figure 3 illustrates the more painful process that physicists follow. By now particle physicists have added numerous quantum fields to the early quantum theory of electrons and photons following the process the figure illustrates. The infinite capacity of theory to expand practically terminates questions of whether such a theory could fail.

### **7. Applications**

As mentioned in the Introduction, our main goal is to take advantage of the efficiency and flexibility of quantum-style data descriptions. We have remarked that description by quantum-style methods is deliberately incomplete. At the same time a great deal of practical experimental information from physics is encoded in wave functions and density matrices by default usage. That is a clue on how to proceed.

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**7.2. The principle of minimum entropy**

How does one evaluate a system, and in particular the *division* of data we mentioned earlier has tremendous flexibility? For that we consider the system's *entropy* S. The definition is

For a normalized *N* × *N* density matrix the absolute minimum entropy is S = 0, if and only if *ρ* has *rank* − 1. The minimum entropy state is the most orderly, least complicated, and is hardly typical. Developing S = 0 from samples requires every single sample pattern to be a multiple of every other. The maximum value of S = *log*(*N*) comes when *<sup>ρ</sup>* → <sup>1</sup>*N*×*N*/*<sup>N</sup>* is

In thermodynamics the entropy is recognized as a logarithmic measure of the phase space-volume occupied by the system. The equilibrium distribution is defined by maximizing that volume, the entropy, subject to all constraints such as a fixed total energy or particle number. Experimental science seeks order where it can be found, under which we express the principle of *minimum* entropy. The principle predicts we should actively use our freedom

10 20 30 40 50 60 70 80

**Figure 4.** Entropy (in bits) of a coherent data record partitioned on dimension *D*. Bottom curve (blue online): the quantum entropy −*tr*(*ρlog*2(*ρ*) is comparatively small, and detects the optimal division with a sharp dip. Middle curve (red online): the classical entropy on the same data resolved to 2*<sup>D</sup>* bit accuracy. Curve highest on left (green online): the entropy of Huffman

**An Experiment:** Figure 4 shows an experiment with Monte Carlo simulation. Two dimensional distributions were defined using sums of Gaussians with randomly generated parameters. A sample from the distribution was extracted from 0.1 unit bins over the interval −<sup>10</sup> < *<sup>x</sup>* < 10, −<sup>3</sup> < *<sup>y</sup>* < 3, making a 12261-point data record *Di J*, which was normalized. Re-partitioning the record on intervals of length *L* created arrays4 Now compare classical and quantum methods. The empirical marginal distribution on index *i* is *di* = ∑*<sup>j</sup> dij*. That leads to the *classical entropy* <sup>S</sup>*cl* <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>i</sup> dilog*(*di*). Compare the density matrix on the same space, *<sup>ρ</sup>ii*′ = *dijdij*′ , with quantum entropy S*<sup>q</sup>* = −*tr*( *<sup>ρ</sup> log*( *<sup>ρ</sup>* ) ). The figure shows that S*<sup>q</sup>* << S*cl* unless the data is very noisy. A dip in S*<sup>q</sup>* occurs at favored divisions (Fig. 4). Indeed if one

compression can be less than the classical unprocessed entropy, but it is larger than the quantum entropy.

<sup>4</sup> Data were rescaled by a constant to be resolved on exactly *L* dimensions.

completely isotropic, and has no preferred basis or pattern associated with it.

to partition data to discover *low-entropy* divisions. They are simple.

1

2

3

4

5

S = −*tr*(*ρ log*(*ρ*)). (38)

#### **7.1. The optimal patterns defined by data**

Given any "data" as a set of numbers, real or complex, and choosing a method of division, one can start making density matrices and classifying data. But in what sense is this intrinsic and why should it be powerful?

As before let *D<sup>J</sup> <sup>x</sup>* stand for the *J*th instance of data *Dx*. The index *x* now may stand for multiple labels *x*1, *x*2...*xn* of any dimension. We are interested in the *patterns* of fluctuations which tend to occur in the entire data set. We define a "pattern " |*e* > as a normalized vector with projections *e*(*x*) =< *x*|*e* >. To quantify the importance of a particular pattern, we calculate its overlap-squared summed over the entire data set:

$$E = \sum\_{I}^{I \text{tot}} | < D^J | e > |^2.$$

Define the optimal pattern as having the largest possible overlap, subject to the normalization constraint. Algebra gives

$$E =  ; \quad \rho = \sum\_{I}^{I \text{tot}} |D^{I} > 
$$\frac{\delta}{\delta |e>} \left( \frac{}{} \right) = 0;$$

$$\rho |e\_{\alpha}> = \Lambda\_{\alpha}^{2} |e\_{\alpha}>$$
$$

The eigenvalue problem tells us that a *complete set* of solutions |*e<sup>α</sup>* > generally exists; since *ρ* = *ρ*†, the patterns are automatically orthogonal, and eigenvalues Λ<sup>2</sup> *<sup>α</sup>* are real and positive. To interpret the eigenvalues note the overlap <sup>Ω</sup>*<sup>β</sup>* =< *<sup>e</sup>β*|*ρ*|*e<sup>β</sup>* >= <sup>Λ</sup><sup>2</sup> *<sup>β</sup>*. Sorting the eigenvalues Λ2 <sup>1</sup> <sup>&</sup>gt; <sup>Λ</sup><sup>2</sup> <sup>2</sup> <sup>&</sup>gt; ...Λ<sup>2</sup> *<sup>N</sup>* to makes an optimally convergent expansion of the data expressed in its own patterns.

This result attributed to *Karhunen-Loeve*[17] is a foundation point of modern signal and image-processing schemes of great effectiveness. Optimally compressing data while retaining a given overlap is done by truncating the expansion of Eq. 37. We think it is no accident that the density matrix appears in Eq. 37.

### **7.2. The principle of minimum entropy**

32 Advances in Quantum Mechanics

by default usage. That is a clue on how to proceed.

we calculate its overlap-squared summed over the entire data set:

*E* =< *e*|*ρ*|*e* >; *ρ* =

*ρ* = *ρ*†, the patterns are automatically orthogonal, and eigenvalues Λ<sup>2</sup>

To interpret the eigenvalues note the overlap <sup>Ω</sup>*<sup>β</sup>* =< *<sup>e</sup>β*|*ρ*|*e<sup>β</sup>* >= <sup>Λ</sup><sup>2</sup>

no accident that the density matrix appears in Eq. 37.

*E* =

*Jtot* ∑ *J*

**7.1. The optimal patterns defined by data**

and why should it be powerful?

As mentioned in the Introduction, our main goal is to take advantage of the efficiency and flexibility of quantum-style data descriptions. We have remarked that description by quantum-style methods is deliberately incomplete. At the same time a great deal of practical experimental information from physics is encoded in wave functions and density matrices

Given any "data" as a set of numbers, real or complex, and choosing a method of division, one can start making density matrices and classifying data. But in what sense is this intrinsic

multiple labels *x*1, *x*2...*xn* of any dimension. We are interested in the *patterns* of fluctuations which tend to occur in the entire data set. We define a "pattern " |*e* > as a normalized vector with projections *e*(*x*) =< *x*|*e* >. To quantify the importance of a particular pattern,


Define the optimal pattern as having the largest possible overlap, subject to the normalization

*Jtot* ∑ *J*

The eigenvalue problem tells us that a *complete set* of solutions |*e<sup>α</sup>* > generally exists; since

This result attributed to *Karhunen-Loeve*[17] is a foundation point of modern signal and image-processing schemes of great effectiveness. Optimally compressing data while retaining a given overlap is done by truncating the expansion of Eq. 37. We think it is

*δ δ*|*e* >

*<sup>x</sup>* stand for the *J*th instance of data *Dx*. The index *x* now may stand for



 = 0;

*ρ*|*e<sup>α</sup>* > = Λ<sup>2</sup>

*<sup>N</sup>* to makes an optimally convergent expansion of the data expressed in its

*<sup>α</sup>*|*e<sup>α</sup>* >

< *e*|*ρ*|*e* > < *e*|*e* >


*<sup>α</sup>* are real and positive.

*<sup>β</sup>*. Sorting the eigenvalues

**7. Applications**

As before let *D<sup>J</sup>*

constraint. Algebra gives

<sup>2</sup> <sup>&</sup>gt; ...Λ<sup>2</sup>

Λ2 <sup>1</sup> <sup>&</sup>gt; <sup>Λ</sup><sup>2</sup>

own patterns.

How does one evaluate a system, and in particular the *division* of data we mentioned earlier has tremendous flexibility? For that we consider the system's *entropy* S. The definition is

$$\mathcal{S} = -tr(\rho \log(\rho)).\tag{38}$$

For a normalized *N* × *N* density matrix the absolute minimum entropy is S = 0, if and only if *ρ* has *rank* − 1. The minimum entropy state is the most orderly, least complicated, and is hardly typical. Developing S = 0 from samples requires every single sample pattern to be a multiple of every other. The maximum value of S = *log*(*N*) comes when *<sup>ρ</sup>* → <sup>1</sup>*N*×*N*/*<sup>N</sup>* is completely isotropic, and has no preferred basis or pattern associated with it.

In thermodynamics the entropy is recognized as a logarithmic measure of the phase space-volume occupied by the system. The equilibrium distribution is defined by maximizing that volume, the entropy, subject to all constraints such as a fixed total energy or particle number. Experimental science seeks order where it can be found, under which we express the principle of *minimum* entropy. The principle predicts we should actively use our freedom to partition data to discover *low-entropy* divisions. They are simple.

**Figure 4.** Entropy (in bits) of a coherent data record partitioned on dimension *D*. Bottom curve (blue online): the quantum entropy −*tr*(*ρlog*2(*ρ*) is comparatively small, and detects the optimal division with a sharp dip. Middle curve (red online): the classical entropy on the same data resolved to 2*<sup>D</sup>* bit accuracy. Curve highest on left (green online): the entropy of Huffman compression can be less than the classical unprocessed entropy, but it is larger than the quantum entropy.

**An Experiment:** Figure 4 shows an experiment with Monte Carlo simulation. Two dimensional distributions were defined using sums of Gaussians with randomly generated parameters. A sample from the distribution was extracted from 0.1 unit bins over the interval −<sup>10</sup> < *<sup>x</sup>* < 10, −<sup>3</sup> < *<sup>y</sup>* < 3, making a 12261-point data record *Di J*, which was normalized. Re-partitioning the record on intervals of length *L* created arrays4 Now compare classical and quantum methods. The empirical marginal distribution on index *i* is *di* = ∑*<sup>j</sup> dij*. That leads to the *classical entropy* <sup>S</sup>*cl* <sup>=</sup> <sup>−</sup> <sup>∑</sup>*<sup>i</sup> dilog*(*di*). Compare the density matrix on the same space, *<sup>ρ</sup>ii*′ = *dijdij*′ , with quantum entropy S*<sup>q</sup>* = −*tr*( *<sup>ρ</sup> log*( *<sup>ρ</sup>* ) ). The figure shows that S*<sup>q</sup>* << S*cl* unless the data is very noisy. A dip in S*<sup>q</sup>* occurs at favored divisions (Fig. 4). Indeed if one

<sup>4</sup> Data were rescaled by a constant to be resolved on exactly *L* dimensions.

makes data with random linear combinations of specified patterns, the entropy will dip at the division of the pattern's periodicity, or approximate periodicity. Minimum entropy finds simplicity.

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′ ,*l* ′ ... *<sup>m</sup>*′′*m*′′′ :

<sup>2</sup> �*j*, *l*, *m*...|

= *tr*(*ρ*0*φ*(*x*1, *t*1)..*φ*(*xn*, *tn*)).

�*k*12(�*x*2−�*x*1)*exp*−*iω*(*t*2−*t*1). The ultimate time

*δll*′ *δmm*′′ ..., .

Use the completeness property of matrix representations *D<sup>j</sup>*

*<sup>d</sup><sup>ω</sup> <sup>D</sup>j*,*l*...

*mm*′(*ω*)*D*¯ *<sup>j</sup>*

′ ,*l* ′ ... *<sup>m</sup>*′′*m*′′′(*ω*) = *<sup>δ</sup>jj*′

*ρ* → |*j*, *l*, *m*...� |�*j*, *l*, *m*... | *a*�|

Only the diagonal elements survive the sums, with the weights just as dictated by quantum

Data which is many copies of some complicated scalar function of angles, such as the shape of a paramecium or the map of the Earth permuted over all orientations will give a density matrix a diagonal sum of spherical harmonics, with probabilities given by the standard formula. It is kinematic because the machinery defining probability was constructed to

**Time Evolution:** Turn to time evolution. Earlier we commented that physics predicts little else, and that our Hamiltonian model was a toy. Bounded linear Hamiltonian time evolution is unitary time evolution, also a toy. We offer the idea these features emerge from the Agreement to categorize things with density matrices. A typical correlation in *QFT* is written

*<sup>C</sup>*(*x*1, *<sup>t</sup>*1, *<sup>x</sup>*2, *<sup>t</sup>*2, ...*xn*, *tn*) = < <sup>0</sup>|*φ*(*x*1, *<sup>t</sup>*1)..*φ*(*xn*, *tn*)|<sup>0</sup> >,

The state of the art of physics consists of reverting all known correlations into a model for the density matrix *ρ*0. It is very beautiful that the model is relatively simple, but whatever

From space - and time-translational symmetry, which physicists desire to arrange, the correlation is a function of differences *<sup>C</sup>* = *<sup>C</sup>*(*x*<sup>2</sup> − *<sup>x</sup>*1, *<sup>t</sup>*<sup>2</sup> − *<sup>t</sup>*1, ...*tn* − *tn*−1). Any function of

evolution is unitary. By standard steps of the convolution theorem the law of Conservation of Frequency (energy) is kinematic, and inventing a Hamiltonian with enough degrees of freedom cannot fail to describe it5, so long as the eigenvalues of *ρ*<sup>0</sup> are conserved: otherwise, add freedoms. Thus we don't need to be too embarrassed about the toy dynamics, whose success ought to be judged on the effectiveness and simplicity of the *model*, not needing

**Many Experiments:** In the Introduction we mentioned that the restriction of the quantum methods to describing micro-physical objects of fundamental physical character is obsolete. We've done many experiments to extend the scope. • In Ref. [20] the radio frequency emissions of relativistic protons were used to construct density matrices from signal data and noise data. A combination of techniques improved the signal to noise ratio by about a factor of more than 100, producing the first detection of virtual Cherenkov radiation from protons. <sup>5</sup> One might discover a time-dependent Hamiltonian, which does occur in physics, yet which is invariably embedded

was obtained in the lab, we cannot see how the representation itself could fail.

*<sup>x</sup>*<sup>2</sup> − *<sup>x</sup>*<sup>1</sup> has a Fourier representation in terms of *exp<sup>i</sup>*

higher principles: in our approach.

in a time-independent larger system.

expedite linear transformations.

rules:

Information theory is like experimental physics in manipulating the encoding of repeated patterns to lower the effective entropy. As *L* increases longer patterns or "words" can be compressed into symbols. Huffman compression is a method to optimize classical information content towards minimum entropy. Huffman coding is a procedure based on *me* class definitions and therefore classical characterization. Figure 4 shows that using Huffman coding produces S*huf f man* < S*cl*. Yet across the board both classical entropies exceed the quantum value. Both miss the optimal division, because no notion of dividing a product space exists. Similar features have been seen in dozens of different types of data.

**An Experiment:** A *symmetry* of a correlation means it is unchanged under a transformation. A symmetry of the density matrix implies it commutes with the generator of the transformation, and then shares eigenstates. Figure 5 shows an experiment in self-organizing or "auto-quantization" of the eigenvectors of the density matrix. To make the figure the first 60 decimal digits of *π* were collected as an array *D*1*i*. Random cyclic permutations of the same list produced the samples *D*2*i*, *D*3*i*...*DJi* for *J* = 1...1000. The eigenvectors of *ρii*′ are found to be nearly pure momentum eigenstates: Each shows a peak in Fourier power at a single wave number. The probability to find *cos*(*πx*/2) sampled on the first 60 integers is about 2.3%.

**Figure 5.** "Auto-quantization" by symmetry. The Fourier power of density matrix eigenvectors made from the first 60 digits of *π* sampled over 1000 random circular permutations is quantized.

Similar results are found with more structured group operations, such as rotations, unitary transformations, or more complicated group operations with elements *R*(*ω*), where *ω* = *<sup>α</sup>J*, *<sup>β</sup>J*, *<sup>γ</sup>J*... are group parameters of sample *<sup>J</sup>*. Let |*l*, *<sup>m</sup>*...� be an irreducible basis of group representations. Let the objects in the sample be copies of a single object transformed under the group, and let <sup>∑</sup>*<sup>J</sup>* <sup>→</sup> *<sup>d</sup>ω*, the invariant group volume. The density matrix approaches a limit:

$$\begin{split} \rho &= \int d\omega \sum\_{j,l',l,l',m,m'...} R(\omega) \left| l \;/\; m... \right\rangle \left\langle l \;/\; m... \right| \\ &\times \left| a \right\rangle \left\langle a \right| \left| l' \;/\; m'... \right\rangle \left\langle l' \;/\; m'... \right| \; R^{\dagger}(\omega) . \end{split}$$

Use the completeness property of matrix representations *D<sup>j</sup>* ′ ,*l* ′ ... *<sup>m</sup>*′′*m*′′′ :

34 Advances in Quantum Mechanics

simplicity.

about 2.3%.

limit:

makes data with random linear combinations of specified patterns, the entropy will dip at the division of the pattern's periodicity, or approximate periodicity. Minimum entropy finds

Information theory is like experimental physics in manipulating the encoding of repeated patterns to lower the effective entropy. As *L* increases longer patterns or "words" can be compressed into symbols. Huffman compression is a method to optimize classical information content towards minimum entropy. Huffman coding is a procedure based on *me* class definitions and therefore classical characterization. Figure 4 shows that using Huffman coding produces S*huf f man* < S*cl*. Yet across the board both classical entropies exceed the quantum value. Both miss the optimal division, because no notion of dividing a product

**An Experiment:** A *symmetry* of a correlation means it is unchanged under a transformation. A symmetry of the density matrix implies it commutes with the generator of the transformation, and then shares eigenstates. Figure 5 shows an experiment in self-organizing or "auto-quantization" of the eigenvectors of the density matrix. To make the figure the first 60 decimal digits of *π* were collected as an array *D*1*i*. Random cyclic permutations of the same list produced the samples *D*2*i*, *D*3*i*...*DJi* for *J* = 1...1000. The eigenvectors of *ρii*′ are found to be nearly pure momentum eigenstates: Each shows a peak in Fourier power at a single wave number. The probability to find *cos*(*πx*/2) sampled on the first 60 integers is

**Figure 5.** "Auto-quantization" by symmetry. The Fourier power of density matrix eigenvectors made from the first 60 digits of

Similar results are found with more structured group operations, such as rotations, unitary transformations, or more complicated group operations with elements *R*(*ω*), where *ω* = *<sup>α</sup>J*, *<sup>β</sup>J*, *<sup>γ</sup>J*... are group parameters of sample *<sup>J</sup>*. Let |*l*, *<sup>m</sup>*...� be an irreducible basis of group representations. Let the objects in the sample be copies of a single object transformed under the group, and let <sup>∑</sup>*<sup>J</sup>* <sup>→</sup> *<sup>d</sup>ω*, the invariant group volume. The density matrix approaches a

*R*(*ω*)|*l*, *m*...� �*l*, *m*...|

state

space exists. Similar features have been seen in dozens of different types of data.

wavenumber

*<sup>d</sup><sup>ω</sup>* ∑ *j*,*j*′ ,*l*,*l*′ ,*m*,*m*′ ...

× |*a*� �*a*|

 *l* ′ , *<sup>m</sup>*′ ... *l* ′ , *<sup>m</sup>*′ ... *<sup>R</sup>*†(*ω*).

*π* sampled over 1000 random circular permutations is quantized.

*ρ* = 

Fourier Power

$$\int d\omega \,\mathcal{D}^{j,l\dots}\_{mm'}(\omega) \bar{\mathcal{D}}^{j',l'\dots}\_{m'm''}(\omega) = \delta^{j\dot{j}'}\delta^{ll'}\delta^{mm''}\dots\dots$$

Only the diagonal elements survive the sums, with the weights just as dictated by quantum rules:

$$\rho \to |j\_{\prime\prime}l\_{\prime\prime}m...\rangle \mid \langle j\_{\prime\prime}l\_{\prime\prime}m...| \; a\rangle \mid^2 \langle j\_{\prime\prime}l\_{\prime\prime}m...| \; $$

Data which is many copies of some complicated scalar function of angles, such as the shape of a paramecium or the map of the Earth permuted over all orientations will give a density matrix a diagonal sum of spherical harmonics, with probabilities given by the standard formula. It is kinematic because the machinery defining probability was constructed to expedite linear transformations.

**Time Evolution:** Turn to time evolution. Earlier we commented that physics predicts little else, and that our Hamiltonian model was a toy. Bounded linear Hamiltonian time evolution is unitary time evolution, also a toy. We offer the idea these features emerge from the Agreement to categorize things with density matrices. A typical correlation in *QFT* is written

$$\begin{aligned} \mathbb{C}(\mathfrak{x}\_{1\prime} \ t\_{1\prime} \ \mathfrak{x}\_{2\prime} \ t\_{2\prime} \dots \mathfrak{x}\_{n\prime} \ t\_{n}) &= < 0 | \phi(\mathfrak{x}\_{1\prime} \ t\_{1}) . \phi(\mathfrak{x}\_{n\prime} \ t\_{n}) | 0 >, \\ &= \operatorname{tr}(\rho\_{0} \phi(\mathfrak{x}\_{1\prime} \ t\_{1}) . \phi(\mathfrak{x}\_{n\prime} \ t\_{n})). \end{aligned}$$

The state of the art of physics consists of reverting all known correlations into a model for the density matrix *ρ*0. It is very beautiful that the model is relatively simple, but whatever was obtained in the lab, we cannot see how the representation itself could fail.

From space - and time-translational symmetry, which physicists desire to arrange, the correlation is a function of differences *<sup>C</sup>* = *<sup>C</sup>*(*x*<sup>2</sup> − *<sup>x</sup>*1, *<sup>t</sup>*<sup>2</sup> − *<sup>t</sup>*1, ...*tn* − *tn*−1). Any function of *<sup>x</sup>*<sup>2</sup> − *<sup>x</sup>*<sup>1</sup> has a Fourier representation in terms of *exp<sup>i</sup>* �*k*12(�*x*2−�*x*1)*exp*−*iω*(*t*2−*t*1). The ultimate time evolution is unitary. By standard steps of the convolution theorem the law of Conservation of Frequency (energy) is kinematic, and inventing a Hamiltonian with enough degrees of freedom cannot fail to describe it5, so long as the eigenvalues of *ρ*<sup>0</sup> are conserved: otherwise, add freedoms. Thus we don't need to be too embarrassed about the toy dynamics, whose success ought to be judged on the effectiveness and simplicity of the *model*, not needing higher principles: in our approach.

**Many Experiments:** In the Introduction we mentioned that the restriction of the quantum methods to describing micro-physical objects of fundamental physical character is obsolete. We've done many experiments to extend the scope. • In Ref. [20] the radio frequency emissions of relativistic protons were used to construct density matrices from signal data and noise data. A combination of techniques improved the signal to noise ratio by about a factor of more than 100, producing the first detection of virtual Cherenkov radiation from protons.

<sup>5</sup> One might discover a time-dependent Hamiltonian, which does occur in physics, yet which is invariably embedded in a time-independent larger system.

• In Refs. [22] data from the cosmic microwave background (*CMB*) was analyzed to test the "cosmological principle" requiring isotropy. The alignment of spherical harmonic multipoles and the entropy of their power distribution contradicts isotropy at a high degree of statistical significance. The origin is unknown, while it cannot be explained by galactic foreground subtractions [23]. • In Ref.[24] the density matrix was constructed from high-dimensional spectroscopic data of a pharmaceutical protein. The principal values were sorted to make projections onto certain subspaces from which the phases of the protein could be determined by inspection. Ref. [25] reviews subsequent progress. By now the method has been used to make empirical phase diagrams towards characterizing the active states, phase transitions and shelf-life of about 100 pharmaceuticals.

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5). The probability

2= 1/20, which is a

√

One such state is written *vi* =(0, 0, 1, 1, 0, 0, 1, 0, 0, 1). Another vector is *ui* =(0, 1, 0, 0, 1, 1, 0, 0, 1, 1). These are not the same so their probability of being the same is *zero*. The probability

Compare the quantum probability that one vector can serve for another. It is not based

much larger probability of coincidence than *zero*. A numerical calculation finds the average overlap-squared of such random vectors with another one is about 0.3. If one generates 1000 normalized vectors, there will be around 180 with a Born probability exceeding 0.5 that look

Two facts are so basic they tend to escapes notice. First, it is not actually possible in principle to sample and then categorize the spaces of most distributions cited for classical physics. If the data uses 16-bit accuracy and 10-element vectors there are (216)<sup>10</sup> = 1.46 × 10<sup>48</sup> mutually exclusive class labels set up in the first step. The entire 19th century conception of multidimensional phase spaces for many particles ("moles" of atoms) does not exist in any physically realizable form. While ignoring all dynamics was a success of thermodynamics, the ambition to keep track of some fragment of the vast *experimental* complexity of many degrees of freedom cannot really be maintained within the framework of distributions: a

Second, the quantum description of any given data is not more complicated, it is vastly less. The language abuse of "quantum particles" greatly confuses this. The 10-dimensional space cited above has no more than 10-1=9 vectors that are mutually-exclusive of any given vector. The simplification of quantum-style data characterization also occurs with infinitely fine resolution, and with no great sensitivity to the resolution. It is grossly misleading to compare a quantum space of 2<sup>10</sup> dimensions, from spin products 1/2 ⊗ 1/2 ⊗ ...1/2 with a

That is why we end reiterating the principle of *minimum* entropy from Section 7.2. Whether or not physics must do it by principle, there is a practical fact that large data sets should be reduced to correlations, and that correlations should be reduced to sub-correlations. When the entropy of dividing data is as low as possible, the experimenter has found order, and statistical regularity, which is the purpose of science. The applications of a new science seeking minimum entropy as defined on flexible quantum-style categories are unlimited.

Research supported in part under DOE Grant Number DE-FG02-04ER14308. We thank Carl Bender, Don Colloday, Danny Marfatia, Phil Mannheim, Doug McKay, Dan

Department of Physics & Astronomy, The University of Kansas, Lawrence KS, USA

for any random 2-state vector is of order 2−<sup>10</sup> ∼ <sup>9</sup> × <sup>10</sup><sup>−</sup>4, which is very small.

on a distribution. The normalized inner product < *u*|*v* >= 1/(2

like a given vector.

**Acknowledgements**

**Author details** John P. Ralston

great deal of classical theory notwithstanding.

classical space of *K*<sup>10</sup> mutually-exclusive categories.

Neusenschwander, and Peter Rolnick for comments.

the vector |*u* > defined above can serve for |*v* > is | < *u*|*v* > |

### **8. Concluding remarks**

Our discussion began with conventional observations that "the framework of quantum mechanics is thought the perfection of fundamental theory" that "predicts an absolute and unvarying law of time evolution." Those observations have been found false in general.

Our Universe seems to need many degrees of freedom for its description. Physics has greatly progressed in the details of models, while giving away credit to a quantum-style framework we claim is independent. The important part of the framework is its extension and enlargement of the definition of probability. In Section 6.1 we stated that the breakthrough of quantum probability lies in generalizing the notion of probability so as not to *insist* on pre-ordained equivalence classes.

We showed that a map exists going from macroscopic information to quantum probability. The map maintains a hidden variable description for quantum systems, and extends the scope of subjects of quantum information theory. We showed that the probabilistic features of quantum mechanics itself come from the process of reducing hidden freedoms. It is no more profound than a certain method of counting. We believe that quantum physics operates by the same procedures, but so long as parts of physics are unknown, that cannot be "derived." Rather than argue with opinions about what the Universe is, we call it "our approach."

No relation to microphysics is needed. Quantum probability can be viewed as an efficient data-management device, a branch of information theory, reversing a perception that quantum information theory should be a science of microphysical objects.

#### *8.0.1. What do we mean by the probability of a vector?*

Both classical probability and quantum probability have a feature that certain independent probabilities multiply, using direct product spaces to organize the mathematics. The difference is illustrated by the different way to discriminate between "different" vectors, which finishes our discussion.

Both approaches will decompose a state space into coordinates *vi* =< *<sup>i</sup>*|*<sup>v</sup>* >, for *<sup>i</sup>* = 1....*<sup>N</sup>* The typical approach to classical probability defines a distribution *f*(*v*) = *dN*/*dv*1*dv*2...*dvN*. Break each dimension into *K* equal bins of resolution ∆*vi*. The distribution for any situation is a list among *K<sup>N</sup> mutually exclusive* possibilities. Completely sampling the distribution needs *K<sup>N</sup>* pieces of information, which may well be impossible. There are practical consequences. The classical device made with 10 2-state *q* − *bit*s has 2<sup>10</sup> possible states, all declared distinct.

One such state is written *vi* =(0, 0, 1, 1, 0, 0, 1, 0, 0, 1). Another vector is *ui* =(0, 1, 0, 0, 1, 1, 0, 0, 1, 1). These are not the same so their probability of being the same is *zero*. The probability for any random 2-state vector is of order 2−<sup>10</sup> ∼ <sup>9</sup> × <sup>10</sup><sup>−</sup>4, which is very small.

Compare the quantum probability that one vector can serve for another. It is not based on a distribution. The normalized inner product < *u*|*v* >= 1/(2 √5). The probability the vector |*u* > defined above can serve for |*v* > is | < *u*|*v* > | 2= 1/20, which is a much larger probability of coincidence than *zero*. A numerical calculation finds the average overlap-squared of such random vectors with another one is about 0.3. If one generates 1000 normalized vectors, there will be around 180 with a Born probability exceeding 0.5 that look like a given vector.

Two facts are so basic they tend to escapes notice. First, it is not actually possible in principle to sample and then categorize the spaces of most distributions cited for classical physics. If the data uses 16-bit accuracy and 10-element vectors there are (216)<sup>10</sup> = 1.46 × 10<sup>48</sup> mutually exclusive class labels set up in the first step. The entire 19th century conception of multidimensional phase spaces for many particles ("moles" of atoms) does not exist in any physically realizable form. While ignoring all dynamics was a success of thermodynamics, the ambition to keep track of some fragment of the vast *experimental* complexity of many degrees of freedom cannot really be maintained within the framework of distributions: a great deal of classical theory notwithstanding.

Second, the quantum description of any given data is not more complicated, it is vastly less. The language abuse of "quantum particles" greatly confuses this. The 10-dimensional space cited above has no more than 10-1=9 vectors that are mutually-exclusive of any given vector. The simplification of quantum-style data characterization also occurs with infinitely fine resolution, and with no great sensitivity to the resolution. It is grossly misleading to compare a quantum space of 2<sup>10</sup> dimensions, from spin products 1/2 ⊗ 1/2 ⊗ ...1/2 with a classical space of *K*<sup>10</sup> mutually-exclusive categories.

That is why we end reiterating the principle of *minimum* entropy from Section 7.2. Whether or not physics must do it by principle, there is a practical fact that large data sets should be reduced to correlations, and that correlations should be reduced to sub-correlations. When the entropy of dividing data is as low as possible, the experimenter has found order, and statistical regularity, which is the purpose of science. The applications of a new science seeking minimum entropy as defined on flexible quantum-style categories are unlimited.

### **Acknowledgements**

Research supported in part under DOE Grant Number DE-FG02-04ER14308. We thank Carl Bender, Don Colloday, Danny Marfatia, Phil Mannheim, Doug McKay, Dan Neusenschwander, and Peter Rolnick for comments.

### **Author details**

John P. Ralston

36 Advances in Quantum Mechanics

and shelf-life of about 100 pharmaceuticals.

**8. Concluding remarks**

pre-ordained equivalence classes.

which finishes our discussion.

• In Refs. [22] data from the cosmic microwave background (*CMB*) was analyzed to test the "cosmological principle" requiring isotropy. The alignment of spherical harmonic multipoles and the entropy of their power distribution contradicts isotropy at a high degree of statistical significance. The origin is unknown, while it cannot be explained by galactic foreground subtractions [23]. • In Ref.[24] the density matrix was constructed from high-dimensional spectroscopic data of a pharmaceutical protein. The principal values were sorted to make projections onto certain subspaces from which the phases of the protein could be determined by inspection. Ref. [25] reviews subsequent progress. By now the method has been used to make empirical phase diagrams towards characterizing the active states, phase transitions

Our discussion began with conventional observations that "the framework of quantum mechanics is thought the perfection of fundamental theory" that "predicts an absolute and unvarying law of time evolution." Those observations have been found false in general.

Our Universe seems to need many degrees of freedom for its description. Physics has greatly progressed in the details of models, while giving away credit to a quantum-style framework we claim is independent. The important part of the framework is its extension and enlargement of the definition of probability. In Section 6.1 we stated that the breakthrough of quantum probability lies in generalizing the notion of probability so as not to *insist* on

We showed that a map exists going from macroscopic information to quantum probability. The map maintains a hidden variable description for quantum systems, and extends the scope of subjects of quantum information theory. We showed that the probabilistic features of quantum mechanics itself come from the process of reducing hidden freedoms. It is no more profound than a certain method of counting. We believe that quantum physics operates by the same procedures, but so long as parts of physics are unknown, that cannot be "derived." Rather than argue with opinions about what the Universe is, we call it "our approach."

No relation to microphysics is needed. Quantum probability can be viewed as an efficient data-management device, a branch of information theory, reversing a perception that

Both classical probability and quantum probability have a feature that certain independent probabilities multiply, using direct product spaces to organize the mathematics. The difference is illustrated by the different way to discriminate between "different" vectors,

Both approaches will decompose a state space into coordinates *vi* =< *<sup>i</sup>*|*<sup>v</sup>* >, for *<sup>i</sup>* = 1....*<sup>N</sup>* The typical approach to classical probability defines a distribution *f*(*v*) = *dN*/*dv*1*dv*2...*dvN*. Break each dimension into *K* equal bins of resolution ∆*vi*. The distribution for any situation is a list among *K<sup>N</sup> mutually exclusive* possibilities. Completely sampling the distribution needs *K<sup>N</sup>* pieces of information, which may well be impossible. There are practical consequences. The classical device made with 10 2-state *q* − *bit*s has 2<sup>10</sup> possible states, all declared distinct.

quantum information theory should be a science of microphysical objects.

*8.0.1. What do we mean by the probability of a vector?*

Department of Physics & Astronomy, The University of Kansas, Lawrence KS, USA

### **References**


10.5772/55954

475

Emergent un-Quantum Mechanics http://dx.doi.org/10.5772/55954

[16] R. F. Werner, Phys. Rev. **A** 42, 4777, (1989). A. Peres, Phys. Rev. Lett. **7**7, 1413 (1996);

[17] H. Karhunen, Ann. Acad. Science. Fenn, Ser A. I. 37, 1947; M. Loeve, supplement to P. Levy, *Processes Stochastic et Mouvement Brownien*, Paris, Gauthier Villars, 1948; H. Hotelling, J. Educ. Psychology 24, 417, 1933; ibid 24, 448, 1933; L. Scharfe, *Statistical*

[18] J. Von Neumann, *Mathematische Grundlagen der Quantenmechanik* Springer, Berlin; English translation in *The Mathematical Foundations of Quantum Mechanics*, Princeton

[19] G. Lindblad, Commun. Math. Phys. **48** 119 (1976) V. Gorini, A. Kossakowski and E C G

[20] A. Bean, J. P. Ralston and J. Snow, Nucl. Instrum. Meth. A **5**96, 172 (2008)

[21] The Bloch-Nordseik treatment of infrared divergences in *QED* is reviewed in many textbooks, including *Ralstivistic Quantum Field Theory,* by J. D. Bjorken and S. D.

[22] P. K. Samal, R. Saha, P. Jain and J. P. Ralston, Mon. Not. Roy. Astron. Soc. **3**85, 1718 (2008) [arXiv:0708.2816 [astro-ph]]; Mon. Not. Roy. Astron. Soc. **3**96, 511 (2009) [arXiv:0811.1639

[23] P. K. Aluri, P. K. Samal, P. Jain and J. .P. Ralston, Mon. Not. Roy. Astron. Soc. **4**14, 1032

[24] L Kueltzo, J. Fan, J. P. Ralston, M. DiBiase, E. Faulkner, and C. R. Middaugh, J. Pharm.

[25] N. Maddux, R Joshi, C. R. Middaugh, J. P. Ralston and R. Volkin, J. Pharm. Sci. **100** 4171

Phys. Scripta **T**76, 52 (1998) [arXiv:quant-ph/9707026].

*Signal Processing*, (Wiley 1990).

University Press, Princeton (1971).

[arXiv:1008.0029 [physics.ins-det]].

(2011) [arXiv:1007.1827 [astro-ph.CO]].

Drell,(Wiley, 1965).

Sci. 94(9), 1893 (2005).

[astro-ph]].

(2011).

Sudarshan, J. Math. Phys. **17** 821 (1976).


[16] R. F. Werner, Phys. Rev. **A** 42, 4777, (1989). A. Peres, Phys. Rev. Lett. **7**7, 1413 (1996); Phys. Scripta **T**76, 52 (1998) [arXiv:quant-ph/9707026].

38 Advances in Quantum Mechanics

[1] R. B. Laughlin and D. Pines, PNAS **9**7, 28 (1999).

(2012) 012003, arXiv:1204.1077 [physics.hist-ph].

[7] J. P. Ralston, J. Phys. A: Math. Theor. **40**, 9883 (2007).

[arXiv:quant-ph/9809072].

Physics, http://www.lth.se/?id=17657.

[12] J. P. Ralston, arXiv:1203.5557 [hep-ph].

[14] A. Caticha, Phys. Rev. A **5**7, 1572 (1998)

[physics.atom-ph]].

[15] J. S. Bell, Physics **1**, 195, (1964).

translated by Morton Mosius, P. Blackiston's Sons, (1914).

Classics, 2007).

[2] There are too many references to cite. A good introduction is the book by S. L. Adler, *Quantum theory as an emergent phenomenon: The statistical mechanics of matrix models as the precursor of quantum field theory*, Cambridge, UK: Univ. Pr. (2004) 225 p. See also G. 't

[3] An extensive blbliography with wide discussion is given by the Archiv preprint of B. Hu, *Emergence: Key Physical Issues for Deeper Philosphical Enquiries*, J. Phys. Conf. Ser. 361

[4] H. -T. Elze, J. Phys. Conf. Ser. **1**74, 012009 (2009) [arXiv:0906.1101 [quant-ph]].; J. Phys. Conf. Ser. **3**3, 399 (2006) [gr-qc/0512016]; J. Phys. Conf. Ser. **1**71, 012034 (2009); M. Blasone, P. Jizba and F. Scardigli, J. Phys. Conf. Ser. **1**74, 012034 (2009)

[5] See, e.g. R. Hedrich, Phys. Phil. **2**010, 016 (2010) [arXiv:0908.0355 [gr-qc]]; B. Koch, AIP Conf. Proc. **1**232, 313 (2010) [arXiv:1004.2879 [hep-th]]; D. Acosta, P. F. de Cordoba,

[6] *Modern Quantum Mechanics*, by J. J. Sakurai, San Fu Tuan, editor (Addison Wesley, 1998).

[8] C. M. Bender and S. Boettcher, Phys. Rev. Lett. **8**0, 5243 (1998) [arXiv:physics/9712001]; C. M. Bender, S. Boettcher and P. Meisinger, J. Math. Phys. **4**0, 2201 (1999)

[9] *Sir George Gabriel Stokes : Memoirs and Scientific Correspondence*, edited by Joseph Larmor (Cambridge University Press, 1907); *The Theory of Electrons*, by H. A. Lorentz (Cosimo

[10] For a review of Rydberg's physics, see "Janne Rydberg his life and work", by I. Martinson and L.J. Curtis, NIM *B 235*, 17 (2005). Graphics is from Lund University

[11] Max Planck , Annalen der Physik **4**, 553 (1901); Translated in http://axion.physics. ubc.ca/200-06/Planck-1901.html. See also Max Planck, *The theory of Heat Radiation*,

[13] P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. **8**0, 633 (2008) [arXiv:0801.0028

[arXiv:0901.3907 [quant-ph]]; G. 't Hooft, Int. J. Mod. Phys. A **2**5, 4385 (2010)

Hooft, AIP Conf. Proc. **9**57, 154 (2007) [arXiv:0707.4568 [hep-th]]

J. M. Isidro and J. L. G. Santander, arXiv:1206.4941 [math-ph].

**References**


**Chapter 20**

**Provisional chapter**

<sup>2</sup> ) [9] and non-relativistic

**The Wigner-Heisenberg Algebra in Quantum**

**The Wigner-Heisenberg Algebra in Quantum**

While the supersymmetry in quantum mechanics (SUSY QM) algebra has thus received much operator applications for potential problems [1–4], another algebra, the general Wigner-Heisenberg(WH) oscillator algebra [5–8], which already possesses an in built structure which generalises the usual oscillators ladder operators, has not, however, in our opinion, received its due attention in the literature as regards its potential for being developed as an effective operator technique for the spectral resolution of oscillator-related

The WH algebraic technique which was super-realized for quantum oscillators [9–11], is related to the paraboson relations and a graded Lie algebra structure analogous to Witten's SUSY QM algebra was realized in which only annihilation operators participate, all expressed in terms of the Wigner annihilation operator of a related super Wigner oscillator system [12]. In this reference, the coherent states are investigate via WH algebra for bound states, which are defined as the eigenstates of the lowering operator, according to the Barut-Girardello approach [13]. Recently, the problem of the construction of coherent states for systems with continuous spectra has been investigated from two viewpoints by Bragov *et al.* [17]. They adopt the approach of Malkin-Manko [18] to systems with continuous spectra that are not oscillator-like systems. On the other hand, they generalize, modify and apply the approach

To illustrate the formalism we consider here simpler types of such potentials only, of the full

The WH algebra has been considered for the three-dimensional non-canonical oscillator to generate a representation of the orthosympletic Lie superalgebra *osp*(3/2), and recently

> ©2012 Rodrigues, 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 Rodrigues; 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 Rodrigues; 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.

**Mechanics**

10.5772/55994

**1. Introduction**

**Mechanics**

Rafael de Lima Rodrigues

Rafael de Lima Rodrigues

http://dx.doi.org/10.5772/55994

Additional information is available at the end of the chapter

potentials. The purpose of the chapter is to bridge this gulf.

3D isotropic harmonic oscillator problem (for a particle of spin <sup>1</sup>

followed in [19] to the same kind of systems.

Coulomb problem for the electron [16].

Additional information is available at the end of the chapter

**Provisional chapter**

### **The Wigner-Heisenberg Algebra in Quantum Mechanics Mechanics**

**The Wigner-Heisenberg Algebra in Quantum**

Rafael de Lima Rodrigues Additional information is available at the end of the chapter

Rafael de Lima Rodrigues

Additional information is available at the end of the chapter 10.5772/55994

http://dx.doi.org/10.5772/55994

### **1. Introduction**

While the supersymmetry in quantum mechanics (SUSY QM) algebra has thus received much operator applications for potential problems [1–4], another algebra, the general Wigner-Heisenberg(WH) oscillator algebra [5–8], which already possesses an in built structure which generalises the usual oscillators ladder operators, has not, however, in our opinion, received its due attention in the literature as regards its potential for being developed as an effective operator technique for the spectral resolution of oscillator-related potentials. The purpose of the chapter is to bridge this gulf.

The WH algebraic technique which was super-realized for quantum oscillators [9–11], is related to the paraboson relations and a graded Lie algebra structure analogous to Witten's SUSY QM algebra was realized in which only annihilation operators participate, all expressed in terms of the Wigner annihilation operator of a related super Wigner oscillator system [12]. In this reference, the coherent states are investigate via WH algebra for bound states, which are defined as the eigenstates of the lowering operator, according to the Barut-Girardello approach [13]. Recently, the problem of the construction of coherent states for systems with continuous spectra has been investigated from two viewpoints by Bragov *et al.* [17]. They adopt the approach of Malkin-Manko [18] to systems with continuous spectra that are not oscillator-like systems. On the other hand, they generalize, modify and apply the approach followed in [19] to the same kind of systems.

To illustrate the formalism we consider here simpler types of such potentials only, of the full 3D isotropic harmonic oscillator problem (for a particle of spin <sup>1</sup> <sup>2</sup> ) [9] and non-relativistic Coulomb problem for the electron [16].

The WH algebra has been considered for the three-dimensional non-canonical oscillator to generate a representation of the orthosympletic Lie superalgebra *osp*(3/2), and recently

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 Rodrigues; 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 Rodrigues; 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 Rodrigues, licensee InTech. This is an open access chapter distributed under the terms of the Creative

Palev have investigated the 3*D* Wigner oscillator under a discrete non-commutative context [20]. Let us now point out the following (anti-)commutation relations ([*A*, *B*]+ ≡ *AB* + *BA* and [*A*, *B*]<sup>−</sup> ≡ *AB* − *BA*).

10.5772/55994

479

http://dx.doi.org/10.5772/55994

<sup>±</sup> (3)

The Wigner-Heisenberg Algebra in Quantum Mechanics

†]+ = 0 → [*R*ˆ, *H*ˆ ]<sup>−</sup> = 0. (5)

<sup>2</sup> (<sup>1</sup> <sup>+</sup> *cR*ˆ) (6)

*P*, *R* = *P*. (8)

. (9)

+]<sup>−</sup> = 1 + *cR*ˆ → [*x*ˆ, *p*ˆ]<sup>−</sup> = *i*(1 + *cR*ˆ) (4)

[*H*ˆ , *a*ˆ

form. The form of this general quantum rule can be given by

abstract operator, Hermitian and unitary, also possessing the properties

*R*ˆ = *R*ˆ † = ˆ *R*−<sup>1</sup> → *R*ˆ <sup>2</sup> = 1, [*R*ˆ, *a*ˆ

*H*ˆ =

*<sup>x</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>x</sup>*, <sup>∞</sup><sup>−</sup> <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>p</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup>*<sup>i</sup> <sup>d</sup>*

<sup>=</sup> <sup>1</sup> √2 ± *d dx* <sup>∓</sup>

*a*ˆ

<sup>±</sup> −→ *<sup>a</sup>*<sup>±</sup> *c* 2

Yang's wave mechanical description was further investigated in [7, 8].

[*a*ˆ <sup>−</sup>, *<sup>a</sup>*<sup>ˆ</sup>

It follows from equations (1) and (4) that

Wigner requires that

ladder operators as given by

<sup>±</sup>]<sup>−</sup> <sup>=</sup> <sup>±</sup>*a*<sup>ˆ</sup>

obtained by also combining the requirement that *x* satisfies the equation of motion of classical

where *c* is a real constant that is related to the ground-state energy *E*(0) of *H*ˆ and *R*ˆ is an

*<sup>a</sup>*ˆ+*a*ˆ<sup>−</sup> + <sup>1</sup>

*<sup>a</sup>*ˆ−*a*ˆ<sup>+</sup> − <sup>1</sup>

Abstractly *<sup>R</sup>*<sup>ˆ</sup> is the Klein operator ±*exp*[*iπ*(*H*<sup>ˆ</sup> − *<sup>E</sup>*0)] while in Schrödinger coordinate representation, first investigated by Yang, R is realised by ±*P* where *P* is the parity operator:

The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that *c* is positive, i.e. *c* =| *c* |> 0. Thus, in coordinate representation the generalized quantization à la

Indeed, following Yang representation [6] we obtain the coordinate representation for the

<sup>2</sup> (<sup>1</sup> <sup>+</sup> *cR*ˆ)

*<sup>P</sup>*|*<sup>x</sup>* <sup>&</sup>gt;<sup>=</sup> ±|*<sup>x</sup>* <sup>&</sup>gt;, *<sup>P</sup>*−<sup>1</sup> <sup>=</sup> *<sup>P</sup>*, *<sup>P</sup>*<sup>2</sup> <sup>=</sup> 1, *PxP*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*x*. (7)

*dx* <sup>+</sup>

*c* 2*x P* − *x* 

*ic* 2*x*

Also, the relevance of WH algebra to quantization in fractional dimension has been also discussed [21] and the properties of Weyl-ordered polynomials in operators *P* and *Q*, in fractional-dimensional quantum mechanics have been developed [22].

The Kustaanheimo-Stiefel mapping [25] yields the Schrödinger equation for the hydrogen atom that has been exactly solved and well-studied in the literature. (See for example, Chen [26], Cornish [27], Chen and Kibler [28], D'Hoker and Vinet [29].)

Kostelecky, Nieto and Truax [30] have studied in a detailed manner the relation of the supersymmetric (SUSY) Coulombian problem [31–33] in D-dimensions with that of SUSY isotropic oscillators in D-dimensions in the radial version.

The vastly simplified algebraic treatment within the framework of the WH algebra of some other oscillator-related potentials like those of certain generalised SUSY oscillator Hamiltonian models of the type of Celka and Hussin which generalise the earlier potentials of Ui and Balantekin have been applied by Jayaraman and Rodrigues [10]. Also, the connection of the WH algebra with the Lie superalgebra *s*ℓ(1|*n*) has been studied in a detailed manner [34].

Also, some super-conformal models are sigma models that describe the propagation of a non-relativistic spinning particle in a curved background [35]. It was conjectured by Gibbons and Townsend that large *n* limit of an *N* = 4 superconformal extension of the *n* particle Calogero model [36] might provide a description of the extreme Reissner-Nordström black hole near the horizon [37]. The superconformal mechanics, black holes and non-linear realizations have also been investigated by Azcárraga *et al.* [38].

### **2. The abstract WH algebra and its super-realisation**

Six decades ago Wigner [5] posed an interesting question as if from the equations of motion determine the quantum mechanical commutation relations and found as an answer a generalised quantum commutation rule for the one-dimensional harmonic oscillator. Starting with the Schrödinger equation *H*|*ψ<sup>n</sup>* >= *En*|*ψ<sup>n</sup>* >, where the Hamiltonian operator becomes

$$\hat{H} = \frac{1}{2}(\hat{\mathfrak{p}}^2 + \mathfrak{k}^2) = \frac{1}{2}[\mathfrak{d}^-, \mathfrak{d}^+]\_+ = \frac{1}{2}(\mathfrak{d}^-\mathfrak{d}^+ + \mathfrak{d}^+\mathfrak{d}^-) \tag{1}$$

(we employ the convention of units such that ¯*h* = *m* = *ω* = 1) where the abstract Wigner Hamiltonian *H*ˆ is expressed in the symmetrised bilinear form in the mutually adjoint abstract operators *<sup>a</sup>*<sup>±</sup> defined by

$$\mathfrak{a}^{\pm} = \frac{1}{\sqrt{2}} (\pm i \mathfrak{p} - \mathfrak{x}) \quad (\mathfrak{a}^{+})^{\dagger} = \mathfrak{a}^{-}. \tag{2}$$

Wigner showed that the Heisenberg equations of motion

$$[\hat{H}, \mathfrak{k}^{\pm}]\_{-} = \pm \mathfrak{k}^{\pm} \tag{3}$$

obtained by also combining the requirement that *x* satisfies the equation of motion of classical form. The form of this general quantum rule can be given by

$$[\mathfrak{i}^-, \mathfrak{i}^+]\_- = 1 + c\mathring{\mathcal{R}} \to [\mathfrak{x}, \mathfrak{j}]\_- = i(1 + c\mathring{\mathcal{R}}) \tag{4}$$

where *c* is a real constant that is related to the ground-state energy *E*(0) of *H*ˆ and *R*ˆ is an abstract operator, Hermitian and unitary, also possessing the properties

$$
\hat{\mathbb{R}} = \hat{\mathbb{R}}^{\dagger} = \hat{\mathbb{R}}^{-1} \to \hat{\mathbb{R}}^{2} = 1,\quad [\hat{\mathbb{R}}, \hat{\mathbb{t}}^{\dagger}]\_{+} = 0 \to [\hat{\mathbb{R}}, \hat{H}]\_{-} = 0.\tag{5}
$$

It follows from equations (1) and (4) that

2 Advances in Quantum Mechanics

and [*A*, *B*]<sup>−</sup> ≡ *AB* − *BA*).

detailed manner [34].

operators *<sup>a</sup>*<sup>±</sup> defined by

Palev have investigated the 3*D* Wigner oscillator under a discrete non-commutative context [20]. Let us now point out the following (anti-)commutation relations ([*A*, *B*]+ ≡ *AB* + *BA*

Also, the relevance of WH algebra to quantization in fractional dimension has been also discussed [21] and the properties of Weyl-ordered polynomials in operators *P* and *Q*, in

The Kustaanheimo-Stiefel mapping [25] yields the Schrödinger equation for the hydrogen atom that has been exactly solved and well-studied in the literature. (See for example, Chen

Kostelecky, Nieto and Truax [30] have studied in a detailed manner the relation of the supersymmetric (SUSY) Coulombian problem [31–33] in D-dimensions with that of SUSY

The vastly simplified algebraic treatment within the framework of the WH algebra of some other oscillator-related potentials like those of certain generalised SUSY oscillator Hamiltonian models of the type of Celka and Hussin which generalise the earlier potentials of Ui and Balantekin have been applied by Jayaraman and Rodrigues [10]. Also, the connection of the WH algebra with the Lie superalgebra *s*ℓ(1|*n*) has been studied in a

Also, some super-conformal models are sigma models that describe the propagation of a non-relativistic spinning particle in a curved background [35]. It was conjectured by Gibbons and Townsend that large *n* limit of an *N* = 4 superconformal extension of the *n* particle Calogero model [36] might provide a description of the extreme Reissner-Nordström black hole near the horizon [37]. The superconformal mechanics, black holes and non-linear

Six decades ago Wigner [5] posed an interesting question as if from the equations of motion determine the quantum mechanical commutation relations and found as an answer a generalised quantum commutation rule for the one-dimensional harmonic oscillator. Starting with the Schrödinger equation *H*|*ψ<sup>n</sup>* >= *En*|*ψ<sup>n</sup>* >, where the Hamiltonian operator becomes

(we employ the convention of units such that ¯*h* = *m* = *ω* = 1) where the abstract Wigner Hamiltonian *H*ˆ is expressed in the symmetrised bilinear form in the mutually adjoint abstract

(±*ip*ˆ − *x*ˆ) (*a*ˆ

+]+ <sup>=</sup> <sup>1</sup> 2 (*a*ˆ −*a*ˆ <sup>+</sup> + *a*ˆ +*a*ˆ

+)† = *a*ˆ

<sup>−</sup>) (1)

<sup>−</sup>. (2)

fractional-dimensional quantum mechanics have been developed [22].

[26], Cornish [27], Chen and Kibler [28], D'Hoker and Vinet [29].)

isotropic oscillators in D-dimensions in the radial version.

realizations have also been investigated by Azcárraga *et al.* [38].

**2. The abstract WH algebra and its super-realisation**

<sup>2</sup>) = <sup>1</sup> 2 [*a*ˆ <sup>−</sup>, *<sup>a</sup>*<sup>ˆ</sup>

*<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>1</sup> 2 (*p*ˆ <sup>2</sup> + *x*ˆ

> *a*ˆ <sup>±</sup> <sup>=</sup> <sup>1</sup> √2

Wigner showed that the Heisenberg equations of motion

$$
\hat{H} = \begin{cases}
\mathfrak{A}^+\mathfrak{A}^- + \frac{1}{2}(1 + c\hat{\mathcal{R}}) \\
\mathfrak{A}^-\mathfrak{A}^+ - \frac{1}{2}(1 + c\hat{\mathcal{R}})
\end{cases}
\tag{6}
$$

Abstractly *<sup>R</sup>*<sup>ˆ</sup> is the Klein operator ±*exp*[*iπ*(*H*<sup>ˆ</sup> − *<sup>E</sup>*0)] while in Schrödinger coordinate representation, first investigated by Yang, R is realised by ±*P* where *P* is the parity operator:

$$P|\mathbf{x}> = \pm|\mathbf{x}>, \quad P^{-1} = P, \quad P^2 = 1,\\ P\mathbf{x}P^{-1} = -\mathbf{x}.\tag{7}$$

The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that *c* is positive, i.e. *c* =| *c* |> 0. Thus, in coordinate representation the generalized quantization à la Wigner requires that

$$\text{if } \mathbf{x} = \mathbf{x}, \quad \infty - < \mathbf{x} < \infty, \quad \mathfrak{p} = -i\frac{d}{d\mathfrak{x}} + \frac{i\mathfrak{c}}{2\mathfrak{x}} P, \quad \mathcal{R} = P. \tag{8}$$

Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by

$$\mathfrak{A}^{\pm} \longrightarrow a\_{\frac{\varepsilon}{2}}^{\pm} = \frac{1}{\sqrt{2}} \left( \pm \frac{d}{d\mathfrak{x}} \mp \frac{c}{2\mathfrak{x}} P - \mathfrak{x} \right). \tag{9}$$

Yang's wave mechanical description was further investigated in [7, 8].

The present author have applied a super-realization so that *R* = Σ<sup>3</sup> to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (*x*, −*i <sup>d</sup> dx* ), the fermionic ones *<sup>b</sup>*∓(= (*b*±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices <sup>Σ</sup>*i*,(*<sup>i</sup>* <sup>=</sup> 1, 2, 3). Indeed, expressing *<sup>a</sup>*±( *<sup>c</sup>* <sup>2</sup> ) in the following respective factorised forms:

$$\mathbf{a}^+(\frac{c}{2}) = \frac{1}{\sqrt{2}} \Sigma\_1 \mathbf{x}^{(1/2)c\Sigma\_3} \exp(\frac{1}{2}\mathbf{x}^2) (\frac{d}{d\mathbf{x}}) \exp(\frac{-1}{2}\mathbf{x}^2) \mathbf{x}^{-(1/2)c\Sigma\_3} \tag{10}$$

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481

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, *c* > 0. (16)

The Wigner-Heisenberg Algebra in Quantum Mechanics

*E*(0) ( *c* 2 ) = <sup>1</sup> 2

*a*− *c* 2 |2*m*, *c* <sup>2</sup> <sup>&</sup>gt; <sup>=</sup> <sup>√</sup>

*a*+ *c* 2 |2*m*, *c* <sup>2</sup> <sup>&</sup>gt; <sup>=</sup>

<sup>|</sup>2*<sup>m</sup>* <sup>+</sup> 1, *<sup>c</sup>*

<sup>|</sup>2*<sup>m</sup>* <sup>+</sup> 1, *<sup>c</sup>*

<sup>2</sup> ) <sup>∝</sup> [*a*+( *<sup>c</sup>*

2 *<sup>a</sup>*±( *<sup>c</sup>* 2 )

(**<sup>1</sup>** <sup>+</sup> <sup>Σ</sup>3)*B*<sup>−</sup> <sup>=</sup> <sup>1</sup>

(**<sup>1</sup>** <sup>+</sup> <sup>Σ</sup>3)*B*<sup>+</sup> <sup>=</sup> <sup>1</sup>

2

 *d*<sup>2</sup> *dx*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>x</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) = <sup>1</sup>

*a*− *c* 2

*a*+ *c* 2

> 1 2

> 1 2

*<sup>B</sup>*−( *c*

for *<sup>H</sup>*( *<sup>c</sup>*

these quantum states are given by

Now, from the role of *<sup>a</sup>*+( *<sup>c</sup>*

respectively given by *<sup>ψ</sup>*(*<sup>n</sup>*)( *<sup>c</sup>*

and

we obtain

and

It is known that the operators ± *<sup>i</sup>*

operators in the even sector with <sup>1</sup>

(1 + *c*) >

At this stage an independent verification of the existence or not of a zero ground-state energy

The question we formulate now is the following: What is the behaviour of the ladder operators on the autokets of the Wigner oscillator quantum states? To answer this question is obtained via WH algebra, note that the Wigner oscillator ladder operators on autokets of

<sup>2</sup>*m*|2*<sup>m</sup>* <sup>−</sup> 1, *<sup>c</sup>*

2(*m* + *E*(0))|2*m*,

<sup>2</sup>(*<sup>m</sup>* <sup>+</sup> *<sup>E</sup>*(0))|2*<sup>m</sup>* <sup>+</sup> 1, *<sup>c</sup>*

<sup>2</sup>(*<sup>m</sup>* <sup>+</sup> <sup>1</sup>)|2*<sup>m</sup>* <sup>+</sup> 2, *<sup>c</sup>*

<sup>2</sup> ) and *<sup>E</sup>*(*<sup>n</sup>*)( *<sup>c</sup>*

<sup>2</sup> *<sup>H</sup>*( *<sup>c</sup>*

<sup>2</sup> and <sup>1</sup>

2 >

*c* 2 >

<sup>2</sup> ) as the energy step-up operator (the upper sign choice)

 *<sup>B</sup>*<sup>−</sup> <sup>0</sup> 0 0

 *B*<sup>+</sup> 0 0 0

<sup>2</sup> <sup>−</sup> <sup>1</sup>) *<sup>c</sup>* 2 *<sup>x</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup>

2 >

<sup>2</sup> <sup>&</sup>gt; . (17)

<sup>2</sup> ) = *<sup>E</sup>*(0) <sup>+</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, 2, ··· .

<sup>2</sup> ) can be chosen as a basis for a

<sup>2</sup> ) are

(18)

2 *a*<sup>±</sup><sup>2</sup>

<sup>2</sup> ) suggested by its positive semi-definite form may be in order.

<sup>2</sup> <sup>&</sup>gt; <sup>=</sup>

<sup>2</sup> <sup>&</sup>gt; <sup>=</sup>

the excited-state energy eigenfunctions and the complete energy spectrum of *<sup>H</sup>*( *<sup>c</sup>*

<sup>2</sup> )]*nψ*(0)( *<sup>c</sup>*

realization of the *so*(2, 1) ∼ *su*(1, 1) ∼ *s*ℓ(2, **R**) Lie algebra. When projected the −<sup>1</sup>

(**1** + Σ3)

(**1** + Σ3)

*d*

 *<sup>a</sup>*<sup>−</sup><sup>2</sup> <sup>=</sup>

 *a*<sup>+</sup><sup>2</sup> =

*dx* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> ( *<sup>c</sup>*

<sup>2</sup> (**<sup>1</sup>** <sup>+</sup> <sup>Σ</sup>3), viz.,

2

2

1 2

$$a^{-}(\frac{c}{2}) = \frac{1}{\sqrt{2}} \Sigma\_1 \mathbf{x}^{(1/2)c\Sigma\_3} \exp(\frac{-1}{2}\mathbf{x}^2)(\frac{-d}{d\mathbf{x}}) \exp(\frac{1}{2}\mathbf{x}^2)\mathbf{x}^{-(1/2)c\Sigma\_3} \tag{11}$$

where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes

$$\begin{split}H(\frac{c}{2}) &= \frac{1}{2} \left[ a^+(\frac{c}{2}), a^-(\frac{c}{2}) \right]\_+ \\ &= \begin{pmatrix} H\_-\left(\frac{c}{2} - 1\right) & 0 \\ 0 & H\_+\left(\frac{c}{2} - 1\right) = H\_-\left(\frac{c}{2}\right) \end{pmatrix} .\end{split} \tag{12}$$

where the even and odd sector Hamiltonians are respectively given by

$$H\_{-}(\frac{c}{2} - 1) = \frac{1}{2} \left\{ -\frac{d^2}{dx^2} + x^2 + \frac{1}{x^2}(\frac{c}{2})(\frac{c}{2} - 1) \right\} \tag{13}$$

and

$$H\_+(\frac{c}{2} - 1) = \frac{1}{2} \left\{ -\frac{d^2}{dx^2} + x^2 + \frac{1}{x^2}(\frac{c}{2})(\frac{c}{2} + 1) \right\} = H\_-(\frac{c}{2}).\tag{14}$$

The time-independent Schrödinger equation for these Hamiltonians of an isotonic oscillator (harmonic plus a centripetal barrier) system becomes the following eigenvalue equation:

$$H\_{\pm}(\frac{c}{2} - 1) \mid m, \frac{c}{2} - 1 > = E\_{\pm}(\frac{c}{2} - 1) \mid m, \frac{c}{2} - 1 > . \tag{15}$$

Thus, from the annihilation condition *<sup>a</sup>*−|<sup>0</sup> >= 0, the ground-state energy is given by

$$E^{(0)}(\frac{c}{2}) = \frac{1}{2}(1+c) > \frac{1}{2}, \quad c > 0. \tag{16}$$

At this stage an independent verification of the existence or not of a zero ground-state energy for *<sup>H</sup>*( *<sup>c</sup>* <sup>2</sup> ) suggested by its positive semi-definite form may be in order.

The question we formulate now is the following: What is the behaviour of the ladder operators on the autokets of the Wigner oscillator quantum states? To answer this question is obtained via WH algebra, note that the Wigner oscillator ladder operators on autokets of these quantum states are given by

$$a\_{\frac{c}{2}}^{-}|2m,\frac{c}{2}> = \sqrt{2m}|2m-1,\frac{c}{2}>$$

$$a\_{\frac{c}{2}}^{-}|2m+1,\frac{c}{2}> = \sqrt{2(m+E^{(0)})}|2m,\frac{c}{2}>$$

$$a\_{\frac{c}{2}}^{+}|2m,\frac{c}{2}> = \sqrt{2(m+E^{(0)})}|2m+1,\frac{c}{2}>$$

$$a\_{\frac{c}{2}}^{+}|2m+1,\frac{c}{2}> = \sqrt{2(m+1)}|2m+2,\frac{c}{2}>.\tag{17}$$

Now, from the role of *<sup>a</sup>*+( *<sup>c</sup>* <sup>2</sup> ) as the energy step-up operator (the upper sign choice) the excited-state energy eigenfunctions and the complete energy spectrum of *<sup>H</sup>*( *<sup>c</sup>* <sup>2</sup> ) are respectively given by *<sup>ψ</sup>*(*<sup>n</sup>*)( *<sup>c</sup>* <sup>2</sup> ) <sup>∝</sup> [*a*+( *<sup>c</sup>* <sup>2</sup> )]*nψ*(0)( *<sup>c</sup>* <sup>2</sup> ) and *<sup>E</sup>*(*<sup>n</sup>*)( *<sup>c</sup>* <sup>2</sup> ) = *<sup>E</sup>*(0) <sup>+</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, 2, ··· .

It is known that the operators ± *<sup>i</sup>* 2 *<sup>a</sup>*±( *<sup>c</sup>* 2 ) <sup>2</sup> and <sup>1</sup> <sup>2</sup> *<sup>H</sup>*( *<sup>c</sup>* <sup>2</sup> ) can be chosen as a basis for a realization of the *so*(2, 1) ∼ *su*(1, 1) ∼ *s*ℓ(2, **R**) Lie algebra. When projected the −<sup>1</sup> 2 *a*<sup>±</sup><sup>2</sup> operators in the even sector with <sup>1</sup> <sup>2</sup> (**<sup>1</sup>** <sup>+</sup> <sup>Σ</sup>3), viz.,

$$\frac{1}{2}(\mathbf{1} + \Sigma\_3)B^- = \frac{1}{2}(\mathbf{1} + \Sigma\_3)\begin{pmatrix} a^- \end{pmatrix}^2 = \begin{pmatrix} B^- & 0\\ 0 & 0 \end{pmatrix}^2$$

and

4 Advances in Quantum Mechanics

forms:

and

The present author have applied a super-realization so that *R* = Σ<sup>3</sup> to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra,

*<sup>b</sup>*∓(= (*b*±)†) that commute with the bosonic set and are represented in terms of the usual

1 2 *<sup>x</sup>*2)( *<sup>d</sup>*

−1 <sup>2</sup> *<sup>x</sup>*2)(−*<sup>d</sup>*

where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10)

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>0</sup> 0 *H*+( *<sup>c</sup>*

*dx*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup>

1 *x*2 ( *c* 2 )( *c* <sup>2</sup> <sup>+</sup> <sup>1</sup>) 

*c*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*,

*c*

*dx* )*exp*(

*dx* )*exp*(

<sup>2</sup> <sup>−</sup> <sup>1</sup>) = *<sup>H</sup>*−( *<sup>c</sup>*

1 *x*2 ( *c* 2 )( *c* <sup>2</sup> <sup>−</sup> <sup>1</sup>) 

2 ) 

> = *H*−( *c* 2

−1

1 2 *dx* ), the fermionic ones

<sup>2</sup> ) in the following respective factorised

<sup>2</sup> *<sup>x</sup>*2)*x*−(1/2)*c*Σ<sup>3</sup> (10)

*<sup>x</sup>*2)*x*−(1/2)*c*Σ<sup>3</sup> , (11)

, (12)

). (14)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; . (15)

(13)

we introduce, in addition to the usual bosonic coordinates (*x*, −*i <sup>d</sup>*

<sup>Σ</sup>1*x*(1/2)*c*Σ<sup>3</sup> *exp*(

<sup>Σ</sup>1*x*(1/2)*c*Σ<sup>3</sup> *exp*(

Pauli matrices <sup>Σ</sup>*i*,(*<sup>i</sup>* <sup>=</sup> 1, 2, 3). Indeed, expressing *<sup>a</sup>*±( *<sup>c</sup>*

) = <sup>1</sup> √2

) = <sup>1</sup> √2

> *H*( *c* 2 ) = <sup>1</sup> 2 *a*+( *c* 2 ), *<sup>a</sup>*−( *c* 2 ) +

*H*−( *c*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) = <sup>1</sup>

*H*+( *c*

> *H*±( *c*

=

*H*−( *<sup>c</sup>*

where the even and odd sector Hamiltonians are respectively given by

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

*c*

*dx*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup>

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;<sup>=</sup> *<sup>E</sup>*±(

Thus, from the annihilation condition *<sup>a</sup>*−|<sup>0</sup> >= 0, the ground-state energy is given by

The time-independent Schrödinger equation for these Hamiltonians of an isotonic oscillator (harmonic plus a centripetal barrier) system becomes the following eigenvalue equation:

<sup>2</sup> <sup>−</sup> <sup>1</sup>) = <sup>1</sup>

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

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*,

*a*+( *c* 2

*<sup>a</sup>*−( *c* 2

the Wigner Hamiltonian becomes

$$\frac{1}{2}(\mathbf{1} + \Sigma\_3)B^+ = \frac{1}{2}(\mathbf{1} + \Sigma\_3)\begin{pmatrix} a^+ \\ \end{pmatrix}^2 = \begin{pmatrix} B^+ & 0 \\ 0 & 0 \\ \end{pmatrix}^2$$

we obtain

$$B^{-}(\frac{c}{2} - 1) = \frac{1}{2} \left\{ \frac{d^2}{dx^2} + 2x\frac{d}{dx} + x^2 - \frac{(\frac{c}{2} - 1)\frac{c}{2}}{x^2} + 1 \right\} \tag{18}$$

and

$$B^{+}(\frac{\mathcal{C}}{2} - 1) = (B^{-})^{\dagger} = \frac{1}{2} \left\{ \frac{d^2}{d\mathbf{x}^2} - 2\mathbf{x}\frac{d}{d\mathbf{x}} + \mathbf{x}^2 - \frac{(\frac{\mathcal{C}}{2} - 1)\frac{\mathcal{C}}{2}}{\mathbf{x}^2} - 1 \right\}.\tag{19}$$

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**3. The 3D Wigner and SUSY systems**

(*Q*−)

The connection between the 3D Wigner Hamiltonian *<sup>H</sup>*(�*<sup>σ</sup>* ·�

(<sup>1</sup> <sup>−</sup> <sup>Σ</sup>3)*a*−(�*<sup>σ</sup>* ·�

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

In this Section, we consider the 3D isotropic spin- <sup>1</sup>

<sup>2</sup>) = <sup>1</sup> 2 (<sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>r*<sup>2</sup> <sup>−</sup> <sup>2</sup> *r ∂ ∂r* + 1 *<sup>r</sup>*<sup>2</sup> (�*<sup>σ</sup>* ·�

for a non-relativistic 3D isotropic oscillator with spin- <sup>1</sup>

*i r <sup>σ</sup>r*(�*<sup>σ</sup>* ·�

�*σ* ·�*r*, *σ*<sup>2</sup>

The 3D fermionic sector Hamiltonian becomes

<sup>2</sup> equalities:

*<sup>r</sup>* <sup>=</sup> 1, [*σr*,�*<sup>σ</sup>* ·�

*L* + **1**), *pr* = −*i*(

Witten [1–4]

odd elements).

harmonic oscillator for spin <sup>1</sup>

sector of a Wigner system

*<sup>L</sup>*) = <sup>1</sup> 2 (*p*<sup>2</sup> + *r*

of the following familiar spin- <sup>1</sup>

*<sup>σ</sup><sup>r</sup>* <sup>=</sup> <sup>1</sup> *r*

�*σ* · �*p* = *σ<sup>r</sup> pr* +

*<sup>H</sup>*−(*<sup>σ</sup>* ·�

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

*Q*<sup>+</sup> = *Q*†

As is well-known, the quantum mechanical (QM) *N* = 2 supersymmetry (SUSY) algebra of

involves bosonic and fermionic sector Hamiltonians of the SUSY Hamiltonian *Hss* (the even element), which get intertwined through the nilpotent charge operators *Q*<sup>−</sup> = (*Q*+)† (the

<sup>2</sup> = 0 → [*Hss*, *Q*∓] = 0, (25)

The Wigner-Heisenberg Algebra in Quantum Mechanics

*L* + **1**), (*Q*−)<sup>2</sup> = (*Q*+)<sup>2</sup> = 0. (26)

<sup>2</sup> oscillator Hamiltonian in the bosonic

<sup>2</sup>) (0 < *r* < ∞) (27)

)*r* = *p*†

*<sup>L</sup>*(�*<sup>σ</sup>* ·�

<sup>2</sup> *σ*. With the use

*r* (28)

*L* + **1**). (29)

<sup>2</sup> is given by anti-commutation relation (25) and the mutually

*<sup>L</sup>*)(�*<sup>σ</sup>* ·�

1 *r* + *∂ ∂r* ) = <sup>1</sup> *r* (−*i ∂ ∂r*

*<sup>L</sup>* <sup>+</sup> **<sup>1</sup>**]+ <sup>=</sup> 0, *<sup>L</sup>*<sup>2</sup> <sup>=</sup>�*<sup>σ</sup>* ·�

*L* + **1**) + *r*

<sup>2</sup> represented here by <sup>1</sup>

*L* + **1**) and a 3D SUSY isotropic

*Hss* = [*Q*−, *Q*+]+

adjoint charge operators *Q*<sup>∓</sup> in terms of the Wigner system ladder operators:

(<sup>1</sup> <sup>+</sup> <sup>Σ</sup>3)*a*+(�*<sup>σ</sup>* ·�

*L* + **1**)

<sup>2</sup> = (*Q*+)

Thus, the Lie algebra becomes

$$[\mathbf{K}\_{0\prime}\mathbf{K}\_{-}]\_{-}=-\mathbf{K}\_{-\prime}\quad[\mathbf{K}\_{0\prime}\mathbf{K}\_{+}]\_{-}=+\mathbf{K}\_{+\prime}\quad[\mathbf{K}\_{-\prime}\mathbf{K}\_{+}]\_{-}=2\mathbf{K}\_{0\prime}\tag{20}$$

where *K*<sup>0</sup> = <sup>1</sup> <sup>2</sup> *<sup>H</sup>*−, *<sup>K</sup>*<sup>−</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> *<sup>B</sup>*<sup>−</sup> and *<sup>K</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>B</sup>*<sup>+</sup> generate once again the *su*(1, 1) Lie algebra. Therefore, these ladder operators obey the following commutation relations:

$$\begin{aligned} \left[\mathcal{B}^{-}(\frac{\mathcal{C}}{2}-1), \mathcal{B}^{+}(\frac{\mathcal{C}}{2}-1)\right]\_{-} &= 4H\_{-}(\frac{\mathcal{C}}{2}-1) \\ \left[H\_{-}, \mathcal{B}^{\pm}(\frac{\mathcal{C}}{2}-1)\right]\_{-} &= \pm 2\mathcal{B}^{\pm}(\frac{\mathcal{C}}{2}-1). \end{aligned} \tag{21}$$

Hence, the quadratic operators *<sup>B</sup>*±( *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup>) acting on the orthonormal basis of eigenstates of *H*−( *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup>), {| *<sup>m</sup>*, *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;} where *<sup>m</sup>* <sup>=</sup> 0, 1, 2, ··· have the effect of raising or lowering the quanta by two units so that we can write

$$B^{-}(\frac{c}{2} - 1) \mid m, \frac{c}{2} - 1 > = \sqrt{2m(2m + c + 1)} \mid m - 1, \frac{c}{2} - 1 > \tag{22}$$

and

$$B^{+}(\frac{c}{2} - 1) \mid m, \frac{c}{2} - 1 > = \sqrt{2(m+1)(2m+c+1)} \mid m+1, \frac{c}{2} - 1 > \tag{23}$$

giving

$$\left| m, \frac{c}{2} - 1 > = 2^{-m} \left\{ \frac{\Gamma(\frac{c+1}{2})}{m! \Gamma(\frac{c+1}{2} + m)} \right\}^{1/2} \left\{ \mathcal{B}^+ (\frac{c}{2} - 1) \right\}^m \right| \left| 0, \frac{c}{2} - 1 > \right. \tag{24}$$

where <sup>Γ</sup>(*x*) is the ordinary Gamma Function. Note that *<sup>B</sup>*±( *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*, *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; are associated with the energy eigenvalues *<sup>E</sup>*(*m*±1) <sup>−</sup> = *<sup>c</sup>*+<sup>1</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>m</sup>* <sup>±</sup> <sup>1</sup>), *<sup>m</sup>* <sup>=</sup> 0, 1, 2, . . . .

Let us to conclude this section presenting the following comments: one can generate the called canonical coherent states, which are defined as the eigenstates of the lowering operator *<sup>B</sup>*−( *<sup>c</sup>* <sup>2</sup> <sup>−</sup> <sup>1</sup>) of the bosonic sector, according to the Barut-Girardello approach [12, 13] and generalized coherent states according to Perelomov [14, 15]. Results of our investigations on these coherent states will be reported separately.

### **3. The 3D Wigner and SUSY systems**

6 Advances in Quantum Mechanics

*B*+( *c*

<sup>2</sup> *<sup>H</sup>*−, *<sup>K</sup>*<sup>−</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

Hence, the quadratic operators *<sup>B</sup>*±( *<sup>c</sup>*

*<sup>B</sup>*−( *c*

*B*+( *c*


with the energy eigenvalues *<sup>E</sup>*(*m*±1)

quanta by two units so that we can write

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*,

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*,

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;<sup>=</sup> <sup>2</sup>−*<sup>m</sup>*

these coherent states will be reported separately.

<sup>2</sup> <sup>−</sup> <sup>1</sup>), {| *<sup>m</sup>*, *<sup>c</sup>*

Thus, the Lie algebra becomes

where *K*<sup>0</sup> = <sup>1</sup>

*H*−( *<sup>c</sup>*

and

giving

*<sup>B</sup>*−( *<sup>c</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup>)=(*B*−)† <sup>=</sup> <sup>1</sup>

[*B*−( *c* 2

<sup>2</sup> *<sup>B</sup>*<sup>−</sup> and *<sup>K</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup>

<sup>2</sup> <sup>−</sup> <sup>1</sup>), *<sup>B</sup>*+(

*<sup>H</sup>*−, *<sup>B</sup>*±(

*c*

*c*

where <sup>Γ</sup>(*x*) is the ordinary Gamma Function. Note that *<sup>B</sup>*±( *<sup>c</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;<sup>=</sup>

Γ( *<sup>c</sup>*+<sup>1</sup>

*<sup>m</sup>*!Γ( *<sup>c</sup>*+<sup>1</sup>

<sup>−</sup> = *<sup>c</sup>*+<sup>1</sup>

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;<sup>=</sup>

Therefore, these ladder operators obey the following commutation relations:

*c*

*c* <sup>2</sup> <sup>−</sup> <sup>1</sup>) 

<sup>2</sup> )

<sup>2</sup> <sup>+</sup> *<sup>m</sup>*)

Let us to conclude this section presenting the following comments: one can generate the called canonical coherent states, which are defined as the eigenstates of the lowering operator

<sup>2</sup> <sup>−</sup> <sup>1</sup>) of the bosonic sector, according to the Barut-Girardello approach [12, 13] and generalized coherent states according to Perelomov [14, 15]. Results of our investigations on

1/2 *B*+( *c* <sup>2</sup> <sup>−</sup> <sup>1</sup>)

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

[*K*0, *<sup>K</sup>*−]<sup>−</sup> = −*K*−, [*K*0, *<sup>K</sup>*+]<sup>−</sup> = +*K*+, [*K*−, *<sup>K</sup>*+]<sup>−</sup> = <sup>2</sup>*K*0, (20)

<sup>−</sup> <sup>=</sup> <sup>±</sup>2*B*±(

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;} where *<sup>m</sup>* <sup>=</sup> 0, 1, 2, ··· have the effect of raising or lowering the

<sup>2</sup>*m*(2*<sup>m</sup>* <sup>+</sup> *<sup>c</sup>* <sup>+</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>* <sup>−</sup> 1, *<sup>c</sup>*

<sup>2</sup>(*<sup>m</sup>* <sup>+</sup> <sup>1</sup>)(2*<sup>m</sup>* <sup>+</sup> *<sup>c</sup>* <sup>+</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>* <sup>+</sup> 1, *<sup>c</sup>*

<sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>m</sup>* <sup>±</sup> <sup>1</sup>), *<sup>m</sup>* <sup>=</sup> 0, 1, 2, . . . .

*c* <sup>2</sup> <sup>−</sup> <sup>1</sup>)

*c*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) acting on the orthonormal basis of eigenstates of

*m* <sup>|</sup> 0, *<sup>c</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) <sup>|</sup> *<sup>m</sup>*, *<sup>c</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup>)]<sup>−</sup> <sup>=</sup> <sup>4</sup>*H*−(

*dx* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> ( *<sup>c</sup>*

<sup>2</sup> <sup>−</sup> <sup>1</sup>) *<sup>c</sup>* 2 *<sup>x</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>

<sup>2</sup> *<sup>B</sup>*<sup>+</sup> generate once again the *su*(1, 1) Lie algebra.

<sup>2</sup> <sup>−</sup> <sup>1</sup>). (21)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; (22)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; (23)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt;, (24)

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>gt; are associated

. (19)

As is well-known, the quantum mechanical (QM) *N* = 2 supersymmetry (SUSY) algebra of Witten [1–4]

$$\begin{aligned} H\_{\text{ss}} &= \left[Q\_{-}, Q\_{+}\right]\_{+} \\ \left(Q\_{-}\right)^{2} &= \left(Q\_{+}\right)^{2} = 0 \quad \rightarrow \quad \left[H\_{\text{ss}}, Q\_{\mp}\right] = 0,\end{aligned} \tag{25}$$

involves bosonic and fermionic sector Hamiltonians of the SUSY Hamiltonian *Hss* (the even element), which get intertwined through the nilpotent charge operators *Q*<sup>−</sup> = (*Q*+)† (the odd elements).

The connection between the 3D Wigner Hamiltonian *<sup>H</sup>*(�*<sup>σ</sup>* ·� *L* + **1**) and a 3D SUSY isotropic harmonic oscillator for spin <sup>1</sup> <sup>2</sup> is given by anti-commutation relation (25) and the mutually adjoint charge operators *Q*<sup>∓</sup> in terms of the Wigner system ladder operators:

$$\begin{aligned} Q\_- &= \frac{1}{2} (1 - \Sigma\_3) a^- (\vec{\sigma} \cdot \vec{L} + \mathbf{1}) \\ Q\_+ &= Q\_-^\dagger = \frac{1}{2} (1 + \Sigma\_3) a^+ (\vec{\sigma} \cdot \vec{L} + \mathbf{1}), \quad (Q\_-)^2 = (Q\_+)^2 = 0. \end{aligned} \tag{26}$$

In this Section, we consider the 3D isotropic spin- <sup>1</sup> <sup>2</sup> oscillator Hamiltonian in the bosonic sector of a Wigner system

$$H\_{-}(\vec{\sigma}\cdot\vec{L}) = \frac{1}{2}(p^2 + r^2) = \frac{1}{2}(-\frac{\partial^2}{\partial r^2} - \frac{2}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}(\vec{\sigma}\cdot\vec{L})(\vec{\sigma}\cdot\vec{L} + \mathbf{1}) + r^2) \quad (0 < r < \infty) \quad (27)$$

for a non-relativistic 3D isotropic oscillator with spin- <sup>1</sup> <sup>2</sup> represented here by <sup>1</sup> <sup>2</sup> *σ*. With the use of the following familiar spin- <sup>1</sup> <sup>2</sup> equalities:

$$\vec{\sigma} \cdot \vec{p} = \sigma\_r p\_r + \frac{i}{r} \sigma\_r (\vec{\sigma} \cdot \vec{L} + \mathbf{1}), \quad p\_r = -i(\frac{1}{r} + \frac{\partial}{\partial r}) = \frac{1}{r}(-i\frac{\partial}{\partial r})r = p\_r^\dagger \tag{28}$$

$$
\sigma\_r = \frac{1}{r}\vec{\sigma} \cdot \vec{\tau}, \quad \sigma\_r^2 = 1, \quad [\sigma\_{r\prime}\vec{\sigma} \cdot \vec{L} + \mathbf{1}]\_+ = 0, \quad L^2 = \vec{\sigma} \cdot \vec{L}(\vec{\sigma} \cdot \vec{L} + \mathbf{1}).\tag{29}
$$

The 3D fermionic sector Hamiltonian becomes

$$H\_{+}(\vec{\sigma}\cdot\vec{L}) = H\_{-}(\vec{\sigma}\cdot\vec{L}+\mathbf{1}) = \frac{1}{2}(-\frac{\partial^{2}}{\partial r^{2}} - \frac{2}{r}\frac{\partial}{\partial r} + \frac{1}{r^{2}}(\vec{\sigma}\cdot\vec{L}+\mathbf{1})(\vec{\sigma}\cdot\vec{L}+\mathbf{2}) + r^{2}).\tag{30}$$

In this case the connection between *Hss* and *H*w is given by

$$H\_{\rm ss} = H\_{\rm W} - \frac{1}{2} \Sigma\_3 \{ 1 + 2(\vec{\sigma} \cdot \vec{L} + \mathbf{1}) \Sigma\_3 \},\tag{31}$$

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= *H*−(ℓ + 1). (37)

<sup>−</sup> <sup>=</sup> <sup>±</sup>*a*±(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>). (39)

<sup>−</sup> = <sup>1</sup> + <sup>2</sup>(ℓ + <sup>1</sup>)Σ<sup>3</sup> (40)

 0 <sup>|</sup> *<sup>m</sup>*, ℓ > (36)

485

(41)

(43)

*<sup>H</sup>*−(ℓ) = <sup>1</sup>

*<sup>H</sup>*+(ℓ) = <sup>1</sup>

2 <sup>−</sup> *<sup>d</sup>*<sup>2</sup> *dr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

In this representation the eigenvalue equation becomes

The WH algebra ladder relations are readily obtained as

for even(odd) quanta *n* = 2*m*(*n* = 2*m* + 1) are given by


and satisfy the following eigenvalue equation

on the ground eigenstate and are given by

and

*<sup>ψ</sup>*(*n*)

*<sup>W</sup>* (*r*) <sup>∝</sup> (*a*+)*<sup>n</sup>ψ*(0)

constitute the WH algebra.

Equations (35) and (39) together with the commutation relation

and

2 <sup>−</sup> *<sup>d</sup>*<sup>2</sup> *dr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

> <sup>2</sup> + 1

*<sup>H</sup>*(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>), *<sup>a</sup>*±(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)

*<sup>a</sup>*−(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>), *<sup>a</sup>*+(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)

 | *m*, ℓ > 0

<sup>2</sup> + 1 *<sup>r</sup>*<sup>2</sup> <sup>ℓ</sup>(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)

*<sup>r</sup>*<sup>2</sup> (<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)(<sup>ℓ</sup> <sup>+</sup> <sup>2</sup>)

The Wigner eigenfunctions that generate the eigenspace associated with even(odd) *σ*3-parity

the non-degenerate energy eigenvalues are obtained by the application of the raising operator

*<sup>W</sup>*,+(*r*)=(*a*+)*<sup>n</sup>χ*(0)

, | *n* = 2*m* + 1, ℓ >=

*<sup>H</sup>*(ℓ + <sup>1</sup>) | *<sup>n</sup>*, ℓ + <sup>1</sup> >= *<sup>E</sup>*(*n*) | *<sup>n</sup>*, ℓ + <sup>1</sup> >, (42)

1,+(*r*) 1 0 

The Wigner-Heisenberg Algebra in Quantum Mechanics

*H*±(ℓ)*χ*(*r*) = *E*±(ℓ)*χ*(*r*), *χ*(*r*) = *rR*(*r*). (38)

where *<sup>H</sup>*<sup>w</sup> <sup>=</sup> *diag*(*H*˜ <sup>−</sup>(�*<sup>σ</sup>* ·� *<sup>L</sup>*), *<sup>H</sup>*˜ +(�*<sup>σ</sup>* ·� *L*)) is the diagonal 3D Wigner Hamiltonian given by

$$H\_{\mathbf{W}} = H\_{\mathbf{W}}(\vec{\boldsymbol{\sigma}} \cdot \vec{\boldsymbol{L}} + \mathbf{1}) = \begin{pmatrix} H\_{-}(\vec{\boldsymbol{\sigma}} \cdot \vec{\boldsymbol{L}}) & \mathbf{0} \\ \mathbf{0} & H\_{-}(\vec{\boldsymbol{\sigma}} \cdot \vec{\boldsymbol{L}} + \mathbf{1}) \end{pmatrix}. \tag{32}$$

Indeed from the role of *a*<sup>+</sup> as the energy step-up operator, the complete excited state wave functions Ψ*<sup>n</sup> <sup>w</sup>* are readily given by the step up operation with *<sup>a</sup>*+:

$$
\psi\_{\mathbf{W}}^{(n)} \propto (a^+)^n \psi\_{\mathbf{W},+}^{(0)}(r,\theta,\varphi) = (a^+)^n \tilde{R}\_{1,+}^{(0)}(r) \begin{pmatrix} 1 \\ 0 \end{pmatrix} \mathfrak{z}\_+(\theta,\varphi). \tag{33}
$$

On the eigenspaces of the operator (�*<sup>σ</sup>* ·� *L* + **1**), the 3D Wigner algebra gets reduced to a 1D from with (�*<sup>σ</sup>* ·� *L* + **1**) replaced by its eigenvalue ∓(ℓ + 1), ℓ = 0, 1, 2,..., where ℓ is the orbital angular momentum quantum number. The eigenfunctions of (�*<sup>σ</sup>* ·� *L* + **1**) for the eigenvalues (ℓ + 1) and −(ℓ + 1) are respectively given by the well known spin-spherical harmonic *y*∓. Thus, from the super-realized first order ladder operators given by

$$a^{\pm}(\ell+1) = \frac{1}{\sqrt{2}} \left\{ \pm \frac{d}{dr} \pm \frac{(\ell+1)}{r} \Sigma\_3 - r \right\} \Sigma\_{1\prime} \tag{34}$$

where *<sup>c</sup>* <sup>2</sup> <sup>=</sup> <sup>ℓ</sup> <sup>+</sup> 1, the Wigner Hamiltonian becomes

$$\begin{split}H(\ell+1) &= \frac{1}{2} \left[ a^+(\ell+1) , a^-(\ell+1) \right]\_+ \\ &= \begin{pmatrix} H\_-(\ell) & 0 \\ 0 & H\_+(\ell) = H\_-(\ell+1) \end{pmatrix} .\end{split} \tag{35}$$

where in the representation of the radial part wave functions, *χ*(*r*) = *rR*(*r*), the even and odd sector Hamiltonians are respectively given by

$$H\_{-}(\ell) = \frac{1}{2} \left\{ -\frac{d^2}{dr^2} + r^2 + \frac{1}{r^2}\ell(\ell+1) \right\} \tag{36}$$

and

8 Advances in Quantum Mechanics

*<sup>H</sup>*+(�*<sup>σ</sup>* ·�

where *<sup>H</sup>*<sup>w</sup> <sup>=</sup> *diag*(*H*˜ <sup>−</sup>(�*<sup>σ</sup>* ·�

functions Ψ*<sup>n</sup>*

from with (�*<sup>σ</sup>* ·�

where *<sup>c</sup>*

*<sup>L</sup>*) = *<sup>H</sup>*−(�*<sup>σ</sup>* ·�

*<sup>L</sup>* <sup>+</sup> **<sup>1</sup>**) = <sup>1</sup>

*Hss* <sup>=</sup> *<sup>H</sup>*<sup>w</sup> <sup>−</sup> <sup>1</sup>

*<sup>L</sup>*), *<sup>H</sup>*˜ +(�*<sup>σ</sup>* ·�

*<sup>H</sup>*<sup>w</sup> <sup>=</sup> *<sup>H</sup>*w(�*<sup>σ</sup>* ·�

<sup>w</sup> <sup>∝</sup> (*a*+)*<sup>n</sup>ψ*(0)

angular momentum quantum number. The eigenfunctions of (�*<sup>σ</sup>* ·�

Thus, from the super-realized first order ladder operators given by

√2 ± *d*

2 

=

*<sup>a</sup>*±(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>) = <sup>1</sup>

*<sup>H</sup>*(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>) = <sup>1</sup>

odd sector Hamiltonians are respectively given by

<sup>2</sup> <sup>=</sup> <sup>ℓ</sup> <sup>+</sup> 1, the Wigner Hamiltonian becomes

*<sup>ψ</sup>*(*n*)

On the eigenspaces of the operator (�*<sup>σ</sup>* ·�

In this case the connection between *Hss* and *H*w is given by

2 (<sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>r*<sup>2</sup> <sup>−</sup> <sup>2</sup> *r ∂ ∂r* + 1 *<sup>r</sup>*<sup>2</sup> (�*<sup>σ</sup>* ·�

2

*L* + **1**) =

*<sup>w</sup>* are readily given by the step up operation with *<sup>a</sup>*+:

<sup>Σ</sup>3{<sup>1</sup> <sup>+</sup> <sup>2</sup>(�*<sup>σ</sup>* ·�

*<sup>H</sup>*−(�*<sup>σ</sup>* ·�

Indeed from the role of *a*<sup>+</sup> as the energy step-up operator, the complete excited state wave

w,+(*r*, *<sup>θ</sup>*, *<sup>ϕ</sup>*)=(*a*+)*nR*(0)

(ℓ + 1) and −(ℓ + 1) are respectively given by the well known spin-spherical harmonic *y*∓.

*<sup>L</sup>* <sup>+</sup> **<sup>1</sup>**)(�*<sup>σ</sup>* ·�

*L*)) is the diagonal 3D Wigner Hamiltonian given by

*L* + **1**)

*L* + **1**), the 3D Wigner algebra gets reduced to a 1D

*L*) 0 <sup>0</sup> *<sup>H</sup>*−(�*<sup>σ</sup>* ·�

> 1,+(*r*) 1 0

> > <sup>Σ</sup><sup>3</sup> − *<sup>r</sup>*

> > > +

*L* + **1**) replaced by its eigenvalue ∓(ℓ + 1), ℓ = 0, 1, 2,..., where ℓ is the orbital

*dr* <sup>±</sup> (<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>) *r*

*<sup>a</sup>*+(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>), *<sup>a</sup>*−(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)

0 *H*+(ℓ) = *H*−(ℓ + 1)

*H*−(ℓ) 0

where in the representation of the radial part wave functions, *χ*(*r*) = *rR*(*r*), the even and

*L* + **2**) + *r*

*<sup>L</sup>* + **<sup>1</sup>**)Σ3}, (31)

<sup>2</sup>). (30)

. (32)

*y*+(*θ*, *ϕ*). (33)

*L* + **1**) for the eigenvalues

Σ1, (34)

, (35)

$$H\_+(\ell) = \frac{1}{2} \left\{ -\frac{d^2}{dr^2} + r^2 + \frac{1}{r^2}(\ell+1)(\ell+2) \right\} = H\_-(\ell+1). \tag{37}$$

In this representation the eigenvalue equation becomes

$$H\_{\pm}(\ell)\chi(r) = E\_{\pm}(\ell)\chi(r), \quad \chi(r) = rR(r). \tag{38}$$

The WH algebra ladder relations are readily obtained as

$$\left[H(\ell+1), a^{\pm}(\ell+1)\right]\_{-}=\pm a^{\pm}(\ell+1). \tag{39}$$

Equations (35) and (39) together with the commutation relation

$$\left[a^{-}(\ell+1), a^{+}(\ell+1)\right]\_{-}=1+2(\ell+1)\Sigma\_{3}\tag{40}$$

constitute the WH algebra.

The Wigner eigenfunctions that generate the eigenspace associated with even(odd) *σ*3-parity for even(odd) quanta *n* = 2*m*(*n* = 2*m* + 1) are given by

$$<\langle n=2m,\ell+1> = \begin{pmatrix} \mid m,\ell> \\ 0 \end{pmatrix}\_{\ell} \quad \mid n=2m+1,\ell> = \begin{pmatrix} 0\\ \mid m,\ell> \end{pmatrix} \tag{41}$$

and satisfy the following eigenvalue equation

$$H(\ell+1) \mid n, \ell+1 > = E^{(n)} \mid n, \ell+1 > , \tag{42}$$

the non-degenerate energy eigenvalues are obtained by the application of the raising operator on the ground eigenstate and are given by

$$
\psi\_{\mathcal{W}}^{(n)}(r) \propto (a^+)^n \psi\_{\mathcal{W},+}^{(0)}(r) = (a^+)^n \chi\_{1,+}^{(0)}(r) \begin{pmatrix} 1 \\ 0 \end{pmatrix} \tag{43}
$$

and

$$E^{(n)} = \ell + \frac{3}{2} + n, \quad n = 0, 1, 2, \dots \tag{44}$$

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Kostelecky, Nieto and Truax have studied in a detailed manner the relation of the SUSY Coulombian problem in D-dimensions with that of SUSY isotropic oscillators in

The bosonic sector of the above eigenvalue equation can immediately be identified with the eigenvalue equation for the Hamiltonian of the 3D Hydrogen-like atom expressed in the

The usual isotropic oscillator in 4*D* has the following eigenvalue equation for it's

*osc*(*y*) = *<sup>E</sup>*<sup>B</sup>

defined, being implemented in the bosonic sector of the super 4*D* Wigner system with unitary frequency. Changing to spherical coordinates in 4-space dimensions, allowing a factorization of the energy eigenfunctions as a product of a radial eigenfunction and spin-spherical

In (52), the coordinates *yi*(*i* = 1, 2, 3, 4) in spherical coordinates in 4D are defined by [26, 29]

cos *<sup>ϕ</sup>* <sup>−</sup> *<sup>ω</sup>* 2

sin *<sup>ϕ</sup>* <sup>−</sup> *<sup>ω</sup>* 2

cos *<sup>ϕ</sup>* <sup>+</sup> *<sup>ω</sup>* 2

sin *<sup>ϕ</sup>* <sup>+</sup> *<sup>ω</sup>* 2

2 

2 

2 

2 

The mapping of the coordinates *yi*(*i* = 1, 2, 3, 4) in 4*D* with the Cartesian coordinates *ρi*(*i* =

*<sup>y</sup>*<sup>1</sup> <sup>=</sup> *<sup>s</sup>* cos *<sup>θ</sup>*

*<sup>y</sup>*<sup>2</sup> <sup>=</sup> *<sup>s</sup>* cos *<sup>θ</sup>*

*<sup>y</sup>*<sup>3</sup> <sup>=</sup> *<sup>s</sup>* sin *<sup>θ</sup>*

*<sup>y</sup>*<sup>4</sup> <sup>=</sup> *<sup>s</sup>* sin *<sup>θ</sup>*

1, 2, 3) in 3*D* is given by the Kustaanheimo-Stiefel transformation

where 0 ≤ *θ* ≤ *π*, 0 ≤ *ϕ* ≤ 2*π* and 0 ≤ *ω* ≤ 4*π*.

osc, described by (employing natural system of units ¯*h* = *m* = 1)

*osc*Ψ<sup>B</sup>

<sup>2</sup> = Σ<sup>4</sup>

*i*=1*y*<sup>2</sup>

osc is in anticipation of the Hamiltonian, with constraint to be

*∂*2 *∂y*<sup>2</sup> *i*

*osc*(*y*), (50)

The Wigner-Heisenberg Algebra in Quantum Mechanics

*<sup>i</sup>* , (51)

, (52)

, (53)

D-dimensions in the radial version. (See also Lahiri *et. al.* [2].)

*H*B *osc*Ψ<sup>B</sup>

*H*B *osc* <sup>=</sup> <sup>−</sup><sup>1</sup> 2 ∇2 <sup>4</sup> + 1 2 *s* 2, *s*

∇2 <sup>4</sup> <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> *∂y*<sup>2</sup> 1 + *∂*2 *∂y*<sup>2</sup> 2 + *∂*2 *∂y*<sup>2</sup> 3 + *∂*2 *∂y*<sup>2</sup> 4 = 4 ∑ *i*=1

equivalent form given by [16]

where the superscript *B* in *H<sup>B</sup>*

time-independent Schrödinger equation

Hamiltonian *H<sup>B</sup>*

with

harmonic.

The ground state energy eigenvalue is determined by the annihilation condition which reads as:

$$a^-\psi\_{\mathbf{W},+}^{(0)} = 0, \quad (\vec{\sigma} \cdot \vec{L} + \mathbf{1}) \to \ell + 1; \tag{45}$$

which, after operation on the fermion spinors and the spin-angular part, turns into

$$
\begin{pmatrix}
\exp(\frac{r^2}{2})r^{-\ell-1}\chi\_{1,+}^{(0)}(r) \\
\exp(\frac{r^2}{2})r^{-\ell+1}\chi\_{2,+}^{(0)}(r)
\end{pmatrix} = \begin{pmatrix}c\_1\\c\_2\end{pmatrix} \tag{46}
$$

Retaining only the non-singular and normalizable *<sup>R</sup>*(0) 1,+(*r*), we simply take the singular solution *<sup>R</sup>*(0) 2,+(*r*), which is physically non-existing, as identically zero. Hence the Wigner's eigenfunction of the ground state becomes

$$
\psi\_{\mathbf{W},+}^{(0)} = \begin{pmatrix} \chi\_{1,+}^{(0)}(r) y\_+ \\ 0 \end{pmatrix}, \quad \chi\_{1,+}^{(0)}(r) \approx r^{\ell+1} \exp(-\frac{r^2}{2}), \tag{47}
$$

where 0 < *r* < ∞. For the radial oscillator the energy eigenvectors satisfy the following eigenvalue equations

$$H\_{-}(\ell) \mid \mathfrak{m}, \ell > = E\_{-}^{(m)} \mid \mathfrak{m}, \ell > , \tag{48}$$

where the eigenvalues are exactly constructed via WH algebra ladder relations and are given by

$$E\_{-}^{(m)} = \ell + \frac{3}{2} + 2m, \qquad m = 0, 1, 2, \dots \tag{49}$$

We stress that similar results can be adequately extended for any physical D-dimensional radial oscillator system by the Hermitian replacement of −*i d dr* <sup>+</sup> <sup>1</sup> *r* → −*i d dr* <sup>+</sup> *<sup>D</sup>*−<sup>1</sup> 2*r* and the Wigner deformation parameter ℓ + <sup>1</sup> → ℓ*<sup>D</sup>* + <sup>1</sup> <sup>2</sup> (*<sup>D</sup>* <sup>−</sup> <sup>1</sup>) where <sup>ℓ</sup>*D*(ℓ*<sup>D</sup>* <sup>=</sup> 0, 1, 2, ···) is the D-dimensional oscillator angular momentum.

#### **4. The constrained Super Wigner Oscillator in** 4*D* **and the hydrogen atom**

In this section, the complete spectrum for the hydrogen atom is found with considerable simplicity. Indeed, the solutions of the time-independent Schrödinger equation for the hydrogen atom were mapped onto the super Wigner harmonic oscillator in 4*D* by using the Kustaanheimo-Stiefel transformation. The Kustaanheimo-Stiefel mapping yields the Schrödinger equation for the hydrogen atom that has been exactly solved and well-studied in the literature. (See for example, [16].)

Kostelecky, Nieto and Truax have studied in a detailed manner the relation of the SUSY Coulombian problem in D-dimensions with that of SUSY isotropic oscillators in D-dimensions in the radial version. (See also Lahiri *et. al.* [2].)

The bosonic sector of the above eigenvalue equation can immediately be identified with the eigenvalue equation for the Hamiltonian of the 3D Hydrogen-like atom expressed in the equivalent form given by [16]

The usual isotropic oscillator in 4*D* has the following eigenvalue equation for it's Hamiltonian *H<sup>B</sup>* osc, described by (employing natural system of units ¯*h* = *m* = 1) time-independent Schrödinger equation

$$H\_{\rm osc}^{\bf B} \Psi\_{\rm osc}^{\bf B}(y) = E\_{\rm osc}^{\bf B} \Psi\_{\rm osc}^{\bf B}(y),\tag{50}$$

with

10 Advances in Quantum Mechanics

as:

solution *<sup>R</sup>*(0)

eigenvalue equations

by

*E*(*n*) = ℓ +

*<sup>a</sup>*−*ψ*(0)

*exp*(*r*<sup>2</sup>

*<sup>χ</sup>*(0) 1,+(*r*)*y*<sup>+</sup> 0

Retaining only the non-singular and normalizable *<sup>R</sup>*(0)

*<sup>E</sup>*(*m*) <sup>−</sup> = ℓ +

radial oscillator system by the Hermitian replacement of −*i*

the Wigner deformation parameter ℓ + <sup>1</sup> → ℓ*<sup>D</sup>* + <sup>1</sup>

D-dimensional oscillator angular momentum.

in the literature. (See for example, [16].)

eigenfunction of the ground state becomes

*<sup>ψ</sup>*(0) w,<sup>+</sup> = *exp*(*r*<sup>2</sup>

3

w,<sup>+</sup> <sup>=</sup> 0, (�*<sup>σ</sup>* ·�

which, after operation on the fermion spinors and the spin-angular part, turns into

<sup>2</sup> )*r*−ℓ−1*χ*(0)

<sup>2</sup> )*r*−ℓ+1*χ*(0)

 , *<sup>χ</sup>*(0)

*<sup>H</sup>*−(ℓ) | *<sup>m</sup>*, ℓ >= *<sup>E</sup>*(*m*)

3

where 0 < *r* < ∞. For the radial oscillator the energy eigenvectors satisfy the following

where the eigenvalues are exactly constructed via WH algebra ladder relations and are given

We stress that similar results can be adequately extended for any physical D-dimensional

**4. The constrained Super Wigner Oscillator in** 4*D* **and the hydrogen atom** In this section, the complete spectrum for the hydrogen atom is found with considerable simplicity. Indeed, the solutions of the time-independent Schrödinger equation for the hydrogen atom were mapped onto the super Wigner harmonic oscillator in 4*D* by using the Kustaanheimo-Stiefel transformation. The Kustaanheimo-Stiefel mapping yields the Schrödinger equation for the hydrogen atom that has been exactly solved and well-studied

1,+(*r*)

 = *c*<sup>1</sup> *c*2 

2,+(*r*)

2,+(*r*), which is physically non-existing, as identically zero. Hence the Wigner's

1,+(*r*) <sup>∝</sup> *<sup>r</sup>*ℓ+1*exp*(−*r*<sup>2</sup>

The ground state energy eigenvalue is determined by the annihilation condition which reads

<sup>2</sup> <sup>+</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, 2, . . . . (44)

*L* + **1**) → ℓ + 1; (45)

1,+(*r*), we simply take the singular

<sup>−</sup> | *m*, ℓ >, (48)

<sup>2</sup> (*<sup>D</sup>* <sup>−</sup> <sup>1</sup>) where <sup>ℓ</sup>*D*(ℓ*<sup>D</sup>* <sup>=</sup> 0, 1, 2, ···) is the

<sup>2</sup> <sup>+</sup> <sup>2</sup>*m*, *<sup>m</sup>* <sup>=</sup> 0, 1, 2, . . . . (49)

 *d dr* <sup>+</sup> <sup>1</sup> *r* → −*i d dr* <sup>+</sup> *<sup>D</sup>*−<sup>1</sup> 2*r* and

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

(46)

$$H\_{\rm osc}^{\rm B} = -\frac{1}{2}\nabla\_4^2 + \frac{1}{2}s^2, \quad s^2 = \Sigma\_{i=1}^4 y\_{i\prime}^2 \tag{51}$$

$$
\nabla\_4^2 = \frac{\partial^2}{\partial y\_1^2} + \frac{\partial^2}{\partial y\_2^2} + \frac{\partial^2}{\partial y\_3^2} + \frac{\partial^2}{\partial y\_4^2} = \sum\_{i=1}^4 \frac{\partial^2}{\partial y\_i^2} \tag{52}
$$

where the superscript *B* in *H<sup>B</sup>* osc is in anticipation of the Hamiltonian, with constraint to be defined, being implemented in the bosonic sector of the super 4*D* Wigner system with unitary frequency. Changing to spherical coordinates in 4-space dimensions, allowing a factorization of the energy eigenfunctions as a product of a radial eigenfunction and spin-spherical harmonic.

In (52), the coordinates *yi*(*i* = 1, 2, 3, 4) in spherical coordinates in 4D are defined by [26, 29]

$$\begin{aligned} y\_1 &= s \cos\left(\frac{\theta}{2}\right) \cos\left(\frac{\varphi - \omega}{2}\right) \\ y\_2 &= s \cos\left(\frac{\theta}{2}\right) \sin\left(\frac{\varphi - \omega}{2}\right) \\ y\_3 &= s \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\varphi + \omega}{2}\right) \\ y\_4 &= s \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{\varphi + \omega}{2}\right) \end{aligned} \tag{53}$$

where 0 ≤ *θ* ≤ *π*, 0 ≤ *ϕ* ≤ 2*π* and 0 ≤ *ω* ≤ 4*π*.

The mapping of the coordinates *yi*(*i* = 1, 2, 3, 4) in 4*D* with the Cartesian coordinates *ρi*(*i* = 1, 2, 3) in 3*D* is given by the Kustaanheimo-Stiefel transformation

$$\rho\_{\dot{i}} = \sum\_{a,b=1}^{2} z\_a^\* \Gamma\_{ab}^i z\_{b\prime} \quad \text{ ( $i = 1, 2, 3$ )}\tag{54}$$

*H*B *osc* <sup>=</sup> <sup>−</sup><sup>1</sup> 2

imposed on *H*B

*osc*, becomes

*H*B *osc* <sup>=</sup> <sup>−</sup><sup>1</sup> 2

*H*B

spin <sup>1</sup>

Now, associating *H*B

− 2 *s*2 1 sin *θ*

dimensional. Note that *ψ*B

 *∂*<sup>2</sup> *<sup>∂</sup>s*<sup>2</sup> <sup>+</sup>

3 *s ∂ ∂s* 

*∂θ* <sup>+</sup>

1 *sin*2*θ*

*∂ ∂ω* <sup>Ψ</sup><sup>B</sup>

*∂*2 *∂ϕ*<sup>2</sup> <sup>+</sup>

We obtain a constraint by projection (or "dimensional reduction") from four to three

1 *sin*2*θ*

*osc* is independent of *ω* provides the constraint condition

*osc*, the expression for this restricted Hamiltonian, which we continue to call as

*∂θ* <sup>+</sup>

1 sin2 *θ*

*∂*2 *∂ϕ*<sup>2</sup> + 1 2 *s*

*L* + 1), (62)

2 

*∂ ∂θ* sin *<sup>θ</sup> <sup>∂</sup>*

Identifying the expression in bracket in (61) with *L*2, the square of the orbital angular

*<sup>L</sup>*)(*<sup>σ</sup>* ·

*osc* the final expression

*<sup>L</sup>*)(*<sup>σ</sup>* ·

*<sup>H</sup>*wΨw(*s*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) = *<sup>E</sup>*wΨw(*s*, *<sup>θ</sup>*, *<sup>ϕ</sup>*), (64)

*osc* with the bosonic sector of the super Wigner system, *H*w, subject to the

*L* + 1) + *s*

which is valid for any system, where *σi*(*i* = 1, 2, 3) are the Pauli matrices representing the

same constraint as in (60), and following the analogy with the Section II of construction of

*<sup>L</sup>*<sup>2</sup> = (*<sup>σ</sup>* ·

3 *s ∂ ∂s* + 4 *<sup>s</sup>*<sup>2</sup> (*<sup>σ</sup>* ·

super Wigner systems, we first must solve the Schrödinger equation

 <sup>2</sup>*cos<sup>θ</sup> <sup>∂</sup> ∂ϕ* <sup>+</sup>

*∂ ∂ω <sup>∂</sup> ∂ω* + 1 2 *s* 2. (59)

The Wigner-Heisenberg Algebra in Quantum Mechanics

*osc*(*s*, *θ*, *ϕ*) = 0, (60)

*∂ ∂θ* sin *<sup>θ</sup> <sup>∂</sup>*

 *∂*<sup>2</sup> *<sup>∂</sup>s*<sup>2</sup> <sup>+</sup>

momentum operator in 3*D*, since we always have

<sup>2</sup> degrees of freedom, we obtain for *<sup>H</sup>*<sup>B</sup>

*H*B *osc* <sup>=</sup> <sup>1</sup> 2 − *∂*<sup>2</sup> *<sup>∂</sup>s*<sup>2</sup> <sup>+</sup>

where the explicit form of *H*w is given by

3 *s ∂ ∂s* − 2 *s*2 1 sin *θ* 10.5772/55994

489

http://dx.doi.org/10.5772/55994

2. (61)

. (63)

$$z\_1 = y\_1 + iy\_{2'} \quad z\_2 = y\_3 + iy\_{4'} \tag{55}$$

where the Γ*<sup>i</sup> ab* are the elements of the usual Pauli matrices. If one defines *z*<sup>1</sup> and *z*<sup>2</sup> as in Eq. (55), *Z* = *z*<sup>1</sup> *z*2 is a two dimensional spinor of *SU*(2) transforming as *<sup>Z</sup>* → *<sup>Z</sup>*′ = *UZ* with *U* a two-by-two matrix of *SU*(2) and of course *Z*†*Z* is invariant. So the transformation (54) is very spinorial. Also, using the standard Euler angles parametrizing *SU*(2) as in transformations (53) and (55) one obtains

$$z\_1 = s \cos\left(\frac{\theta}{2}\right) e^{\frac{i}{2}(\varphi - \omega)}$$

$$z\_2 = s \sin\left(\frac{\theta}{2}\right) e^{\frac{i}{2}(\varphi + \omega)}.\tag{56}$$

Note that the angles in these equations are divided by two. However, in 3D, the angles are not divided by two, viz., *ρ*<sup>3</sup> = *ρcos*2( *<sup>θ</sup>* <sup>2</sup> ) <sup>−</sup> *<sup>ρ</sup>sin*2( *<sup>θ</sup>* <sup>2</sup> ) = *ρcosθ*. Indeed, from (54) and (56), we obtain

$$
\rho\_1 = \rho \sin \theta \cos \varphi, \quad \rho\_2 = \rho \sin \theta \sin \varphi, \quad \rho\_3 = \rho \cos \theta \tag{57}
$$

and also that

$$\rho = \left\{\rho\_1^2 + \rho\_2^2 + \rho\_3^2\right\}^{\frac{1}{2}} = \left\{ (\rho\_1 + i\rho\_2)(\rho\_1 - i\rho\_2) + \rho\_3^2 \right\}^{\frac{1}{2}}$$

$$= \left\{ (2z\_1^\*z\_2)(2z\_1z\_2^\*) + (z\_1^\*z\_1 - z\_2^\*z\_2)^2 \right\}^{\frac{1}{2}}$$

$$= (z\_1z\_1^\* + z\_2z\_2^\*) = \sum\_{i=1}^4 y\_i^2 = s^2. \tag{58}$$

The complex form of the Kustaanheimo-Stiefel transformation was given by Cornish [27]. Thus, the expression for *H*B *osc* in (51) can be written in the form

$$\begin{split} H\_{\mathrm{osc}}^{\mathrm{B}} &= -\frac{1}{2} \left( \frac{\partial^2}{\partial s^2} + \frac{3}{s} \frac{\partial}{\partial s} \right) \\ &- \frac{2}{s^2} \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial\theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2} + \frac{1}{\sin^2\theta} \left( 2\cos\theta \frac{\partial}{\partial\varphi} + \frac{\partial}{\partial\omega} \right) \frac{\partial}{\partial\omega} \right] + \frac{1}{2} s^2. \end{split} \tag{59}$$

We obtain a constraint by projection (or "dimensional reduction") from four to three dimensional. Note that *ψ*B *osc* is independent of *ω* provides the constraint condition

$$
\frac{
\partial
}{
\partial\omega
}\Psi\_{osc}^{\mathbf{B}}(s,\theta,\varphi) = 0,\tag{60}
$$

imposed on *H*B *osc*, the expression for this restricted Hamiltonian, which we continue to call as *H*B *osc*, becomes

$$H\_{\rm osc}^{\bf B} = -\frac{1}{2} \left( \frac{\partial^2}{\partial s^2} + \frac{3}{s} \frac{\partial}{\partial s} \right) - \frac{2}{s^2} \left[ \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \sin \theta \frac{\partial}{\partial \theta} + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] + \frac{1}{2} s^2. \tag{61}$$

Identifying the expression in bracket in (61) with *L*2, the square of the orbital angular momentum operator in 3*D*, since we always have

$$L^2 = (\vec{\sigma} \cdot \vec{L})(\vec{\sigma} \cdot \vec{L} + 1),\tag{62}$$

which is valid for any system, where *σi*(*i* = 1, 2, 3) are the Pauli matrices representing the spin <sup>1</sup> <sup>2</sup> degrees of freedom, we obtain for *<sup>H</sup>*<sup>B</sup> *osc* the final expression

$$H\_{\rm osc}^{\rm B} = \frac{1}{2} \left[ -\left( \frac{\partial^2}{\partial s^2} + \frac{3}{s} \frac{\partial}{\partial s} \right) + \frac{4}{s^2} (\vec{\sigma} \cdot \vec{L})(\vec{\sigma} \cdot \vec{L} + 1) + s^2 \right]. \tag{63}$$

Now, associating *H*B *osc* with the bosonic sector of the super Wigner system, *H*w, subject to the same constraint as in (60), and following the analogy with the Section II of construction of super Wigner systems, we first must solve the Schrödinger equation

$$H\_{\mathbf{W}}\Psi\_{\mathbf{W}}(\mathbf{s},\boldsymbol{\theta},\boldsymbol{\varphi}) = E\_{\mathbf{W}}\Psi\_{\mathbf{W}}(\mathbf{s},\boldsymbol{\theta},\boldsymbol{\varphi}),\tag{64}$$

where the explicit form of *H*w is given by

12 Advances in Quantum Mechanics

where the Γ*<sup>i</sup>*

Eq. (55), *Z* =

obtain

and also that

 *z*<sup>1</sup> *z*2 

transformations (53) and (55) one obtains

not divided by two, viz., *ρ*<sup>3</sup> = *ρcos*2( *<sup>θ</sup>*

*ρ* = *ρ*2 <sup>1</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup>

> = (2*z*<sup>∗</sup>

Thus, the expression for *H*B

= (*z*1*z*<sup>∗</sup>

*ρ<sup>i</sup>* =

2 ∑ *a*,*b*=1

*z*∗ *a*Γ*i*

*abzb*, (*i* = 1, 2, 3) (54)

. (56)

<sup>2</sup> ) = *ρcosθ*. Indeed, from (54) and (56), we

3 1 2

2. (58)

*z*<sup>1</sup> = *y*<sup>1</sup> + *iy*2, *z*<sup>2</sup> = *y*<sup>3</sup> + *iy*4, (55)

*ab* are the elements of the usual Pauli matrices. If one defines *z*<sup>1</sup> and *z*<sup>2</sup> as in

with *U* a two-by-two matrix of *SU*(2) and of course *Z*†*Z* is invariant. So the transformation (54) is very spinorial. Also, using the standard Euler angles parametrizing *SU*(2) as in

> *θ* 2 *e i* <sup>2</sup> (*ϕ*−*ω*)

 *θ* 2 *e i* <sup>2</sup> (*ϕ*+*ω*)

Note that the angles in these equations are divided by two. However, in 3D, the angles are

<sup>2</sup> ) <sup>−</sup> *<sup>ρ</sup>sin*2( *<sup>θ</sup>*

*z*<sup>1</sup> = *s* cos

*z*<sup>2</sup> = *s* sin

<sup>2</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup> 3 1 2 = 

<sup>1</sup> *<sup>z</sup>*2)(2*z*1*z*<sup>∗</sup>

<sup>2</sup> ) =

<sup>1</sup> <sup>+</sup> *<sup>z</sup>*2*z*<sup>∗</sup>

<sup>2</sup> )+(*z*<sup>∗</sup>

4 ∑ *i*=1 *y*2 *<sup>i</sup>* = *s*

The complex form of the Kustaanheimo-Stiefel transformation was given by Cornish [27].

*osc* in (51) can be written in the form

<sup>1</sup> *<sup>z</sup>*<sup>1</sup> <sup>−</sup> *<sup>z</sup>*<sup>∗</sup>

is a two dimensional spinor of *SU*(2) transforming as *<sup>Z</sup>* → *<sup>Z</sup>*′ = *UZ*

*ρ*<sup>1</sup> = *ρ* sin *θ* cos *ϕ*, *ρ*<sup>2</sup> = *ρ* sin *θ* sin *ϕ*, *ρ*<sup>3</sup> = *ρ* cos *θ* (57)

(*ρ*<sup>1</sup> + *<sup>i</sup>ρ*2)(*ρ*<sup>1</sup> − *<sup>i</sup>ρ*2) + *<sup>ρ</sup>*<sup>2</sup>

<sup>2</sup> *<sup>z</sup>*2)<sup>2</sup> 1 2

$$\begin{pmatrix} H\_{\mathbf{W}}(2\vec{r}\cdot\vec{L}+\frac{3}{2}) = \\\\ \begin{pmatrix} -\frac{1}{2}(\frac{\partial}{\partial s}+\frac{3}{2s})^2+\frac{1}{2}s^2+\frac{(2\vec{r}\cdot\vec{L}+\frac{1}{2})(2\vec{r}\cdot\vec{L}+\frac{3}{2})}{2s^2} & 0\\ 0 & -\frac{1}{2}(\frac{\partial}{\partial s}+\frac{3}{2s})^2+\frac{1}{2}s^2+\frac{(2\vec{r}\cdot\vec{L}+\frac{3}{2})(2\vec{r}\cdot\vec{L}+\frac{5}{2})}{2s^2} \end{pmatrix}. \text{(65)}$$

<sup>2</sup> ) while

491

<sup>2</sup> ) and <sup>−</sup>(2<sup>ℓ</sup> <sup>+</sup> <sup>5</sup>

http://dx.doi.org/10.5772/55994

The Wigner-Heisenberg Algebra in Quantum Mechanics

*y*±(*θ*, *ϕ*) (71)

(*θ*, *ϕ*) (72)

<sup>2</sup> )*y*<sup>+</sup> and (2�*<sup>σ</sup>* ·�

Σ1. (73)

= 0. (74)

*L* +

2 )

osc

commutes with the basic elements *<sup>a</sup>*±, <sup>Σ</sup><sup>3</sup> and *<sup>H</sup>*<sup>w</sup> of the WH

<sup>2</sup> )*y*<sup>+</sup> = (2<sup>ℓ</sup> <sup>+</sup> <sup>3</sup>

<sup>2</sup> ) acquiring respectively the forms *<sup>H</sup>*w(2<sup>ℓ</sup> <sup>+</sup> <sup>3</sup>

Since the operator

so that, we obtain: (�*<sup>σ</sup>* ·�

<sup>2</sup> )*y*<sup>−</sup> <sup>=</sup> <sup>−</sup>[2(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>) + <sup>1</sup>

a formal 1D radial form with *<sup>H</sup>*w(2�*<sup>σ</sup>* ·�

*H*w 

the complete energy spectrum and eigenfunctions given by

 *EB osc*(*m*)

3

and

Ψ(*n*)

by *a*+(2ℓ + <sup>3</sup>

fermion number operator *N* = <sup>1</sup>

<sup>2</sup>�*<sup>σ</sup>* ·� *L* + <sup>3</sup> 2 

acting on the respective eigenspace in the from

<sup>Ψ</sup>osc(*s*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) =

algebra (67), (68) and (69) it can be replaced by its eigenvalues (2ℓ + <sup>3</sup>

ΨB

ΨF

in the notation where *y*±(*θ*, *ϕ*) are the spin-spherical harmonics [43],

*y*+(*θ*, *ϕ*) = *y*ℓ <sup>1</sup>

*y*−(*θ*, *ϕ*) = *y*ℓ+<sup>1</sup> <sup>1</sup>

<sup>−</sup>2<sup>ℓ</sup> <sup>−</sup> <sup>5</sup> 2 

*<sup>a</sup>*−(2<sup>ℓ</sup> <sup>+</sup>

3 2 ) *RB*(0) osc(*s*) *RF*(0) osc(*s*)

*<sup>L</sup>* <sup>+</sup> <sup>1</sup>)*y*<sup>±</sup> <sup>=</sup> <sup>±</sup>(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)*y*±, (2�*<sup>σ</sup>* ·�

*L* + <sup>3</sup>

osc(*s*, *θ*, *ϕ*)

 = *R*B osc(*s*) *R*F osc(*s*)

osc(*s*, *θ*, *ϕ*)

<sup>2</sup> ;*j*=ℓ<sup>+</sup> <sup>1</sup> <sup>2</sup> ,*mj*

<sup>=</sup> <sup>Σ</sup>1*H*<sup>W</sup>

Thus, the positive finite form of *H*w in (67) together with the ladder relations (68) and the form (69) leads to the direct determination of the state energies and the corresponding Wigner ground state wave functions by the simple application of the annihilation conditions

Then, the complete energy spectrum for *H*w and the whole set of energy eigenfunctions

with fermion number *nf* = 0 and even orbital angular momentum ℓ<sup>4</sup> = <sup>2</sup>ℓ,(ℓ = 0, 1, 2, . . .),

osc(*s*, *<sup>θ</sup>*, *<sup>ϕ</sup>*)(*<sup>n</sup>* <sup>=</sup> <sup>2</sup>*m*, 2*<sup>m</sup>* <sup>+</sup> 1, *<sup>m</sup>* <sup>=</sup> 0, 1, 2, ···) follows from the step up operation provided

<sup>2</sup> ) acting on the ground state, which are also simultaneous eigenfunctions of the

<sup>2</sup> (**<sup>1</sup>** <sup>−</sup> <sup>Σ</sup>3). We obtain for the bosonic sector Hamiltonian *<sup>H</sup><sup>B</sup>*

<sup>ℓ</sup>4=2<sup>ℓ</sup> <sup>=</sup> <sup>2</sup><sup>ℓ</sup> <sup>+</sup> <sup>2</sup> <sup>+</sup> <sup>2</sup>*m*, (*<sup>m</sup>* <sup>=</sup> 0, 1, 2, . . .), (75)

 2ℓ + 3 2 

<sup>2</sup> ;*j*=(ℓ+1)<sup>−</sup> <sup>1</sup>

(*θ*, *ϕ*)

<sup>2</sup> ,*mj*

*L* + <sup>3</sup>

<sup>2</sup> ]*y*−. Note that on these subspaces the 3D WH algebra is reduced to

Using the operator technique in references [9, 10], we begin with the following super-realized mutually adjoint operators

$$a\_{\mathbf{W}}^{\pm} \equiv a^{\pm}(2\vec{\sigma} \cdot \vec{L} + \frac{3}{2}) = \frac{1}{\sqrt{2}} \left[ \pm \left( \frac{\partial}{\partial \mathbf{s}} + \frac{3}{2 \mathbf{s}} \right) \Sigma\_1 \mp \frac{1}{\mathbf{s}} (2\vec{\sigma} \cdot \vec{L} + \frac{3}{2}) \Sigma\_1 \Sigma\_3 - \Sigma\_1 \mathbf{s} \right],\tag{66}$$

where Σ*i*(*i* = 1, 2, 3) constitute a set of Pauli matrices that provide the fermionic coordinates commuting with the similar Pauli set *σi*(*i* = 1, 2, 3) already introduced representing the spin 1 <sup>2</sup> degrees of freedom.

It is checked, after some algebra, that *<sup>a</sup>*<sup>+</sup> and *<sup>a</sup>*<sup>−</sup> of (66) are indeed the raising and lowering operators for the spectra of the super Wigner Hamiltonian *H*w and they satisfy the following (anti-)commutation relations of the WH algebra:

$$\begin{aligned} H\_{\mathbf{W}} &= \frac{1}{2} [a\_{\mathbf{W}'}^- a\_{\mathbf{W}}^+]\_+ \\ &= a\_{\mathbf{W}}^+ a\_{\mathbf{W}}^- + \frac{1}{2} \left[ 1 + 2(2\vec{\sigma} \cdot \vec{L} + \frac{3}{2}) \Sigma\_3 \right] \\ &= a\_{\mathbf{W}}^- a\_{\mathbf{W}}^+ - \frac{1}{2} \left[ 1 + 2(2\vec{\sigma} \cdot \vec{L} + \frac{3}{2}) \Sigma\_3 \right] \end{aligned} \tag{67}$$

$$[H\_{\mathbf{W}}, a\_{\mathbf{W}}^{\pm}]\_{-}=\pm a\_{\mathbf{W}}^{\pm} \tag{68}$$

$$[a\_{\mathbf{W'}}^{-}a\_{\mathbf{W}}^{+}]\_{-}=1+2(2\vec{\sigma}\cdot\vec{L}+\frac{3}{2})\Sigma\_{3\prime}\tag{69}$$

$$[\Sigma\_{\mathfrak{B}'} a\_{\mathbf{W}}^{\pm}]\_{+} = 0 \Rightarrow [\Sigma\_{\mathfrak{B}'} H\_{\mathbf{W}}]\_{-} = 0. \tag{70}$$

Since the operator <sup>2</sup>�*<sup>σ</sup>* ·� *L* + <sup>3</sup> 2 commutes with the basic elements *<sup>a</sup>*±, <sup>Σ</sup><sup>3</sup> and *<sup>H</sup>*<sup>w</sup> of the WH algebra (67), (68) and (69) it can be replaced by its eigenvalues (2ℓ + <sup>3</sup> <sup>2</sup> ) and <sup>−</sup>(2<sup>ℓ</sup> <sup>+</sup> <sup>5</sup> <sup>2</sup> ) while acting on the respective eigenspace in the from

$$\Psi\_{\rm OSC}(s,\theta,\varphi) = \begin{pmatrix} \Psi\_{\rm OSC}^{\rm B}(s,\theta,\varphi) \\ \Psi\_{\rm OSC}^{\rm F}(s,\theta,\varphi) \end{pmatrix} = \begin{pmatrix} R\_{\rm OSC}^{\rm B}(s) \\ R\_{\rm OSC}^{\rm F}(s) \end{pmatrix} y\_{\pm}(\theta,\varphi) \tag{71}$$

in the notation where *y*±(*θ*, *ϕ*) are the spin-spherical harmonics [43],

14 Advances in Quantum Mechanics

*<sup>H</sup>*w(<sup>2</sup>*<sup>σ</sup>* ·

 −1 <sup>2</sup> ( *<sup>∂</sup> <sup>∂</sup><sup>s</sup>* <sup>+</sup> <sup>3</sup>

*L* + 3 2 ) =

mutually adjoint operators

<sup>w</sup> <sup>≡</sup> *<sup>a</sup>*±(<sup>2</sup>*<sup>σ</sup>* ·

*a*±

<sup>2</sup> degrees of freedom.

1

<sup>2</sup>*<sup>s</sup>* )<sup>2</sup> <sup>+</sup> <sup>1</sup>

*L* + 3 2

(anti-)commutation relations of the WH algebra:

<sup>2</sup> *<sup>s</sup>*<sup>2</sup> <sup>+</sup> (<sup>2</sup>*<sup>σ</sup>*·

) = <sup>1</sup> √2 � ± � *∂ ∂s* + 3 2*s* � <sup>Σ</sup><sup>1</sup> ∓ 1 *s* (<sup>2</sup>*<sup>σ</sup>* · *L* + 3 2

*<sup>H</sup>*<sup>w</sup> <sup>=</sup> <sup>1</sup> 2 [*a*− <sup>w</sup>, *<sup>a</sup>*<sup>+</sup> w]+

= *a*<sup>+</sup> w*a*<sup>−</sup> w + 1 2 �

<sup>=</sup> *<sup>a</sup>*<sup>−</sup> w*a*<sup>+</sup> <sup>w</sup> <sup>−</sup> <sup>1</sup> 2 �

[*a*− <sup>w</sup>, *<sup>a</sup>*<sup>+</sup>

[Σ3, *<sup>a</sup>*<sup>±</sup>

 *L*+ <sup>1</sup> <sup>2</sup> )(<sup>2</sup>*<sup>σ</sup>*· *L*+ <sup>3</sup> 2 )

0 −<sup>1</sup>

<sup>2</sup>*s*<sup>2</sup> 0

<sup>2</sup> *<sup>s</sup>*<sup>2</sup> <sup>+</sup> (<sup>2</sup>*<sup>σ</sup>*·

 *L*+ <sup>3</sup> <sup>2</sup> )(<sup>2</sup>*<sup>σ</sup>*· *L*+ <sup>5</sup> 2 )

2*s*<sup>2</sup>

)Σ1Σ<sup>3</sup> − <sup>Σ</sup>1*<sup>s</sup>*

w (68)

)Σ3, (69)

�

, (66)

(67)

 .(65)

<sup>2</sup> ( *<sup>∂</sup> <sup>∂</sup><sup>s</sup>* <sup>+</sup> <sup>3</sup> 2*s* ) 2 + <sup>1</sup>

Using the operator technique in references [9, 10], we begin with the following super-realized

where Σ*i*(*i* = 1, 2, 3) constitute a set of Pauli matrices that provide the fermionic coordinates commuting with the similar Pauli set *σi*(*i* = 1, 2, 3) already introduced representing the spin

It is checked, after some algebra, that *<sup>a</sup>*<sup>+</sup> and *<sup>a</sup>*<sup>−</sup> of (66) are indeed the raising and lowering operators for the spectra of the super Wigner Hamiltonian *H*w and they satisfy the following

<sup>1</sup> <sup>+</sup> <sup>2</sup>(<sup>2</sup>*<sup>σ</sup>* ·

<sup>1</sup> <sup>+</sup> <sup>2</sup>(<sup>2</sup>*<sup>σ</sup>* ·

<sup>w</sup>]<sup>−</sup> <sup>=</sup> <sup>±</sup>*a*<sup>±</sup>

[*H*w, *<sup>a</sup>*<sup>±</sup>

<sup>w</sup>]<sup>−</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>(<sup>2</sup>*<sup>σ</sup>* ·

*L* + 3 2 )Σ<sup>3</sup> �

*L* + 3 2 )Σ<sup>3</sup> �

*L* + 3 2

<sup>w</sup>]+ <sup>=</sup> <sup>0</sup> <sup>⇒</sup> [Σ3, *<sup>H</sup>*w]<sup>−</sup> <sup>=</sup> 0. (70)

$$y\_{+}(\theta,\varphi) = y\_{\ell\frac{1}{2};j=\ell+\frac{1}{2},m\_{j}}(\theta,\varphi)$$

$$y\_{-}(\theta,\varphi) = y\_{\ell+1\frac{1}{2};j=(\ell+1)-\frac{1}{2},m\_{j}}(\theta,\varphi) \tag{72}$$

so that, we obtain: (�*<sup>σ</sup>* ·� *<sup>L</sup>* <sup>+</sup> <sup>1</sup>)*y*<sup>±</sup> <sup>=</sup> <sup>±</sup>(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>)*y*±, (2�*<sup>σ</sup>* ·� *L* + <sup>3</sup> <sup>2</sup> )*y*<sup>+</sup> = (2<sup>ℓ</sup> <sup>+</sup> <sup>3</sup> <sup>2</sup> )*y*<sup>+</sup> and (2�*<sup>σ</sup>* ·� *L* + 3 <sup>2</sup> )*y*<sup>−</sup> <sup>=</sup> <sup>−</sup>[2(<sup>ℓ</sup> <sup>+</sup> <sup>1</sup>) + <sup>1</sup> <sup>2</sup> ]*y*−. Note that on these subspaces the 3D WH algebra is reduced to a formal 1D radial form with *<sup>H</sup>*w(2�*<sup>σ</sup>* ·� *L* + <sup>3</sup> <sup>2</sup> ) acquiring respectively the forms *<sup>H</sup>*w(2<sup>ℓ</sup> <sup>+</sup> <sup>3</sup> 2 ) and

$$H\_{\mathbf{W}}\left(-2\ell-\frac{5}{2}\right)=\Sigma\_1 H\_{\mathbf{W}}\left(2\ell+\frac{3}{2}\right)\Sigma\_1.\tag{73}$$

Thus, the positive finite form of *H*w in (67) together with the ladder relations (68) and the form (69) leads to the direct determination of the state energies and the corresponding Wigner ground state wave functions by the simple application of the annihilation conditions

$$a^{-} (2\ell + \frac{3}{2}) \begin{pmatrix} R\_{\rm OSC}^{B^{(0)}}(s) \\ R\_{\rm OSC}^{F^{(0)}}(s) \end{pmatrix} = 0. \tag{74}$$

Then, the complete energy spectrum for *H*w and the whole set of energy eigenfunctions Ψ(*n*) osc(*s*, *<sup>θ</sup>*, *<sup>ϕ</sup>*)(*<sup>n</sup>* <sup>=</sup> <sup>2</sup>*m*, 2*<sup>m</sup>* <sup>+</sup> 1, *<sup>m</sup>* <sup>=</sup> 0, 1, 2, ···) follows from the step up operation provided by *a*+(2ℓ + <sup>3</sup> <sup>2</sup> ) acting on the ground state, which are also simultaneous eigenfunctions of the fermion number operator *N* = <sup>1</sup> <sup>2</sup> (**<sup>1</sup>** <sup>−</sup> <sup>Σ</sup>3). We obtain for the bosonic sector Hamiltonian *<sup>H</sup><sup>B</sup>* osc with fermion number *nf* = 0 and even orbital angular momentum ℓ<sup>4</sup> = <sup>2</sup>ℓ,(ℓ = 0, 1, 2, . . .), the complete energy spectrum and eigenfunctions given by

$$\left[E\_{\rm osc}^{B}\right]\_{\ell=2\ell}^{(m)} = 2\ell + 2 + 2m, \quad (m = 0, 1, 2, \dots), \tag{75}$$

$$\left[\Psi\_{\rm osc}^{B}(s,\theta,\varphi)\right]\_{\ell=2\ell}^{(m)} \propto s^{2\ell} \exp\left(-\frac{1}{2}s^{2}\right) L\_{m}^{(2\ell+1)}(s^{2}) \begin{cases} y\_{+}(\theta,\varphi) \\ y\_{-}(\theta,\varphi) \end{cases} \tag{76}$$

where *L<sup>α</sup> <sup>m</sup>*(*s*2) are generalized Laguerre polynomials [9]. Now, to relate the mapping of the 4D super Wigner system with the corresponding system in 3D, we make use of the substitution of *s*<sup>2</sup> = *ρ*, Eq. (60) and the following substitutions

$$\frac{\partial}{\partial s} = 2\sqrt{\rho}\frac{\partial}{\partial \rho'} \quad \frac{\partial^2}{\partial s^2} = 4\rho \frac{\partial^2}{\partial \rho^2} + 2\frac{\partial}{\partial \rho'} \tag{77}$$

and

where *E<sup>B</sup>*

representation

Ψ†Ψ = 1.

where *B* =

Ψ†, Ψ  [*ψ*(*ρ*, *θ*, *ϕ*)]

*osc* is given by Eq. (75).

may be realized in atomic systems [44].

superconformal quantum mechanics [39–42, 46].

(*m*)

<sup>ℓ</sup>;æ,*mj* <sup>∝</sup> *<sup>ρ</sup>*<sup>ℓ</sup> exp (−*<sup>ρ</sup>*

**5. The superconformal quantum mechanics from WH algebra**

2

Here, *N* = ℓ + *m* + 1 (ℓ = 0, 1, 2, ··· , *N* − 1; *m* = 0, 1, 2, ···) is the principal quantum number. Kostelecky and Nieto shown that the supersymmetry in non-relativistic quantum mechanics

The superconformal quantum mechanics has been examined in [35]. Another application for these models is in the study of the radial motion of test particle near the horizon of extremal Reissner-Nordström black holes [35, 37]. Also, another interesting application of the superconformal symmetry is the treatment of the Dirac oscillator, in the context of the

In this section we introduce the explicit supersymmetry for the conformal Hamiltonian in the WH-algebra picture. Let us consider the supersymmetric generalization of *H*, given by

{*Qc*, *Q*†

√*g x* Ψ†,

√*g x* 

**<sup>1</sup>***<sup>g</sup>* <sup>+</sup> <sup>√</sup>*gB*(<sup>1</sup> <sup>−</sup> *<sup>c</sup>***P**)

<sup>−</sup> , so that the parity operator is conserved, i.e., [H, **<sup>P</sup>**]<sup>−</sup> = 0.

where the new supercharge operators are given in terms of the momentum Yang

<sup>H</sup> <sup>=</sup> <sup>1</sup> 2

> *ipx* +

with Ψ and Ψ† being Grassmannian operators so that its anticommutator is {Ψ, Ψ†} = ΨΨ† +

*Qc* = −*ipx* +

*Q*† *<sup>c</sup>* = Ψ

<sup>H</sup> <sup>=</sup> <sup>1</sup> 2 (**1***p*<sup>2</sup> *x* +

Explicitly the superconformal Hamiltonian becomes

When one introduces the following operators

)*L*(2ℓ+1) *<sup>m</sup>* (*ρ*)

*y*+(*θ*, *ϕ*)

The Wigner-Heisenberg Algebra in Quantum Mechanics

10.5772/55994

493

*<sup>y</sup>*−(*θ*, *<sup>ϕ</sup>*) (82)

http://dx.doi.org/10.5772/55994

*<sup>c</sup>* }, (83)

, (84)

*<sup>x</sup>*<sup>2</sup> ) (85)

in (65) and divide the eigenvalue equation for *H*w in (64) by 4*s*<sup>2</sup> = 4*ρ*, obtaining

$$
\begin{pmatrix}
0 & -\frac{1}{2}\left(\frac{\partial^2}{\partial\rho^2} + \frac{2}{\rho}\frac{\partial}{\partial\rho}\right) - \frac{1}{2}\left[-\frac{1}{4} - \frac{(\vec{\sigma}\cdot\vec{L}+\frac{1}{2})(\vec{\sigma}\cdot\vec{L}+\frac{3}{2})}{\rho^2}\right] \\
0 = \frac{1}{4\rho}E\_{\rm W}\left(\frac{\mathbf{\Psi}^B}{\mathbf{\Psi}^F}\right).
\end{pmatrix}
\tag{78}
$$

The bosonic sector of the above eigenvalue equation can immediately be identified with the eigenvalue equation for the Hamiltonian of the 3D Hydrogen-like atom expressed in the equivalent form given by

$$\left\{-\frac{1}{2}\left(\frac{\partial^2}{\partial\rho^2} + \frac{2}{\rho}\frac{\partial}{\partial\rho}\right) - \frac{1}{2}\left[-\frac{1}{4} - \frac{\vec{\sigma}\cdot\vec{L}(\vec{\sigma}\cdot\vec{L}+1)}{\rho^2}\right]\right\}\psi(\rho,\theta,\varphi) = \frac{\ell}{2\rho}\psi(\rho,\theta,\varphi),\tag{79}$$

where Ψ*<sup>B</sup>* = *ψ*(*ρ*, *θ*, *ϕ*) and the connection between the dimensionless and dimensionfull eigenvalues, respectively, ℓ and *Ea* with *e* = 1 = *m* = *h*¯ is given by [43]

$$\ell = \frac{Z}{\sqrt{-2E\_a}}, \quad \rho = ar, \quad a = \sqrt{-8E\_a} \tag{80}$$

where *Ea* is the energy of the electron Hydrogen-like atom, (*r*, *θ*, *ϕ*) stand for the spherical polar coordinates of the position vector �*<sup>r</sup>* = (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) of the electron in relative to the nucleons of charge *Z* together with *s*<sup>2</sup> = *ρ*. We see then from equations (75), (76), (79) and (80) that the complete energy spectrum and eigenfunctions for the Hydrogen-like atom given by

$$\frac{\lambda}{2} = \frac{E\_{\rm osc}^{B}}{4} \Rightarrow \left[\mathbf{E}\_{\rm a}\right]\_{\ell}^{(m)} = \left[\mathbf{E}\_{\rm a}\right]^{(N)} = -\frac{Z^{2}}{2N^{2}}, \quad (N = 1, 2, \ldots) \tag{81}$$

and

16 Advances in Quantum Mechanics

where *L<sup>α</sup>*

−1 2 � *∂*<sup>2</sup> *∂ρ*<sup>2</sup> <sup>+</sup> <sup>2</sup> *ρ ∂ ∂ρ* � − 1 2 � −1 <sup>4</sup> <sup>−</sup> �*σ*· � *L*(�*σ*· � *L*+1) *ρ*2

<sup>=</sup> <sup>1</sup> 4*ρ E*w � Ψ*<sup>B</sup>* Ψ*F* �

equivalent form given by

� *∂*<sup>2</sup> *∂ρ*<sup>2</sup> <sup>+</sup>

> *λ* <sup>2</sup> <sup>=</sup> *<sup>E</sup><sup>B</sup> osc* <sup>4</sup> <sup>⇒</sup> [*Ea*]

2 *ρ ∂ ∂ρ* � − 1 2 � −1 <sup>4</sup> <sup>−</sup> �*<sup>σ</sup>* ·�

� −1 2

by

� Ψ*B*

*osc*(*s*, *θ*, *ϕ*)

of *s*<sup>2</sup> = *ρ*, Eq. (60) and the following substitutions

*∂ <sup>∂</sup><sup>s</sup>* <sup>=</sup> <sup>2</sup>

0 −<sup>1</sup>

√*ρ ∂ ∂ρ* , *<sup>∂</sup>*<sup>2</sup>

in (65) and divide the eigenvalue equation for *H*w in (64) by 4*s*<sup>2</sup> = 4*ρ*, obtaining

�

2 � *∂*<sup>2</sup> *∂ρ*<sup>2</sup> <sup>+</sup> <sup>2</sup> *ρ ∂ ∂ρ* � − 1 2 � −1 <sup>4</sup> <sup>−</sup> (�*σ*· � *L*+ <sup>1</sup> <sup>2</sup> )(�*σ*· � *L*+ <sup>3</sup> 2 )

The bosonic sector of the above eigenvalue equation can immediately be identified with the eigenvalue equation for the Hamiltonian of the 3D Hydrogen-like atom expressed in the

*<sup>L</sup>*(�*<sup>σ</sup>* ·�

where Ψ*<sup>B</sup>* = *ψ*(*ρ*, *θ*, *ϕ*) and the connection between the dimensionless and dimensionfull

where *Ea* is the energy of the electron Hydrogen-like atom, (*r*, *θ*, *ϕ*) stand for the spherical polar coordinates of the position vector �*<sup>r</sup>* = (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) of the electron in relative to the nucleons of charge *Z* together with *s*<sup>2</sup> = *ρ*. We see then from equations (75), (76), (79) and (80) that the complete energy spectrum and eigenfunctions for the Hydrogen-like atom given

(*N*) <sup>=</sup> <sup>−</sup> *<sup>Z</sup>*<sup>2</sup>

eigenvalues, respectively, ℓ and *Ea* with *e* = 1 = *m* = *h*¯ is given by [43]

(*m*) <sup>ℓ</sup> = [*Ea*]

<sup>ℓ</sup> <sup>=</sup> *<sup>Z</sup>* √−2*Ea* *ρ*2

*L* + 1)

��

*<sup>ψ</sup>*(*ρ*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) = <sup>ℓ</sup>

, *<sup>ρ</sup>* <sup>=</sup> *<sup>α</sup>r*, *<sup>α</sup>* <sup>=</sup> √−8*Ea*, (80)

�(*m*) <sup>ℓ</sup>4=2<sup>ℓ</sup> <sup>∝</sup> *<sup>s</sup>*

<sup>2</sup><sup>ℓ</sup> exp � −1 2 *s* 2 �

*<sup>L</sup>*(2ℓ+1) *<sup>m</sup>* (*<sup>s</sup>*

*<sup>m</sup>*(*s*2) are generalized Laguerre polynomials [9]. Now, to relate the mapping of the 4D

*<sup>∂</sup>s*<sup>2</sup> <sup>=</sup> <sup>4</sup>*<sup>ρ</sup>*

*∂*2 *∂ρ*<sup>2</sup> <sup>+</sup> <sup>2</sup> *<sup>∂</sup>*

. (78)

0

super Wigner system with the corresponding system in 3D, we make use of the substitution

2)

� *y*+(*θ*, *ϕ*)

*<sup>y</sup>*−(*θ*, *<sup>ϕ</sup>*) (76)

*∂ρ* , (77)

*ρ*2

2*ρ*

<sup>2</sup>*N*<sup>2</sup> , (*<sup>N</sup>* <sup>=</sup> 1, 2, . . .) (81)

�

� Ψ*<sup>B</sup>* Ψ*F* �

*ψ*(*ρ*, *θ*, *ϕ*), (79)

$$\left[\psi(\rho,\theta,\varphi)\right]\_{\ell;\mathbf{e},m\_{\parallel}}^{(m)} \propto \rho^{\ell} \exp\left(-\frac{\rho}{2}\right) L\_{m}^{(2\ell+1)}(\rho) \begin{cases} y\_{+}(\theta,\varphi) \\ y\_{-}(\theta,\varphi) \end{cases} \tag{82}$$

where *E<sup>B</sup> osc* is given by Eq. (75).

Here, *N* = ℓ + *m* + 1 (ℓ = 0, 1, 2, ··· , *N* − 1; *m* = 0, 1, 2, ···) is the principal quantum number. Kostelecky and Nieto shown that the supersymmetry in non-relativistic quantum mechanics may be realized in atomic systems [44].

#### **5. The superconformal quantum mechanics from WH algebra**

The superconformal quantum mechanics has been examined in [35]. Another application for these models is in the study of the radial motion of test particle near the horizon of extremal Reissner-Nordström black holes [35, 37]. Also, another interesting application of the superconformal symmetry is the treatment of the Dirac oscillator, in the context of the superconformal quantum mechanics [39–42, 46].

In this section we introduce the explicit supersymmetry for the conformal Hamiltonian in the WH-algebra picture. Let us consider the supersymmetric generalization of *H*, given by

$$\mathcal{H} = \frac{1}{2} \{ \mathbb{Q}\_{\mathcal{C}}, \mathbb{Q}\_{\mathcal{C}}^{\dagger} \}, \tag{83}$$

where the new supercharge operators are given in terms of the momentum Yang representation

$$Q\_c = \left(-ip\_x + \frac{\sqrt{\mathcal{S}}}{\mathfrak{x}}\right) \Psi^\dagger \,,$$

$$Q\_c^\dagger = \Psi \left(ip\_x + \frac{\sqrt{\mathcal{S}}}{\mathfrak{x}}\right) \,. \tag{84}$$

with Ψ and Ψ† being Grassmannian operators so that its anticommutator is {Ψ, Ψ†} = ΨΨ† + Ψ†Ψ = 1.

Explicitly the superconformal Hamiltonian becomes

$$\mathcal{H} = \frac{1}{2} (\mathbf{1}p\_x^2 + \frac{\mathbf{1}g + \sqrt{g}B(1 - c\mathbf{P})}{x^2}) \tag{85}$$

where *B* = Ψ†, Ψ <sup>−</sup> , so that the parity operator is conserved, i.e., [H, **<sup>P</sup>**]<sup>−</sup> = 0. When one introduces the following operators

$$\begin{aligned} \mathbf{S} &= \mathbf{x} \mathbf{Y}^{\dagger}, \\ \mathbf{S}^{\dagger} &= \mathbf{Y} \mathbf{x}, \end{aligned} \tag{86}$$

495

<sup>2</sup> <sup>&</sup>gt; (<sup>|</sup> <sup>2</sup>*m*, *<sup>c</sup>*

<sup>2</sup> <sup>&</sup>gt;)

*<sup>c</sup>* , *S*, *S*† are fermionic ones. The

http://dx.doi.org/10.5772/55994

The Wigner-Heisenberg Algebra in Quantum Mechanics

(H + *<sup>K</sup>*), [H0, **<sup>P</sup>**]<sup>−</sup> = 0. (89)

where, H, *D*, *K*, *B* are bosonic operators and *Qc*, *Q*†

in the black hole geometry.

oscillator-related potentials.

oscillator for spin <sup>1</sup>

**6. Summary and conclusion**

technique developed in this chapter.

<sup>H</sup><sup>0</sup> <sup>=</sup> <sup>1</sup> 2

supersymmetric extension of the Hamiltonian *L*<sup>0</sup> (presented in the previous section) is

In general, superconformal quantum mechanics has interesting applications in supersymmetric black holes, for example in the problem of a quantum test particle moving

In this chapter, firstly we start by summarizing the R-deformed Heisenberg algebra or Wigner-Heisenberg algebraic technique for the Wigner quantum oscillator, based on the super-realization of the ladder operators effective spectral resolutions of general

We illustrate the applications of our operator method to the cases of the Hamiltonians of an isotonic oscillator (harmonic plus a centripetal barrier) system and a 3D isotropic harmonic

Also, the energy eigenvalues and eigenfunctions of the hydrogen atom via Wigner-Heisenberg (WH) algebra in non-relativistic quantum mechanics, from the ladder operators for the 4-dimensional (4*D*) super Wigner system, ladder operators for the mapped super 3*D* system, and hence for hydrogen-like atom in bosonic sector, are deduced. The complete spectrum for the hydrogen atom is found with considerable simplicity by using the Kustaanheimo-Stiefel transformation. From the ladder operators for the four-dimensional (4D) super-Wigner system, ladder operators for the mapped super 3D system, and hence for the hydrogen-like atom in bosonic sector, can be deduced. Results of

For future directions, such a direct algebraic method considered in this chapter proves highly profitable for simpler algebraic treatment, as we shall show in subsequent publications, of other quantum mechanical systems with underlying oscillator connections like for example those of a relativistic electron in a Coulomb potential or of certain 3D SUSY oscillator models of the type of Celka and Hussin. This SUSY model has been reported in nonrelativistic context by Jayaraman and Rodrigues [10]. We will also demonstrate elsewhere the application of our method for a spectral resolution complete of the Pöschl-Teller I and II potentials by virtue of their hidden oscillator connections using the WH algebra operator

In the work of the Ref. [46], we analyze the Wigner-Heisenberg algebra to bosonic systems in connection with oscillators and, thus, we find a new representation for the Virasoro algebra.

the eigenvalue of the ground state of the Wigner oscillator appears only in the excited states associated with the even(odd) quanta. We show that only in the case associated with one

present investigations on these ladder operators will be reported separately.

Acting the annihilation operator(creation operator) in the Fock basis | <sup>2</sup>*<sup>m</sup>* +1, *<sup>c</sup>*

even index and one odd index in the operator *Ln* the Virasoro algebra is changed.

<sup>2</sup> embedded in the bosonic sector of a corresponding Wigner system.

it can be shown that these operators together with the conformal quantum mechanics operators *D* and *K*

$$D = \frac{1}{2} (x p\_{\text{x}} + p\_{\text{x}} x)\_{\text{\textdegree}}$$

$$K = \frac{1}{2} x^2 \text{\textdegree} \tag{87}$$

satisfy the deformed superalgebra *osp*(2|2) (Actually, this superalgebra is *osp*(2|2) when we fix **P** = 1 or **P** = −1.), viz.,

$$\begin{aligned} \left[\mathcal{H},\mathcal{D}\right]\_{-} &= -2i\mathcal{H},\\ \left[\mathcal{H},\mathcal{K}\right]\_{-} &= -iD,\\ \left[\mathcal{K},\mathcal{D}\right]\_{+} &= 2i\mathcal{H},\\ \left[\mathcal{Q},\mathcal{Q}^{\dagger}\right]\_{+} &= 2\mathcal{H},\\ \left[\mathcal{Q},\mathcal{S}^{\dagger}\right]\_{+} &= -iD - \frac{1}{2}B(1+c\mathcal{P}) + \sqrt{\mathcal{S}},\\ \left[\mathcal{Q}^{\dagger}\_{+},\mathcal{S}\right]\_{+} &= iD - \frac{1}{2}B(1+c\mathcal{P}) + \sqrt{\mathcal{S}},\\ \left[\mathcal{Q}^{\dagger}\_{+},\mathcal{D}\right]\_{-} &= -i\mathcal{Q}^{\dagger}\_{c},\\ \left[\mathcal{Q}^{\dagger}\_{+},\mathcal{S}\right]\_{-} &= \mathcal{S}^{\dagger},\\ \left[\mathcal{Q}^{\dagger}\_{-},\mathcal{S}\right]\_{-} &= 2Q^{\dagger},\\ \left[\mathcal{Q}\_{c},\mathcal{S}\right]\_{-} &= -S\_{c},\\ \left[\mathcal{Q}\_{c},\mathcal{B}\right]\_{-} &= -2Q\_{c},\\ \left[\mathcal{Q}\_{c},\mathcal{S}\right]\_{-} &= -iQ\_{c},\\ \left[\mathcal{H},\mathcal{S}^{\dagger}\right]\_{-} &= -2S\_{c},\\ \left[\mathcal{B},\mathcal{S}^{\dagger}\right]\_{-} &= -2S\_{c}^{\dagger}, \qquad \left[\mathcal{B},\mathcal{S}^{\dagger}\right]\_{-} = 2S\_{c},\\ \left[\mathcal{B},\mathcal{S}\right]\_{+} &= -iS\_{c}, \qquad \left[\mathcal{D},\mathcal{S}^{\dagger}\right]\_{-} = -i\mathcal{S}^{\dagger},\\ \left[\mathcal{S}^{$$

where, H, *D*, *K*, *B* are bosonic operators and *Qc*, *Q*† *<sup>c</sup>* , *S*, *S*† are fermionic ones. The supersymmetric extension of the Hamiltonian *L*<sup>0</sup> (presented in the previous section) is

$$
\mathcal{H}\_0 = \frac{1}{2}(\mathcal{H} + K)\_\prime \quad \left[\mathcal{H}\_0, \mathbf{P}\right]\_- = 0. \tag{89}
$$

In general, superconformal quantum mechanics has interesting applications in supersymmetric black holes, for example in the problem of a quantum test particle moving in the black hole geometry.

### **6. Summary and conclusion**

18 Advances in Quantum Mechanics

operators *D* and *K*

fix **P** = 1 or **P** = −1.), viz.,

 *Qc*, *Q*† *c* 

 *Qc*, *S*† 

> *Q*† *<sup>c</sup>* , *S*

 *Q*† *<sup>c</sup>* , *D* 

 *Q*† *<sup>c</sup>* , *K* <sup>−</sup> <sup>=</sup> *<sup>S</sup>*†,

 *Q*† *<sup>c</sup>* , *B* 

 *B*, *S*† 

 *S*†, *S*  *S* = *x*Ψ†,

it can be shown that these operators together with the conformal quantum mechanics

satisfy the deformed superalgebra *osp*(2|2) (Actually, this superalgebra is *osp*(2|2) when we

(*xpx* + *px x*),

*<sup>D</sup>* <sup>=</sup> <sup>1</sup> 2

*<sup>K</sup>* <sup>=</sup> <sup>1</sup> 2

[H, *D*]<sup>−</sup> = −2*ı*H, [H, *K*]<sup>−</sup> = −*ıD*, [*K*, *D*]<sup>−</sup> = 2*ıK*,

<sup>+</sup> <sup>=</sup> <sup>2</sup>H,

<sup>+</sup> <sup>=</sup> <sup>−</sup>*ıD* <sup>−</sup> <sup>1</sup>

<sup>+</sup> <sup>=</sup> *ıD* <sup>−</sup> <sup>1</sup>

<sup>−</sup> <sup>=</sup> <sup>−</sup>*ıQ*†

<sup>−</sup> <sup>=</sup> <sup>2</sup>*Q*† *c* ,

[*Qc*, *K*]<sup>−</sup> = −*S*, [*Qc*, *B*]<sup>−</sup> = −2*Qc*, [*Qc*, *D*]<sup>−</sup> = −*ıQc*, [H, *S*]<sup>−</sup> = *Qc*,

[*D*, *S*]<sup>−</sup> = −*ıS*,

2

2

*c* ,

*<sup>B</sup>*(<sup>1</sup> <sup>+</sup> *<sup>c</sup>***P**) + <sup>√</sup>*g*,

 H, *S*† 

> *D*, *S*†

<sup>−</sup> <sup>=</sup> <sup>−</sup>2*S*†, [*B*, *<sup>S</sup>*]<sup>−</sup> <sup>=</sup> <sup>2</sup>*S*,

<sup>−</sup> <sup>=</sup> <sup>−</sup>*Q*† *c* ,

<sup>−</sup> <sup>=</sup> <sup>−</sup>*ıS*†,

<sup>+</sup> <sup>=</sup> <sup>2</sup>*K*, (88)

*<sup>B</sup>*(<sup>1</sup> <sup>+</sup> *<sup>c</sup>***P**) + <sup>√</sup>*g*,

*S*† = Ψ*x*, (86)

*x*2, (87)

In this chapter, firstly we start by summarizing the R-deformed Heisenberg algebra or Wigner-Heisenberg algebraic technique for the Wigner quantum oscillator, based on the super-realization of the ladder operators effective spectral resolutions of general oscillator-related potentials.

We illustrate the applications of our operator method to the cases of the Hamiltonians of an isotonic oscillator (harmonic plus a centripetal barrier) system and a 3D isotropic harmonic oscillator for spin <sup>1</sup> <sup>2</sup> embedded in the bosonic sector of a corresponding Wigner system.

Also, the energy eigenvalues and eigenfunctions of the hydrogen atom via Wigner-Heisenberg (WH) algebra in non-relativistic quantum mechanics, from the ladder operators for the 4-dimensional (4*D*) super Wigner system, ladder operators for the mapped super 3*D* system, and hence for hydrogen-like atom in bosonic sector, are deduced. The complete spectrum for the hydrogen atom is found with considerable simplicity by using the Kustaanheimo-Stiefel transformation. From the ladder operators for the four-dimensional (4D) super-Wigner system, ladder operators for the mapped super 3D system, and hence for the hydrogen-like atom in bosonic sector, can be deduced. Results of present investigations on these ladder operators will be reported separately.

For future directions, such a direct algebraic method considered in this chapter proves highly profitable for simpler algebraic treatment, as we shall show in subsequent publications, of other quantum mechanical systems with underlying oscillator connections like for example those of a relativistic electron in a Coulomb potential or of certain 3D SUSY oscillator models of the type of Celka and Hussin. This SUSY model has been reported in nonrelativistic context by Jayaraman and Rodrigues [10]. We will also demonstrate elsewhere the application of our method for a spectral resolution complete of the Pöschl-Teller I and II potentials by virtue of their hidden oscillator connections using the WH algebra operator technique developed in this chapter.

In the work of the Ref. [46], we analyze the Wigner-Heisenberg algebra to bosonic systems in connection with oscillators and, thus, we find a new representation for the Virasoro algebra. Acting the annihilation operator(creation operator) in the Fock basis | <sup>2</sup>*<sup>m</sup>* +1, *<sup>c</sup>* <sup>2</sup> <sup>&</sup>gt; (<sup>|</sup> <sup>2</sup>*m*, *<sup>c</sup>* <sup>2</sup> <sup>&</sup>gt;) the eigenvalue of the ground state of the Wigner oscillator appears only in the excited states associated with the even(odd) quanta. We show that only in the case associated with one even index and one odd index in the operator *Ln* the Virasoro algebra is changed.

Recently, we have analyzed the connection between the conformal quantum mechanics and the Wigner-Heisenberg (WH) algebra [46]. With an appropriate relationship between the coupling constant *g* and Wigner parameter *c* one can identify the Wigner Hamiltonian with the simple Calogero Hamiltonian.

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The Wigner-Heisenberg Algebra in Quantum Mechanics

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[12] J. Jayaraman, R. de Lima Rodrigues and A. N. Vaidya, *J. Phys. A: Math. Gen.* 32, 6643

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[18] I. A. Malkin and V. I. ManŠko, *Dynamical Symmetries and Coherent States of Quantum*

[19] J. P. Gazeau and J. Klauder, *Coherent states for systems with discrete and continuous spectrum*

[20] R. C. King, T. D. Palev, N. I. Stoilova and J. Van der Jeugt, *J. Phys. A: Math. Gen.* 36, 4337

Sharma J K, Mehta C L and Sudarshan E C G 1978 *J. Math. Phys.* 19 2089

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*Coherent States and Their Applications* (Berlin Springer-Verlag, 1986).

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125306 (2012).

The important result is that the introduction of the WH algebra in the conformal quantum mechanics is still consistent with the conformal symmetry, and a realization of superconformal quantum mechanics in terms of deformed WH algebra is discussed. The spectra for the Casimir operator and the Hamiltonian *L*<sup>0</sup> depend on the parity operator. The ladder operators depend on the parity operator, too. It is shown, for example, that the eigenvalues of Calogero-type Hamiltonian is dependent of the Wigner parameter *c* and the eigenvalues of the parity operator *P*. When *c*=0 we obtain the usual conformal Hamiltonian structure.

We also investigated the supersymmetrization of this model, in that case we obtain a new spectrum for the supersymmetric Hamiltonian of the Calogero interaction's type.

In this case the spectra for the super-Casimir operator and the superhamiltonian depend also on the parity operator. Therefore, we have found a new realization of supersymmetric Calogero-type model on the quantum mechanics context in terms of deformed WH algebra.

### **Author details**

Rafael de Lima Rodrigues

UFCG-Campus Cuité-PB, Brazil

### **References**


20 Advances in Quantum Mechanics

structure.

**Author details**

**References**

Rafael de Lima Rodrigues

Berlin (1996).

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UFCG-Campus Cuité-PB, Brazil

the simple Calogero Hamiltonian.

Recently, we have analyzed the connection between the conformal quantum mechanics and the Wigner-Heisenberg (WH) algebra [46]. With an appropriate relationship between the coupling constant *g* and Wigner parameter *c* one can identify the Wigner Hamiltonian with

The important result is that the introduction of the WH algebra in the conformal quantum mechanics is still consistent with the conformal symmetry, and a realization of superconformal quantum mechanics in terms of deformed WH algebra is discussed. The spectra for the Casimir operator and the Hamiltonian *L*<sup>0</sup> depend on the parity operator. The ladder operators depend on the parity operator, too. It is shown, for example, that the eigenvalues of Calogero-type Hamiltonian is dependent of the Wigner parameter *c* and the eigenvalues of the parity operator *P*. When *c*=0 we obtain the usual conformal Hamiltonian

We also investigated the supersymmetrization of this model, in that case we obtain a new

In this case the spectra for the super-Casimir operator and the superhamiltonian depend also on the parity operator. Therefore, we have found a new realization of supersymmetric Calogero-type model on the quantum mechanics context in terms of deformed WH algebra.

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spectrum for the supersymmetric Hamiltonian of the Calogero interaction's type.


**Chapter 21**

**Provisional chapter**

**New System-Specific Coherent States by**

**New System-Specific Coherent States by**

Chia-Chun Chou, Mason T. Biamonte,

Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri

Bernhard G. Bodmann and Donald J. Kouri

Additional information is available at the end of the chapter

using the standard Rayleigh-Ritz variational method ([5, 19, 20]).

Additional information is available at the end of the chapter

**Calculations**

**State Calculations**

http://dx.doi.org/10.5772/54010

10.5772/54010

**1. Introduction**

**Supersymmetric Quantum Mechanics for Bound State**

Supersymmetric quantum mechanics (SUSY-QM) has been developed as an elegant analytical approach to one-dimensional problems. The SUSY-QM formalism generalizes the ladder operator approach used in the treatment of the harmonic oscillator. In analogy with the harmonic oscillator Hamiltonian, the factorization of a one-dimensional Hamiltonian can be achieved by introducing "charge operators". For the one-dimensional harmonic oscillator, the charge operators are the usual raising and lowering operators. The SUSY charge operators not only allow the factorization of a one-dimensional Hamiltonian but also form a Lie algebra structure. This structure leads to the generation of isospectral SUSY partner Hamiltonians. The eigenstates of the various partner Hamiltonians are connected by application of the charge operators. As an analytical approach, the SUSY-QM approach has been utilized to study a number of quantum mechanics problems including the Morse oscillator ([16]) and the radial hydrogen atom equation ([24]). In addition, SUSY-QM has been applied to the discovery of new exactly solvable potentials, the development of a more accurate WKB approximation, and the improvement of large *N* expansions and variational methods ([7, 11]). Developments and applications of one-dimensional SUSY-QM can be found in relevant reviews and books ([7, 9, 11, 15, 26, 32, 33]). Recently, SUSY-QM has been developed as a computational tool to provide much more accurate excitation energies

The harmonic oscillator is fundamental to a wide range of physics, including the electromagnetic field, spectroscopy, solid state physics, coherent state theory, and SUSY-QM.

> ©2012 Chou et al., 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 Chou 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 Chou 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.

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

**Supersymmetric Quantum Mechanics for Bound**


**Provisional chapter**

#### **New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations**

Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri

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

http://dx.doi.org/10.5772/54010 10.5772/54010

### **1. Introduction**

22 Advances in Quantum Mechanics

498 Advances in Quantum Mechanics

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Supersymmetric quantum mechanics (SUSY-QM) has been developed as an elegant analytical approach to one-dimensional problems. The SUSY-QM formalism generalizes the ladder operator approach used in the treatment of the harmonic oscillator. In analogy with the harmonic oscillator Hamiltonian, the factorization of a one-dimensional Hamiltonian can be achieved by introducing "charge operators". For the one-dimensional harmonic oscillator, the charge operators are the usual raising and lowering operators. The SUSY charge operators not only allow the factorization of a one-dimensional Hamiltonian but also form a Lie algebra structure. This structure leads to the generation of isospectral SUSY partner Hamiltonians. The eigenstates of the various partner Hamiltonians are connected by application of the charge operators. As an analytical approach, the SUSY-QM approach has been utilized to study a number of quantum mechanics problems including the Morse oscillator ([16]) and the radial hydrogen atom equation ([24]). In addition, SUSY-QM has been applied to the discovery of new exactly solvable potentials, the development of a more accurate WKB approximation, and the improvement of large *N* expansions and variational methods ([7, 11]). Developments and applications of one-dimensional SUSY-QM can be found in relevant reviews and books ([7, 9, 11, 15, 26, 32, 33]). Recently, SUSY-QM has been developed as a computational tool to provide much more accurate excitation energies using the standard Rayleigh-Ritz variational method ([5, 19, 20]).

The harmonic oscillator is fundamental to a wide range of physics, including the electromagnetic field, spectroscopy, solid state physics, coherent state theory, and SUSY-QM.

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 Chou 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, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Chou 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.

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

The broad application of the harmonic oscillator stems from the raising and lowering ladder operators which are used to factor the system Hamiltonian. For example, canonical coherent states are defined as the eigenstates of the lowering operator of the harmonic oscillator, and they are also minimum uncertainty states which minimize the Heisenberg uncertainty product for position and momentum. In addition, several different approaches have been employed to study generalized and approximate coherent states for systems other than the harmonic oscillator ([3, 12, 17, 18, 27–31, 34, 37]). Furthermore, algebraic treatments have been applied to the extension of coherent states for shape-invariant systems ([1, 4, 8, 10]).

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*mω*2*x*2, (1)

, (2)

<sup>2</sup>*mh*¯ *<sup>ω</sup>* , (3)

<sup>2</sup>*mh*¯ *<sup>ω</sup>* . (4)

(*x*ˆ + *ip*ˆ*x*) *ψ*<sup>0</sup> = 0. (5)

*<sup>x</sup>*|*ψ*� ≥ |�*ψ*|*x*ˆ*p*ˆ*x*|*ψ*�|2, (7)

*<sup>ψ</sup>*0(*x*) = �*x*|0� <sup>=</sup> *Ne*−*x*2/2, (6)

**2. Theoretical formulation**

operator *p*ˆ*x* by

operator

state wave function

Schwarz's inequality ([25])

where *N* is the normalization constant.

**2.1. Harmonic oscillator and conventional coherent states**

can be written in terms of the raising and lowering operators as

*<sup>H</sup>* <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*m*

*H* = *h*¯ *ω*

*a*ˆ =

*<sup>a</sup>*ˆ*ψ*<sup>0</sup> <sup>=</sup> <sup>1</sup> √2

�*ψ*|*x*ˆ

<sup>2</sup>|*ψ*��*ψ*|*p*ˆ

2

*a*ˆ † = *mω* 2¯*<sup>h</sup> <sup>x</sup>*<sup>ˆ</sup> <sup>+</sup>

*mω*

*d*2 *dx*<sup>2</sup> <sup>+</sup>

where *m* is the particle's mass, *ω* is the angular frequency of the oscillator. The Hamiltonian

 *a*ˆ †*a*ˆ + 1 2 

where *a*ˆ† is the raising operator and *a*ˆ is the lowering operator. These two operators can be expressed in terms of the position operator *x*ˆ and its canonically conjugate momentum

*ip*ˆ*<sup>x</sup>* √

2¯*<sup>h</sup> <sup>x</sup>*<sup>ˆ</sup> <sup>−</sup> *ip*ˆ*<sup>x</sup>* <sup>√</sup>

Without loss of generality, we set ¯*h* = 2*m* = 1 throughout this study and *ω* = 2 for this case. In particular, the ground state of the harmonic oscillator is annihilated by the lowering

By solving this differential equation in the position representation, we obtain the ground

One of the important properties for the ground state of the harmonic oscillator is that the ground state is a minimum uncertainty state, which minimizes the Heisenberg uncertainty product ∆*x*ˆ∆*p*ˆ*x*. The usual derivation of the Heisenberg uncertainty principle makes use of

1 2

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

The Hamiltonian of the harmonic oscillator is expressed by

The lowering operator of the harmonic oscillator annihilates the ground state, and the ground state minimizes the Heisenberg uncertainty product. Conventional harmonic oscillator coherent states correspond to those states which minimize the position-momentum uncertainty relation. However, these harmonic oscillator coherent states are also constructed by applying shift operators labeled with points of the discrete phase space to the ground state of the harmonic oscillator, termed the "fiducial state" ([18]). Indeed, Klauder and Skagerstam choose to define coherent states in the broadest sense in precisely this manner ([21]). Analogously, the charge operator in SUSY-QM annihilates the ground state of the corresponding system. We therefore expect that the ground state wave function should provide the ideal fiducial function for constructing efficient, overcomplete coherent states for computations of excited states of the system.

In our recent study ([6]), we construct system-specific coherent states for any bound quantum system by making use of the similarity between the treatment of the harmonic oscillator and SUSY-QM. First, since the charge operator annihilates the ground state, the superpotential that arises in SUSY-QM can be regarded as a SUSY-displacement operator or a generalized displacement variable. We show that the ground state for any bound quantum system minimizes the SUSY-displacement-standard momentum uncertainty product. Then, we use the ground state of the system as a fiducial function to generate new system-specific dynamically-adapted coherent states. Moreover, the discretized system-specific coherent states can be utilized as a dynamically-adapted basis for calculations of excited state energies and wave functions for bound quantum systems. Computational results demonstrate that these discretized system-specific coherent states provide more rapidly-converging expansions for excited state energies and wave functions than the conventional coherent states and the standard harmonic oscillator basis.

The organization of the remainder of this chapter is as follows. In Sec. 2, we briefly review the harmonic oscillator, conventional coherent states, and SUSY-QM. We also show that the ground state of a quantum system minimizes the SUSY-displacement-standard momentum uncertainty product. We then construct system-specific coherent states by applying shift operators to the ground state of the system. In Sec. 3, the discretized system-specific coherent state basis is developed for and applied to the Morse oscillator, the double well potential, and the two-dimensional anharmonic oscillator system for calculations of the excited state energies and wave functions. In Sec. 4, we summarize our results and conclude with some comments.

### **2. Theoretical formulation**

2 Quantum Mechanics

The broad application of the harmonic oscillator stems from the raising and lowering ladder operators which are used to factor the system Hamiltonian. For example, canonical coherent states are defined as the eigenstates of the lowering operator of the harmonic oscillator, and they are also minimum uncertainty states which minimize the Heisenberg uncertainty product for position and momentum. In addition, several different approaches have been employed to study generalized and approximate coherent states for systems other than the harmonic oscillator ([3, 12, 17, 18, 27–31, 34, 37]). Furthermore, algebraic treatments have been applied to the extension of coherent states for shape-invariant systems ([1, 4, 8, 10]).

The lowering operator of the harmonic oscillator annihilates the ground state, and the ground state minimizes the Heisenberg uncertainty product. Conventional harmonic oscillator coherent states correspond to those states which minimize the position-momentum uncertainty relation. However, these harmonic oscillator coherent states are also constructed by applying shift operators labeled with points of the discrete phase space to the ground state of the harmonic oscillator, termed the "fiducial state" ([18]). Indeed, Klauder and Skagerstam choose to define coherent states in the broadest sense in precisely this manner ([21]). Analogously, the charge operator in SUSY-QM annihilates the ground state of the corresponding system. We therefore expect that the ground state wave function should provide the ideal fiducial function for constructing efficient, overcomplete coherent states

In our recent study ([6]), we construct system-specific coherent states for any bound quantum system by making use of the similarity between the treatment of the harmonic oscillator and SUSY-QM. First, since the charge operator annihilates the ground state, the superpotential that arises in SUSY-QM can be regarded as a SUSY-displacement operator or a generalized displacement variable. We show that the ground state for any bound quantum system minimizes the SUSY-displacement-standard momentum uncertainty product. Then, we use the ground state of the system as a fiducial function to generate new system-specific dynamically-adapted coherent states. Moreover, the discretized system-specific coherent states can be utilized as a dynamically-adapted basis for calculations of excited state energies and wave functions for bound quantum systems. Computational results demonstrate that these discretized system-specific coherent states provide more rapidly-converging expansions for excited state energies and wave functions than the conventional coherent

The organization of the remainder of this chapter is as follows. In Sec. 2, we briefly review the harmonic oscillator, conventional coherent states, and SUSY-QM. We also show that the ground state of a quantum system minimizes the SUSY-displacement-standard momentum uncertainty product. We then construct system-specific coherent states by applying shift operators to the ground state of the system. In Sec. 3, the discretized system-specific coherent state basis is developed for and applied to the Morse oscillator, the double well potential, and the two-dimensional anharmonic oscillator system for calculations of the excited state energies and wave functions. In Sec. 4, we summarize our results and conclude with some

for computations of excited states of the system.

states and the standard harmonic oscillator basis.

comments.

#### **2.1. Harmonic oscillator and conventional coherent states**

The Hamiltonian of the harmonic oscillator is expressed by

$$H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2,\tag{1}$$

where *m* is the particle's mass, *ω* is the angular frequency of the oscillator. The Hamiltonian can be written in terms of the raising and lowering operators as

$$H = \hbar\omega\left(\hbar^\dagger\hbar + \frac{1}{2}\right),\tag{2}$$

where *a*ˆ† is the raising operator and *a*ˆ is the lowering operator. These two operators can be expressed in terms of the position operator *x*ˆ and its canonically conjugate momentum operator *p*ˆ*x* by

$$
\hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\hat{\mathbf{x}} + \frac{i\hat{p}\_{\mathbf{x}}}{\sqrt{2m\hbar\omega}}\,,\tag{3}
$$

$$\mathfrak{A}^{\dagger} = \sqrt{\frac{m\omega}{2\hbar}}\hat{\mathfrak{x}} - \frac{i\mathfrak{p}\_{\text{x}}}{\sqrt{2m\hbar\omega}}\text{.}\tag{4}$$

Without loss of generality, we set ¯*h* = 2*m* = 1 throughout this study and *ω* = 2 for this case. In particular, the ground state of the harmonic oscillator is annihilated by the lowering operator

$$
\hat{a}\psi\_0 = \frac{1}{\sqrt{2}} \left( \hat{\mathfrak{x}} + i\hat{p}\_x \right) \psi\_0 = 0. \tag{5}
$$

By solving this differential equation in the position representation, we obtain the ground state wave function

$$
\psi\_0(\mathbf{x}) = \langle \mathbf{x} | \mathbf{0} \rangle = \text{Ne}^{-\mathbf{x}^2/2}\text{ }\tag{6}
$$

where *N* is the normalization constant.

One of the important properties for the ground state of the harmonic oscillator is that the ground state is a minimum uncertainty state, which minimizes the Heisenberg uncertainty product ∆*x*ˆ∆*p*ˆ*x*. The usual derivation of the Heisenberg uncertainty principle makes use of Schwarz's inequality ([25])

$$
\langle \langle \Psi | \hat{\mathfrak{x}}^2 | \Psi \rangle \langle \Psi | \hat{p}\_{\mathbf{x}}^2 | \Psi \rangle \geq |\langle \Psi | \hat{\mathfrak{x}} \hat{p}\_{\mathbf{x}} | \Psi \rangle|^2. \tag{7}
$$

where zero expectation values of the position and momentum operators are assumed for convenience. The equality holds for the state |*ψ*�, which satisfies the condition

$$
\langle \pounds | \psi \rangle = -i\sigma^2 \not p\_{\text{x}} | \psi \rangle,\tag{8}
$$

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<sup>0</sup> (*x*). (13)

. (15)

<sup>0</sup> = 0. (16)

<sup>1</sup> (*x*) + *V*1(*x*) = 0. (14)

*<sup>W</sup>*1(*x*) = <sup>−</sup> *<sup>d</sup>*

*dW*1(*x*) *dx* <sup>−</sup> *<sup>W</sup>*<sup>2</sup>

*<sup>V</sup>*1(*x*) =

On the other hand, if *W*1(*x*) is known, then *V*1(*x*) is given by

Obviously, the Schrödinger equation in Eq. (12) is equivalent to

<sup>−</sup> *<sup>d</sup>*2*ψ*(1) 0 *dx*<sup>2</sup> <sup>+</sup>

*<sup>Q</sup>*<sup>1</sup> <sup>=</sup> *<sup>d</sup>*

*Q*†

= *Q*1*Q*† 1 *<sup>Q</sup>*1*ψ*(1) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(1) *<sup>n</sup> <sup>Q</sup>*1*ψ*(1) *<sup>n</sup>* 

*Q*† <sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>d</sup>*

introducing the "charge" operator and its adjoint

<sup>1</sup>*Q*1. Since *<sup>E</sup>*(1)

*H*<sup>2</sup> *<sup>Q</sup>*1*ψ*(1) *<sup>n</sup>* 

where the sector two Hamiltonian is defined as *H*<sup>2</sup> = *Q*1*Q*†

 *W*<sup>2</sup> <sup>1</sup> <sup>−</sup> *dW*<sup>1</sup> *dx*

Analogous to the harmonic oscillator, the Hamiltonian operator can be factorized by

where *p*ˆ*<sup>x</sup>* = −*i*(*d*/*dx*) is the coordinate representation of the momentum operator. Throughout this study, the ground state wave function *ψ*0(*x*) is assumed to be purely real; hence, the superpotential *W*(*x*) is self-adjoint. Then, the sector one Hamiltonian is defined

where *<sup>ψ</sup>*(1) *<sup>n</sup>* is an eigenstate of *<sup>H</sup>*<sup>1</sup> with *<sup>E</sup>*(1) *<sup>n</sup>* �<sup>=</sup> 0. Applying *<sup>Q</sup>*<sup>1</sup> to this equation, we obtain

*<sup>H</sup>*<sup>2</sup> with the same energy *<sup>E</sup>*(1) *<sup>n</sup>* as the state *<sup>ψ</sup>*(1) *<sup>n</sup>* . Analogously, we consider the eigenstates of

for the superpotential

as *H*<sup>1</sup> = *Q*†

*n* > 0

*H*<sup>2</sup>

*dx* ln *<sup>ψ</sup>*(1)

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

*<sup>W</sup>*1(*x*)<sup>2</sup> <sup>−</sup> *dW*1(*x*)

*dx*

*<sup>ψ</sup>*(1)

<sup>0</sup> = 0 for *n* = 0, it follows from the Schrödinger equation that for

*dx* <sup>+</sup> *<sup>W</sup>*<sup>1</sup> <sup>=</sup> *<sup>W</sup>*<sup>1</sup> <sup>+</sup> *ip*ˆ*x*, (17)

*dx* <sup>+</sup> *<sup>W</sup>*<sup>1</sup> <sup>=</sup> *<sup>W</sup>*<sup>1</sup> <sup>−</sup> *ip*ˆ*x*, (18)

<sup>1</sup>*Q*1*ψ*(1) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(1) *<sup>n</sup> <sup>ψ</sup>*(1) *<sup>n</sup>* , (19)

, (20)

<sup>1</sup>. Thus, *<sup>Q</sup>*1*ψ*(1) *<sup>n</sup>* is an eigenstate of

Substituting Eq. (11) into the Schrödinger equation in Eq. (12), we obtain the Riccati equation

where *σ*<sup>2</sup> is real and greater than zero. As noted in Eq. (5), the ground state of the harmonic oscillator satisfies the relation with *σ*<sup>2</sup> = 1, and hence it is a minimum uncertainty state . In fact, the ground state corresponds to a state centered in the phase space at *x* = 0 and *k* = 0. Harmonic oscillator coherent states can be constructed by applying shift operators labeled with points of the discrete phase space to a fiducial state, which is taken as the ground state of the harmonic oscillator ([18, 21]). In this sense, harmonic oscillator coherent states are generated by |*α*� = *D*ˆ (*α*)|0�. The shift operator is given by

*D*ˆ (*α*) = *e <sup>α</sup>a*ˆ†−*α*<sup>∗</sup> *<sup>a</sup>*<sup>ˆ</sup> , (9)

where

$$\alpha = \frac{1}{\sqrt{2}} \left[ \frac{\chi}{\sigma} + ik\sigma \right]. \tag{10}$$

Here *α* is a complex-number representation of the phase point *x* and *k*, and the quantity *σ* is a scaling parameter with the dimensions of length. Thus, the harmonic oscillator coherent states can be constructed by applying the shift operator to the ground state of the harmonic oscillator.

#### **2.2. Supersymmetric quantum mechanics**

For one-dimensional SUSY-QM, the superpotential *W* is defined in terms of the ground state wave function by the Riccati substitution

$$\psi\_0^{(1)}(\mathbf{x}) = N \exp\left[-\int\_0^\mathbf{x} \mathcal{W}\_1(\mathbf{x'}) d\mathbf{x'}\right],\tag{11}$$

where *N* is the normalization constant. The index "(1)" indicates that the ground state wave function and the superpotential are associated with the sector one Hamiltonian. It is assumed that Eq. (11) solves the Schrödinger equation with energy equal to zero

$$-\frac{d^2\psi\_0^{(1)}}{d\mathbf{x}^2} + V\_1\psi\_0^{(1)} = 0.\tag{12}$$

This does not impose any restriction since the energy can be changed by adding any constant to the Hamiltonian. From Eq. (11), the superpotential can be expressed in terms of the ground state wave function by

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$$W\_1(\mathbf{x}) = -\frac{d}{d\mathbf{x}} \ln \psi\_0^{(1)}(\mathbf{x}).\tag{13}$$

Substituting Eq. (11) into the Schrödinger equation in Eq. (12), we obtain the Riccati equation for the superpotential

$$\frac{d\mathcal{W}\_1(\mathbf{x})}{d\mathbf{x}} - \mathcal{W}\_1^2(\mathbf{x}) + V\_1(\mathbf{x}) = \mathbf{0}.\tag{14}$$

On the other hand, if *W*1(*x*) is known, then *V*1(*x*) is given by

4 Quantum Mechanics

where

oscillator.

where zero expectation values of the position and momentum operators are assumed for

where *σ*<sup>2</sup> is real and greater than zero. As noted in Eq. (5), the ground state of the harmonic oscillator satisfies the relation with *σ*<sup>2</sup> = 1, and hence it is a minimum uncertainty state . In fact, the ground state corresponds to a state centered in the phase space at *x* = 0 and *k* = 0. Harmonic oscillator coherent states can be constructed by applying shift operators labeled with points of the discrete phase space to a fiducial state, which is taken as the ground state of the harmonic oscillator ([18, 21]). In this sense, harmonic oscillator coherent states are

*D*ˆ (*α*) = *e*

Here *α* is a complex-number representation of the phase point *x* and *k*, and the quantity *σ* is a scaling parameter with the dimensions of length. Thus, the harmonic oscillator coherent states can be constructed by applying the shift operator to the ground state of the harmonic

For one-dimensional SUSY-QM, the superpotential *W* is defined in terms of the ground state

 − *<sup>x</sup>* 0

where *N* is the normalization constant. The index "(1)" indicates that the ground state wave function and the superpotential are associated with the sector one Hamiltonian. It is assumed

This does not impose any restriction since the energy can be changed by adding any constant to the Hamiltonian. From Eq. (11), the superpotential can be expressed in terms of the ground

*<sup>W</sup>*1(*x*′

)*dx*′ 

*<sup>α</sup>* <sup>=</sup> <sup>1</sup> √2 *x σ* + *ikσ* 

*<sup>α</sup>a*ˆ†−*α*<sup>∗</sup> *<sup>a</sup>*<sup>ˆ</sup>

*x*ˆ|*ψ*� = −*iσ*<sup>2</sup> *p*ˆ*x*|*ψ*�, (8)

, (9)

. (10)

, (11)

<sup>0</sup> = 0. (12)

convenience. The equality holds for the state |*ψ*�, which satisfies the condition

generated by |*α*� = *D*ˆ (*α*)|0�. The shift operator is given by

**2.2. Supersymmetric quantum mechanics**

*<sup>ψ</sup>*(1)

<sup>0</sup> (*x*) = *N* exp

that Eq. (11) solves the Schrödinger equation with energy equal to zero

<sup>−</sup> *<sup>d</sup>*2*ψ*(1) 0 *dx*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*1*ψ*(1)

wave function by the Riccati substitution

state wave function by

$$W\_1(\mathbf{x}) = \left(W\_1(\mathbf{x})^2 - \frac{dW\_1(\mathbf{x})}{d\mathbf{x}}\right). \tag{15}$$

Obviously, the Schrödinger equation in Eq. (12) is equivalent to

$$-\frac{d^2\psi\_0^{(1)}}{dx^2} + \left(W\_1^2 - \frac{d\mathcal{W}\_1}{dx}\right)\psi\_0^{(1)} = 0.\tag{16}$$

Analogous to the harmonic oscillator, the Hamiltonian operator can be factorized by introducing the "charge" operator and its adjoint

$$Q\_1 = \frac{d}{d\mathbf{x}} + W\_1 = W\_1 + i\mathfrak{p}\_{\mathbf{x}\prime} \tag{17}$$

$$Q\_1^\dagger = -\frac{d}{d\mathbf{x}} + W\_1 = W\_1 - i\mathfrak{p}\_{\mathbf{x}\prime} \tag{18}$$

where *p*ˆ*<sup>x</sup>* = −*i*(*d*/*dx*) is the coordinate representation of the momentum operator. Throughout this study, the ground state wave function *ψ*0(*x*) is assumed to be purely real; hence, the superpotential *W*(*x*) is self-adjoint. Then, the sector one Hamiltonian is defined as *H*<sup>1</sup> = *Q*† <sup>1</sup>*Q*1. Since *<sup>E</sup>*(1) <sup>0</sup> = 0 for *n* = 0, it follows from the Schrödinger equation that for *n* > 0

$$\mathcal{Q}\_1^\dagger \mathcal{Q}\_1 \psi\_n^{(1)} = E\_n^{(1)} \psi\_n^{(1)} \,\,\,\,\,\tag{19}$$

where *<sup>ψ</sup>*(1) *<sup>n</sup>* is an eigenstate of *<sup>H</sup>*<sup>1</sup> with *<sup>E</sup>*(1) *<sup>n</sup>* �<sup>=</sup> 0. Applying *<sup>Q</sup>*<sup>1</sup> to this equation, we obtain

$$H\_2\left(Q\_1\psi\_n^{(1)}\right) = Q\_1Q\_1^\dagger\left(Q\_1\psi\_n^{(1)}\right) = E\_n^{(1)}\left(Q\_1\psi\_n^{(1)}\right),\tag{20}$$

where the sector two Hamiltonian is defined as *H*<sup>2</sup> = *Q*1*Q*† <sup>1</sup>. Thus, *<sup>Q</sup>*1*ψ*(1) *<sup>n</sup>* is an eigenstate of *<sup>H</sup>*<sup>2</sup> with the same energy *<sup>E</sup>*(1) *<sup>n</sup>* as the state *<sup>ψ</sup>*(1) *<sup>n</sup>* . Analogously, we consider the eigenstates of *H*<sup>2</sup>

$$H\_2 \psi\_n^{(2)} = Q\_1 Q\_1^\dagger \psi\_n^{(2)} = E\_n^{(2)} \psi\_n^{(2)}.\tag{21}$$

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*<sup>x</sup>*|*ψ*�, (24)

*<sup>x</sup>*|*ψ*� ≥ |�*ψ*|*W*˜ *<sup>p</sup>*˜*x*|*ψ*�|2. (25)

(*W*<sup>ˆ</sup> − *<sup>λ</sup>p*ˆ*x*)|*ψ*� = (*W*<sup>0</sup> − *<sup>λ</sup>p*0)|*ψ*�. (26)

(*W*<sup>ˆ</sup> + *ip*ˆ*x*)|*ψ*� = (*W*<sup>0</sup> + *ip*0)|*ψ*�. (27)

 <sup>∞</sup> −∞ *ψ*∗ <sup>0</sup> (*x*)

*dψ*0(*x*)

*dψ*0(*x*)

*dx dx*. (29)

*dx dx*, (28)

∆*W*ˆ ∆*p*ˆ*x*. For an arbitrary normalized wave function, we consider the square of the

where *<sup>W</sup>*˜ = *<sup>W</sup>*<sup>ˆ</sup> − *<sup>W</sup>*<sup>0</sup> and *<sup>p</sup>*˜*<sup>x</sup>* = *<sup>p</sup>*ˆ*<sup>x</sup>* − *<sup>p</sup>*0. The quantities *<sup>W</sup>*<sup>0</sup> = �*W*� and *<sup>p</sup>*<sup>0</sup> = �*p*ˆ*x*� correspond to the averaged SUSY-displacement and momentum values, respectively. In order to obtain a lower bound on the uncertainty product in Eq. (24), we employ the Cauchy-Schwarz

The equality is satisfied when the two vectors *W*˜ |*ψ*� and *p*˜*x*|*ψ*� are collinear. From this

It follows from Eq. (23) that (*W*<sup>0</sup> + *ip*0) = �*ψ*0|*W*<sup>ˆ</sup> + *ip*ˆ*x*|*ψ*0� = 0 for the ground state of the system. Thus, Eq. (23) implies that the ground state satisfies the condition in Eq. (27). Therefore, the ground state of a bound quantum system minimizes the

We present some properties of the averaged SUSY-displacement and standard momentum values for the ground state. The averaged SUSY-displacement for the ground state is

<sup>0</sup> (*x*)*W*(*x*)*ψ*0(*x*)*dx* <sup>=</sup> <sup>−</sup>

 <sup>∞</sup> −∞ *ψ*∗ <sup>0</sup> (*x*)

Again, from Eqs. (28) and (29), *W*<sup>0</sup> + *ip*<sup>0</sup> = 0 for the ground state of the system, as indicated in Eq. (23). Furthermore, when the ground state wave function is purely real, it follows from integration by parts that the integral in Eqs. (28) and (29) is equal to zero. Thus, the averaged SUSY-displacement and momentum values for the real-valued ground state wave function

The ground state of a quantum system is the minimizer of the SUSY Heisenberg uncertainty product. We can derive the minimum value for the SUSY Heisenberg uncertainty product

where Eq. (13) has been used. The averaged momentum for the ground state is given by

2

<sup>2</sup> <sup>=</sup> �*ψ*|*W*˜ <sup>2</sup>|*ψ*��*ψ*|*p*˜

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

2

SUSY-displacement-standard momentum uncertainty product

 ∆*W*ˆ ∆*p*ˆ*<sup>x</sup>*

�*ψ*|*W*˜ <sup>2</sup>|*ψ*��*ψ*|*p*˜

condition, we obtain *W*˜ |*ψ*� = *λp*˜*x*|*ψ*�. Rearranging this equation yields

SUSY-displacement-standard momentum uncertainty product ∆*W*ˆ ∆*p*ˆ*x*.

 <sup>∞</sup> −∞ *ψ*∗

*<sup>p</sup>*<sup>0</sup> = �*ψ*0|*p*ˆ*x*|*ψ*0� = −*<sup>i</sup>*

As a special case for *λ* = −*i*, this equation becomes

*<sup>W</sup>*<sup>0</sup> = �*ψ*0|*W*|*ψ*0� =

are equal to zero, *W*<sup>0</sup> = *p*<sup>0</sup> = 0.

inequality

evaluated by

Applying *Q*† <sup>1</sup> to this equation, we notice that *<sup>Q</sup>*† <sup>1</sup>*ψ*(2) *<sup>n</sup>* is an eigenstate of *<sup>H</sup>*<sup>1</sup>

$$H\_1\left(\mathbf{Q}\_1^\dagger \boldsymbol{\psi}\_n^{(2)}\right) = \left(\mathbf{Q}\_1^\dagger \mathbf{Q}\_1\right)\left(\mathbf{Q}\_1^\dagger \boldsymbol{\psi}\_n^{(2)}\right) = E\_n^{(2)}\left(\mathbf{Q}\_1^\dagger \boldsymbol{\psi}\_n^{(2)}\right). \tag{22}$$

It follows that the Hamiltonians *H*<sup>1</sup> and *H*<sup>2</sup> have identical spectra with the exception of the ground state with *<sup>E</sup>*(1) <sup>0</sup> <sup>=</sup> 0. For the ground state, *<sup>Q</sup>*1*ψ*(1) <sup>0</sup> = 0, and this shows that the quantity *<sup>Q</sup>*1*ψ*(1) <sup>0</sup> cannot be used to generate the ground state of the sector two Hamiltonian. Because of the uniqueness of the ground state with *<sup>E</sup>*(1) <sup>0</sup> = 0, the indexing of the first and second sector levels must be modified. It is clear that the eigenvalues and eigenfunctions of the two Hamiltonians *H*<sup>1</sup> and *H*<sup>2</sup> are related by

$$\begin{aligned} E\_n^{(2)} &= E\_{n+1'}^{(1)} & E\_0^{(1)} &= 0, \\ \psi\_n^{(2)} &= \frac{\mathbb{Q}\_1 \psi\_{n+1}^{(1)}}{\sqrt{E\_{n+1}^{(1)}}}, & \psi\_{n+1}^{(1)} &= \frac{\mathbb{Q}\_1^{\dagger} \psi\_n^{(2)}}{\sqrt{E\_n^{(2)}}}. \end{aligned}$$

Analogously, starting from *<sup>H</sup>*<sup>2</sup> whose ground state energy is *<sup>E</sup>*(2) <sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1) <sup>1</sup> , we can generate the sector three Hamiltonian *H*<sup>3</sup> as a SUSY partner of *H*2. This procedure can be continued until the number of bound excited states supported by *H*<sup>1</sup> is exhausted.

#### **2.3. SUSY Heisenberg uncertainty products**

It follows from Eq. (13) that the charge operator annihilates the corresponding ground state

$$Q\psi\_0 = \left(\hat{\mathcal{W}} + i\mathfrak{p}\_x\right)\psi\_0 = 0.\tag{23}$$

Because we concentrate only on the sector one Hamiltonian in the present study, we suppress the sector index. For the harmonic oscillator, the charge operators correspond to the raising and lowering operators for the harmonic oscillator with *W*(*x*) = *x*. From the similarity, the superpotential *W*ˆ can be regarded as a "SUSY-displacement" operator although such a displacement would, in general, not be generated by the standard momentum operator *p*ˆ*x*. In fact, *W*ˆ and *p*ˆ*<sup>x</sup>* are not canonically conjugate variables.

The ground state of the harmonic oscillator is a minimum uncertainty state, which minimizes the Heisenberg uncertainty product ∆*x*ˆ∆*p*ˆ*x*. Analogously, it is expected that the ground state for a bound quantum system minimizes the SUSY Heisenberg uncertainty product ∆*W*ˆ ∆*p*ˆ*x*. For an arbitrary normalized wave function, we consider the square of the SUSY-displacement-standard momentum uncertainty product

$$\left(\Delta \hat{\mathcal{W}} \Delta \hat{p}\_{\text{x}}\right)^{2} = \langle \psi | \tilde{\mathcal{W}}^{2} | \psi \rangle \langle \psi | \tilde{p}\_{\text{x}}^{2} | \psi \rangle,\tag{24}$$

where *<sup>W</sup>*˜ = *<sup>W</sup>*<sup>ˆ</sup> − *<sup>W</sup>*<sup>0</sup> and *<sup>p</sup>*˜*<sup>x</sup>* = *<sup>p</sup>*ˆ*<sup>x</sup>* − *<sup>p</sup>*0. The quantities *<sup>W</sup>*<sup>0</sup> = �*W*� and *<sup>p</sup>*<sup>0</sup> = �*p*ˆ*x*� correspond to the averaged SUSY-displacement and momentum values, respectively. In order to obtain a lower bound on the uncertainty product in Eq. (24), we employ the Cauchy-Schwarz inequality

$$<\langle \psi | \tilde{\mathcal{W}}^2 | \psi \rangle \langle \psi | \tilde{p}\_x^2 | \psi \rangle \ge |\langle \psi | \tilde{\mathcal{W}} \tilde{p}\_x | \psi \rangle|^2. \tag{25}$$

The equality is satisfied when the two vectors *W*˜ |*ψ*� and *p*˜*x*|*ψ*� are collinear. From this condition, we obtain *W*˜ |*ψ*� = *λp*˜*x*|*ψ*�. Rearranging this equation yields

$$(\hat{\mathcal{W}} - \lambda \pounds\_{\mathcal{X}}) |\psi\rangle = (\mathcal{W}\_0 - \lambda \pounds\_0) |\psi\rangle. \tag{26}$$

As a special case for *λ* = −*i*, this equation becomes

6 Quantum Mechanics

Applying *Q*†

ground state with *<sup>E</sup>*(1)

quantity *<sup>Q</sup>*1*ψ*(1)

*<sup>H</sup>*2*ψ*(2) *<sup>n</sup>* <sup>=</sup> *<sup>Q</sup>*1*Q*†

It follows that the Hamiltonians *H*<sup>1</sup> and *H*<sup>2</sup> have identical spectra with the exception of the

second sector levels must be modified. It is clear that the eigenvalues and eigenfunctions of

<sup>0</sup> = 0,

*<sup>n</sup>*+<sup>1</sup> <sup>=</sup> *<sup>Q</sup>*†

<sup>1</sup>*ψ*(2) *<sup>n</sup> E*(2) *<sup>n</sup>* .

, *<sup>ψ</sup>*(1)

the sector three Hamiltonian *H*<sup>3</sup> as a SUSY partner of *H*2. This procedure can be continued

It follows from Eq. (13) that the charge operator annihilates the corresponding ground state

*W*ˆ + *ip*ˆ*<sup>x</sup>* 

Because we concentrate only on the sector one Hamiltonian in the present study, we suppress the sector index. For the harmonic oscillator, the charge operators correspond to the raising and lowering operators for the harmonic oscillator with *W*(*x*) = *x*. From the similarity, the superpotential *W*ˆ can be regarded as a "SUSY-displacement" operator although such a displacement would, in general, not be generated by the standard momentum operator *p*ˆ*x*.

The ground state of the harmonic oscillator is a minimum uncertainty state, which minimizes the Heisenberg uncertainty product ∆*x*ˆ∆*p*ˆ*x*. Analogously, it is expected that the ground state for a bound quantum system minimizes the SUSY Heisenberg uncertainty product

*<sup>n</sup>*+1, *<sup>E</sup>*(1)

<sup>0</sup> cannot be used to generate the ground state of the sector two Hamiltonian.

<sup>0</sup> <sup>=</sup> 0. For the ground state, *<sup>Q</sup>*1*ψ*(1)

<sup>1</sup> to this equation, we notice that *<sup>Q</sup>*†

Because of the uniqueness of the ground state with *<sup>E</sup>*(1)

*E*(2) *<sup>n</sup>* = *E*(1)

Analogously, starting from *<sup>H</sup>*<sup>2</sup> whose ground state energy is *<sup>E</sup>*(2)

until the number of bound excited states supported by *H*<sup>1</sup> is exhausted.

*Qψ*<sup>0</sup> =

*<sup>ψ</sup>*(2) *<sup>n</sup>* <sup>=</sup> *<sup>Q</sup>*1*ψ*(1) *<sup>n</sup>*+<sup>1</sup> *E*(1) *n*+1

the two Hamiltonians *H*<sup>1</sup> and *H*<sup>2</sup> are related by

**2.3. SUSY Heisenberg uncertainty products**

In fact, *W*ˆ and *p*ˆ*<sup>x</sup>* are not canonically conjugate variables.

*H*1 *Q*† <sup>1</sup>*ψ*(2) *<sup>n</sup>* = *Q*† <sup>1</sup>*Q*<sup>1</sup> *Q*† <sup>1</sup>*ψ*(2) *<sup>n</sup>* = *E*(2) *<sup>n</sup> Q*† <sup>1</sup>*ψ*(2) *<sup>n</sup>* 

<sup>1</sup>*ψ*(2) *<sup>n</sup>* <sup>=</sup> *<sup>E</sup>*(2) *<sup>n</sup> <sup>ψ</sup>*(2) *<sup>n</sup>* . (21)

. (22)

<sup>1</sup> , we can generate

<sup>0</sup> = 0, and this shows that the

<sup>0</sup> = 0, the indexing of the first and

<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(1)

*ψ*<sup>0</sup> = 0. (23)

<sup>1</sup>*ψ*(2) *<sup>n</sup>* is an eigenstate of *<sup>H</sup>*<sup>1</sup>

$$(\hat{\mathcal{W}} + i\mathfrak{p}\_{\mathcal{X}})|\psi\rangle = (\mathcal{W}\_0 + i\mathfrak{p}\_0)|\psi\rangle. \tag{27}$$

It follows from Eq. (23) that (*W*<sup>0</sup> + *ip*0) = �*ψ*0|*W*<sup>ˆ</sup> + *ip*ˆ*x*|*ψ*0� = 0 for the ground state of the system. Thus, Eq. (23) implies that the ground state satisfies the condition in Eq. (27). Therefore, the ground state of a bound quantum system minimizes the SUSY-displacement-standard momentum uncertainty product ∆*W*ˆ ∆*p*ˆ*x*.

We present some properties of the averaged SUSY-displacement and standard momentum values for the ground state. The averaged SUSY-displacement for the ground state is evaluated by

$$\mathcal{W}\_0 = \langle \psi\_0 | \mathcal{W} | \psi\_0 \rangle = \int\_{-\infty}^{\infty} \psi\_0^\*(\mathbf{x}) \mathcal{W}(\mathbf{x}) \psi\_0(\mathbf{x}) d\mathbf{x} = -\int\_{-\infty}^{\infty} \psi\_0^\*(\mathbf{x}) \frac{d\psi\_0(\mathbf{x})}{d\mathbf{x}} d\mathbf{x},\tag{28}$$

where Eq. (13) has been used. The averaged momentum for the ground state is given by

$$p\_0 = \langle \psi\_0 | \not p\_x | \psi\_0 \rangle = -i \int\_{-\infty}^{\infty} \psi\_0^\*(\mathbf{x}) \frac{d\psi\_0(\mathbf{x})}{d\mathbf{x}} d\mathbf{x}.\tag{29}$$

Again, from Eqs. (28) and (29), *W*<sup>0</sup> + *ip*<sup>0</sup> = 0 for the ground state of the system, as indicated in Eq. (23). Furthermore, when the ground state wave function is purely real, it follows from integration by parts that the integral in Eqs. (28) and (29) is equal to zero. Thus, the averaged SUSY-displacement and momentum values for the real-valued ground state wave function are equal to zero, *W*<sup>0</sup> = *p*<sup>0</sup> = 0.

The ground state of a quantum system is the minimizer of the SUSY Heisenberg uncertainty product. We can derive the minimum value for the SUSY Heisenberg uncertainty product in Eq. (25). For the real-valued ground state wave function, *<sup>W</sup>*˜ = *<sup>W</sup>*<sup>ˆ</sup> − *<sup>W</sup>*<sup>0</sup> = *<sup>W</sup>*<sup>ˆ</sup> and *<sup>p</sup>*˜*<sup>x</sup>* = *<sup>p</sup>*ˆ*<sup>x</sup>* − *<sup>p</sup>*<sup>0</sup> = *<sup>p</sup>*ˆ*x*. The right side of the uncertainty product in Eq. (25) becomes

$$
\langle \psi\_0 | \hat{W} \hat{p}\_{\ge} | \psi\_0 \rangle = i \langle \psi\_0 | \hat{W}^2 | \psi\_0 \rangle \,. \tag{30}
$$

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For the harmonic oscillator, *W*(*x*) = *x* and *dW*/*dx* = 1. We recover the conventional Heisenberg uncertainty product for the ground state ∆*x*ˆ∆*p*ˆ*<sup>x</sup>* = 1/2. As a special case, a similar derivation has been employed to determine exact minimum uncertainty coherent

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

Analogous to the harmonic oscillator coherent state, the analysis of a bound quantum system in terms of the SUSY Heisenberg uncertainty principle suggests the construction of system-specific coherent states based on the SUSY-QM ground state. Similarly, the procedure for creating an overcomplete set of such coherent states is to apply the shift operator to the

where *N* is the normalization constant. The raising and lowering operators for the shift

state. Thus, the functions *ψα* form an overcomplete set of the coherent states in the standard phase space which are specifically associated with the quantum-mechanical system described

We now consider a coordinate transformation given by *<sup>x</sup>*′ <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> for the system-specific

also normalized. The momentum operator is invariant under the coordinate transformation

) = *e ik*<sup>0</sup> *<sup>x</sup>*′ *<sup>ψ</sup>*0(*x*′

The averaged SUSY-displacement for the system-specific coherent state is given by

 <sup>∞</sup> −∞ *ψ*∗ *α*(*x*′

= − <sup>∞</sup> −∞ )*W*(*x*′

*<sup>ψ</sup>*0(*x*′ ) *<sup>d</sup>ψ*0(*x*′ ) *dx*′ *dx*′

*e*

√

) + *ip*ˆ*x*′)|*ψα*� = *ik*0|*ψα*�. (40)

)*ψα*(*x*′

)*dx*′

. (41)

<sup>=</sup> *Neik*0(*x*−*x*0)

2 is a point in the phase space which completely describes the coherent

2 and *a*ˆ = (*x*ˆ + *ip*ˆ*x*)/

) is the normalized real-valued ground state wave function, and thus *ψα*(*x*′

−*x*0(*d*/*dx*)

*ψ*0(*x*)

), (39)

) is

*<sup>ψ</sup>*0(*<sup>x</sup>* − *<sup>x</sup>*0), (38)

2, respectively. The quantity

*ψα*(*x*) = �*x*|*α*� <sup>=</sup> �*x*|*D*<sup>ˆ</sup> (*α*)|*ψ*0� <sup>=</sup> *Neik*0(*x*−*x*0)

√

coherent states in Eq. (38). The system-specific coherent state becomes

(*W*<sup>ˆ</sup> (*x*′

*<sup>W</sup>*0,*<sup>α</sup>* = �*ψα*|*W*|*ψα*� =

*ψα*(*x*′

states for the Morse oscillator ([8]).

**2.4. System-specific coherent states**

ground state as a fiducial function ([18, 21])

operator are given by *a*ˆ† = (*x*ˆ − *ip*ˆ*x*)/

(i.e., *p*ˆ*x*′ = *p*ˆ*x*). It is straightforward to show that

√

by the SUSY-displacement *W*(*x*).

*α* = (*x*<sup>0</sup> + *ik*0)/

where *<sup>ψ</sup>*0(*x*′

where *<sup>p</sup>*ˆ*x*|*ψ*0� = *iW*<sup>ˆ</sup> |*ψ*0� from Eq. (23) has been used. Thus, the right side of the uncertainty product in Eq. (25) is given by

$$|\langle \psi\_0 | \hat{W} \hat{\rho}\_x | \psi\_0 \rangle|^2 = \langle \hat{W}^2 \rangle^2. \tag{31}$$

Similarly, the left side of the uncertainty product in Eq. (25) is given by

$$
\langle \psi\_0 | \hat{\mathcal{W}}^2 | \psi\_0 \rangle \langle \psi\_0 | \hat{\mathcal{P}}\_x^2 | \psi\_0 \rangle = \langle \hat{\mathcal{W}}^2 \rangle \langle \hat{\mathcal{W}}^2 \rangle. \tag{32}
$$

Therefore, the equality in Eq. (25) holds for the ground state, and the SUSY Heisenberg uncertainty product is equal to ∆*W*ˆ ∆*p*ˆ*<sup>x</sup>* = �*W*ˆ <sup>2</sup>�.

The expectation value of *W*ˆ <sup>2</sup> for the ground state is evaluated by

$$
\langle \hat{\mathcal{W}}^2 \rangle = \int\_{-\infty}^{\infty} \psi\_0(\mathbf{x}) \mathcal{W}(\mathbf{x})^2 \psi\_0(\mathbf{x}) d\mathbf{x} = -\int\_{-\infty}^{\infty} \psi\_0(\mathbf{x}) \mathcal{W}(\mathbf{x}) \frac{d\psi\_0(\mathbf{x})}{d\mathbf{x}} d\mathbf{x},\tag{33}
$$

where Eq. (13) has been used. From integration by parts, the integral can be expressed by

$$\int\_{-\infty}^{\infty} \psi\_0(\mathbf{x}) \mathcal{W}(\mathbf{x}) \frac{d\psi\_0(\mathbf{x})}{d\mathbf{x}} d\mathbf{x} = -\frac{1}{2} \int\_{-\infty}^{\infty} \psi\_0(\mathbf{x}) \frac{d\mathcal{W}(\mathbf{x})}{d\mathbf{x}} \psi\_0(\mathbf{x}) d\mathbf{x}.\tag{34}$$

Thus, the expectation value of *W*ˆ <sup>2</sup> for the ground state is equal to one half of the expectation value for the derivative of the superpotential

$$
\langle \hat{\mathcal{W}}^2 \rangle = \frac{1}{2} \left\langle \frac{d\hat{\mathcal{W}}}{d\mathbf{x}} \right\rangle. \tag{35}
$$

Moreover, the commutation relation of the SUSY-displacement and the momentum operator is given by

$$\mathbb{E}\left[\hat{\mathcal{W}}\_{\prime}\hat{\mathcal{P}}\_{\mathbb{X}}\right] = i\frac{d\hat{\mathcal{W}}}{d\boldsymbol{x}}.\tag{36}$$

Therefore, the SUSY Heisenberg uncertainty product for the ground state becomes

$$
\Delta \hat{\mathcal{W}} \Delta \hat{\mathcal{p}}\_{\text{X}} = \left< \hat{\mathcal{W}}^2 \right> = \frac{1}{2} \left< \frac{d \hat{\mathcal{W}}}{d \mathbf{x}} \right> = \frac{1}{2i} \left< \left[ \hat{\mathcal{W}} \,\hat{\mathcal{p}}\_{\text{x}} \right] \right>. \tag{37}
$$

For the harmonic oscillator, *W*(*x*) = *x* and *dW*/*dx* = 1. We recover the conventional Heisenberg uncertainty product for the ground state ∆*x*ˆ∆*p*ˆ*<sup>x</sup>* = 1/2. As a special case, a similar derivation has been employed to determine exact minimum uncertainty coherent states for the Morse oscillator ([8]).

### **2.4. System-specific coherent states**

8 Quantum Mechanics

product in Eq. (25) is given by

in Eq. (25). For the real-valued ground state wave function, *<sup>W</sup>*˜ = *<sup>W</sup>*<sup>ˆ</sup> − *<sup>W</sup>*<sup>0</sup> = *<sup>W</sup>*<sup>ˆ</sup> and *<sup>p</sup>*˜*<sup>x</sup>* =

where *<sup>p</sup>*ˆ*x*|*ψ*0� = *iW*<sup>ˆ</sup> |*ψ*0� from Eq. (23) has been used. Thus, the right side of the uncertainty

2

Therefore, the equality in Eq. (25) holds for the ground state, and the SUSY Heisenberg

where Eq. (13) has been used. From integration by parts, the integral can be expressed by

2 <sup>∞</sup> −∞

Thus, the expectation value of *W*ˆ <sup>2</sup> for the ground state is equal to one half of the expectation

2 *dW*ˆ *dx*

Moreover, the commutation relation of the SUSY-displacement and the momentum operator

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

�*W*<sup>ˆ</sup> <sup>2</sup>� <sup>=</sup> <sup>1</sup>

 *W*ˆ , *p*ˆ*<sup>x</sup>* = *i dW*ˆ

<sup>∆</sup>*W*<sup>ˆ</sup> <sup>∆</sup>*p*ˆ*<sup>x</sup>* <sup>=</sup> �*W*<sup>ˆ</sup> <sup>2</sup>� <sup>=</sup> <sup>1</sup>

Therefore, the SUSY Heisenberg uncertainty product for the ground state becomes

2 *dW*ˆ *dx*

 <sup>=</sup> <sup>1</sup> 2*i*

*W*ˆ , *p*ˆ*<sup>x</sup>*

 <sup>∞</sup> −∞

*ψ*0(*x*)

�*ψ*0|*W*<sup>ˆ</sup> *<sup>p</sup>*ˆ*x*|*ψ*0� = *<sup>i</sup>*�*ψ*0|*W*<sup>ˆ</sup> <sup>2</sup>|*ψ*0�, (30)


*ψ*0(*x*)*W*(*x*)

*dW*(*x*)

*<sup>x</sup>*|*ψ*0� <sup>=</sup> �*W*<sup>ˆ</sup> <sup>2</sup>��*W*<sup>ˆ</sup> <sup>2</sup>�. (32)

*dψ*0(*x*)

*dx dx*, (33)

*dx <sup>ψ</sup>*0(*x*)*dx*. (34)

. (35)

*dx* . (36)

. (37)

*<sup>p</sup>*ˆ*<sup>x</sup>* − *<sup>p</sup>*<sup>0</sup> = *<sup>p</sup>*ˆ*x*. The right side of the uncertainty product in Eq. (25) becomes

Similarly, the left side of the uncertainty product in Eq. (25) is given by

�*ψ*0|*W*<sup>ˆ</sup> <sup>2</sup>|*ψ*0��*ψ*0|*p*<sup>ˆ</sup>

*<sup>ψ</sup>*0(*x*)*W*(*x*)2*ψ*0(*x*)*dx* = −

*dψ*0(*x*)

The expectation value of *W*ˆ <sup>2</sup> for the ground state is evaluated by

uncertainty product is equal to ∆*W*ˆ ∆*p*ˆ*<sup>x</sup>* = �*W*ˆ <sup>2</sup>�.

 <sup>∞</sup> −∞

*ψ*0(*x*)*W*(*x*)

�*W*ˆ <sup>2</sup>� =

 <sup>∞</sup> −∞

is given by

value for the derivative of the superpotential

Analogous to the harmonic oscillator coherent state, the analysis of a bound quantum system in terms of the SUSY Heisenberg uncertainty principle suggests the construction of system-specific coherent states based on the SUSY-QM ground state. Similarly, the procedure for creating an overcomplete set of such coherent states is to apply the shift operator to the ground state as a fiducial function ([18, 21])

$$\begin{split} \psi\_{\mathbf{a}}(\mathbf{x}) = \langle \mathbf{x} | \mathbf{a} \rangle = \langle \mathbf{x} | \hat{D}(\mathbf{a}) | \psi\_0 \rangle = N e^{i\mathbf{k}\_0(\mathbf{x} - \mathbf{x}\_0)} e^{-\mathbf{x}\_0(d/d\mathbf{x})} \psi\_0(\mathbf{x}) \\ = N e^{i\mathbf{k}\_0(\mathbf{x} - \mathbf{x}\_0)} \psi\_0(\mathbf{x} - \mathbf{x}\_0) \end{split} \tag{38}$$

where *N* is the normalization constant. The raising and lowering operators for the shift operator are given by *a*ˆ† = (*x*ˆ − *ip*ˆ*x*)/ √2 and *a*ˆ = (*x*ˆ + *ip*ˆ*x*)/ √2, respectively. The quantity *α* = (*x*<sup>0</sup> + *ik*0)/ √2 is a point in the phase space which completely describes the coherent state. Thus, the functions *ψα* form an overcomplete set of the coherent states in the standard phase space which are specifically associated with the quantum-mechanical system described by the SUSY-displacement *W*(*x*).

We now consider a coordinate transformation given by *<sup>x</sup>*′ <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> for the system-specific coherent states in Eq. (38). The system-specific coherent state becomes

$$
\psi\_a(\mathbf{x'}) = e^{i\mathbf{k}\_0 \mathbf{x'}} \psi\_0(\mathbf{x'}),
\tag{39}
$$

where *<sup>ψ</sup>*0(*x*′ ) is the normalized real-valued ground state wave function, and thus *ψα*(*x*′ ) is also normalized. The momentum operator is invariant under the coordinate transformation (i.e., *p*ˆ*x*′ = *p*ˆ*x*). It is straightforward to show that

$$(\hat{\mathcal{W}}(\mathbf{x'}) + i\mathfrak{p}\_{\mathbf{x'}})|\psi\_{\mathfrak{a}}\rangle = i\mathfrak{k}\_0|\psi\_{\mathfrak{a}}\rangle. \tag{40}$$

The averaged SUSY-displacement for the system-specific coherent state is given by

$$\begin{split} W\_{0,\mathfrak{a}} = \langle \psi\_{\mathfrak{a}} | W | \psi\_{\mathfrak{a}} \rangle &= \int\_{-\infty}^{\infty} \psi\_{\mathfrak{a}}^{\*}(\mathbf{x}') W(\mathbf{x}') \psi\_{\mathfrak{a}}(\mathbf{x}') d\mathbf{x}' \\ &= -\int\_{-\infty}^{\infty} \psi\_{\mathfrak{0}}(\mathbf{x}') \frac{d\psi\_{\mathfrak{0}}(\mathbf{x}')}{d\mathbf{x}'} d\mathbf{x}'. \end{split} \tag{41}$$

Again, it follows from integration by parts that *W*0,*<sup>α</sup>* = 0 for all system-specific coherent states. Analogously, the averaged momentum for the system-specific coherent state is given by

$$p\_{0\mathcal{A}} = \langle \psi\_{\mathfrak{k}} | \hat{p}\_{\mathbf{x'}} | \psi\_{\mathfrak{k}} \rangle = k\_0 - i \int\_{-\infty}^{\infty} \psi\_0(\mathbf{x'}) \frac{d\psi\_0(\mathbf{x'})}{d\mathbf{x'}} d\mathbf{x'}.\tag{42}$$

Because the integral is equal to zero, *p*0,*<sup>α</sup>* = *k*0. Thus, Eq. (40) can be written as

$$(\hat{\mathcal{W}}(\mathbf{x'}) + i\mathfrak{p}\_{\mathbf{x'}})|\psi\_{\mathbf{a}}\rangle = (\mathcal{W}\_{0,\mathbf{a}} + i p\_{0,\mathbf{a}})|\psi\_{\mathbf{a}}\rangle. \tag{43}$$

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*E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

**Exact -56.25 -42.25 -30.25 -20.25** SSCS (*M* = 9) -56.25 -42.2499824 -30.2270611 -19.52261 SSCS (*M* = 15) -56.25 -42.2499999 -30.2499343 -20.23502 HOCS (*M* = 9) -54.95 -37.00 -21.08 -10.22 HOCS (*M* = 15) -56.13 -41.62 -28.61 -17.62 HO (*M* = 9) -53.79 -33.34 -16.45 -6.40 HO (*M* = 15) -55.54 -39.03 -23.84 -12.30

**Table 1.** Comparison of the energy eigenvalues for the Morse oscillator obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis functions (HO) with the exact results.

0.2 0.4 0.6 0.8 1 1.2

∆ x

**Figure 1.** Logarithm of the relative error, versus grid spacing ∆*x*, of the first (◦), second (⋄), and third () excited state energies for the Morse oscillator using the system-specific coherent states (—) and the harmonic oscillator coherent states (- - -) with 15

To use the Rayleigh-Ritz variational principle, we construct a trial wave function in terms of

*M* ∑ *i*=1

where *ci* are the coefficients. Because of the non-orthogonality of the system-specific coherent states, the energy eigenvalues and wave functions are determined by solving the generalized

where*Hij* = �*αi*|*H*|*αj*� is the matrix element of the Hamiltonian, *Sij* = �*αi*|*αj*� is the overlap matrix, and *C* is a vector of linear combination coefficients for the eigenvector. Therefore, solving Eq. (47) yields the variational approximation to the eigenvalues and eigenvectors of

*ci*|*αi*�, (46)

*HC* = *ESC*, (47)


−9 −8 −7 −6 −5 −4 −3 −2 −1 0

a linear combination of the system-specific coherent states

basis functions.

eigenvalue problem ([36])

the Hamiltonian operator.

Log10(Err)

Analogous to the uncertainty condition for the ground state in Eq. (27), this equation implies that the system-specific coherent state |*ψα*� minimizes the SUSY-displacement-momentum uncertainty product <sup>∆</sup>*W*<sup>ˆ</sup> <sup>∆</sup>*p*ˆ*x*′ for the displaced coordinate *<sup>x</sup>*′ <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0.

#### **2.5. Discretized system-specific coherent states**

A discretized SUSY-QM coherent state basis can be constructed by discretizing the continuous label *α* = (*q* + *ik*)/ √2 and setting up a von Neumann lattice in phase space with an appropriate density *D*. The discretized system-specific coherent state basis is given by

$$\psi\_{\mathbf{a}\_{i}}(\mathbf{x}) = \langle \mathbf{x} | \mathbf{a}\_{i} \rangle = N e^{i \mathbf{k}\_{i} (\mathbf{x} - q\_{i})} \exp \left[ - \int\_{0}^{\mathbf{x} - q\_{i}} W(\mathbf{x}') d\mathbf{x}' \right],\tag{44}$$

where *i* = 1, . . . , *M* and *M* is the number of basis functions. The phase space grid points are defined as ([2])

$$\{(q\_i, k\_i)\} = \left\{ \left( m \Delta x \sqrt{\frac{2\pi}{D}}, \frac{n}{\Delta x} \sqrt{\frac{2\pi}{D}} \right) \right\} \quad m, n \in \mathbb{Z} \tag{45}$$

where *m* and *n* run over all integers, hence *i* can be thought of as a joint index consisting of *m* and *n*. The quantity *D* is the density of grid points in units of 2*πh*¯. As discussed in Klauder and Skagerstam's book ([18]), generalized coherent states constructed by applying displacement operators to a fiducial state are overcomplete; however, completeness of the discretized system-specific coherent states in Eq. (44) has not been established here.

Since the ground state solves the time-independent Schrödinger equation for the corresponding Hamiltonian, the system-specific coherent states build in the dynamics of the system under investigation. This property leads to the expectation that these dynamically-adapted and system-specific coherent states will prove more rapidly convergent in calculations of the excited state energies and wave functions for quantum systems using variational methods.

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10 Quantum Mechanics

by

by

defined as ([2])

variational methods.

Again, it follows from integration by parts that *W*0,*<sup>α</sup>* = 0 for all system-specific coherent states. Analogously, the averaged momentum for the system-specific coherent state is given

Analogous to the uncertainty condition for the ground state in Eq. (27), this equation implies that the system-specific coherent state |*ψα*� minimizes the SUSY-displacement-momentum

A discretized SUSY-QM coherent state basis can be constructed by discretizing the

with an appropriate density *D*. The discretized system-specific coherent state basis is given

where *i* = 1, . . . , *M* and *M* is the number of basis functions. The phase space grid points are

where *m* and *n* run over all integers, hence *i* can be thought of as a joint index consisting of *m* and *n*. The quantity *D* is the density of grid points in units of 2*πh*¯. As discussed in Klauder and Skagerstam's book ([18]), generalized coherent states constructed by applying displacement operators to a fiducial state are overcomplete; however, completeness of the

Since the ground state solves the time-independent Schrödinger equation for the corresponding Hamiltonian, the system-specific coherent states build in the dynamics of the system under investigation. This property leads to the expectation that these dynamically-adapted and system-specific coherent states will prove more rapidly convergent in calculations of the excited state energies and wave functions for quantum systems using

2*π <sup>D</sup>* , *<sup>n</sup>* ∆*x*

discretized system-specific coherent states in Eq. (44) has not been established here.

 − *<sup>x</sup>*−*qi* 0

2*π D*

 <sup>∞</sup> −∞ *<sup>ψ</sup>*0(*x*′ ) *<sup>d</sup>ψ*0(*x*′ ) *dx*′ *dx*′

) + *ip*ˆ*x*′)|*ψα*� = (*W*0,*<sup>α</sup>* + *ip*0,*α*)|*ψα*�. (43)

2 and setting up a von Neumann lattice in phase space

*<sup>W</sup>*(*x*′ )*dx*′ 

. (42)

, (44)

*m*, *n* ∈ Z (45)

*<sup>p</sup>*0,*<sup>α</sup>* = �*ψα*|*p*ˆ*x*′|*ψα*� = *<sup>k</sup>*<sup>0</sup> − *<sup>i</sup>*

uncertainty product <sup>∆</sup>*W*<sup>ˆ</sup> <sup>∆</sup>*p*ˆ*x*′ for the displaced coordinate *<sup>x</sup>*′ <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0.

√

*ψαi*(*x*) = �*x*|*αi*� <sup>=</sup> *Neiki*(*x*−*qi*) exp

 *m*∆*x*

(*W*<sup>ˆ</sup> (*x*′

**2.5. Discretized system-specific coherent states**

{(*qi*, *ki*)} =

continuous label *α* = (*q* + *ik*)/

Because the integral is equal to zero, *p*0,*<sup>α</sup>* = *k*0. Thus, Eq. (40) can be written as

**Table 1.** Comparison of the energy eigenvalues for the Morse oscillator obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis functions (HO) with the exact results.

**Figure 1.** Logarithm of the relative error, versus grid spacing ∆*x*, of the first (◦), second (⋄), and third () excited state energies for the Morse oscillator using the system-specific coherent states (—) and the harmonic oscillator coherent states (- - -) with 15 basis functions.

To use the Rayleigh-Ritz variational principle, we construct a trial wave function in terms of a linear combination of the system-specific coherent states

$$|\psi\rangle = \sum\_{i=1}^{M} c\_i |\alpha\_i\rangle\_\prime \tag{46}$$

where *ci* are the coefficients. Because of the non-orthogonality of the system-specific coherent states, the energy eigenvalues and wave functions are determined by solving the generalized eigenvalue problem ([36])

$$HC = ESC\_{\prime} \tag{47}$$

where*Hij* = �*αi*|*H*|*αj*� is the matrix element of the Hamiltonian, *Sij* = �*αi*|*αj*� is the overlap matrix, and *C* is a vector of linear combination coefficients for the eigenvector. Therefore, solving Eq. (47) yields the variational approximation to the eigenvalues and eigenvectors of the Hamiltonian operator.

### **3. Computational results**

#### **3.1. Morse oscillator**

In order to demonstrate features of system-specific coherent states, computational results will be presented for three quantum systems. The first of these concerns the Morse oscillator. The Hamiltonian of the Morse oscillator is given by

$$H = -\frac{d^2}{dx^2} + V(\mathbf{x}) = -\frac{d^2}{dx^2} + 64\left(e^{-2\mathbf{x}} - 2e^{-\mathbf{x}}\right). \tag{48}$$

The exact energy eigenvalues are *En* = −(*n* − 15/2)<sup>2</sup> where *n* = 0, . . . , 7, and the analytical expression of the ground state wave function is given by

$$\psi\_0(\mathbf{x}) = N \exp\left[-8e^{-\mathbf{x}} - \frac{15}{2}\mathbf{x}\right],\tag{49}$$

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�3 �2 �1 0 1 2 3 x

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

**Figure 2.** The ground state wave function of the double well potential obtained by the imaginary time propagation method is

√*π*

where *Hn*(*x*) is the Hermite polynomial. Compared with the results obtained from these two basis sets, the computational results from the system-specific coherent states achieve significantly higher accuracy using a small number of basis functions. Thus, the system-specific coherent states provide more accurate approximations of the excited state

Figure 1 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different

*Err* <sup>=</sup> *Enumerical* <sup>−</sup> *Eexact*

As shown in this figure, the system-specific coherent states yield excellent results for the excited state energies. The relative error of the first-excited state energy can even reach 10−<sup>9</sup> for a wide range of ∆*x*. Additionally, compared with the harmonic oscillator coherent states, the system-specific coherent states give much more accurate results for the first three excited state energies. Also, the system-specific coherent states yield stable computational results for

As an example of quantum systems without exact analytical solutions, we consider a

*Hn*(*x*)*e*

*<sup>φ</sup>n*(*x*) = <sup>1</sup>

values for ∆*x* with 15 basis functions. The relative error is defined by

2*nn*!

Vs(x)

<sup>−</sup>*<sup>x</sup>*2/2, (52)

<sup>|</sup>*Eexact*<sup>|</sup> . (53)

*V*(*x*) = 3*x*<sup>4</sup> − 8*x*2. (54)

�0.2

0

0.2

Ψ

shown with the scaled potential *Vs*(*x*) = *V*(*x*)/20.

energies for the Morse oscillator.

a wide range of ∆*x*.

**3.2. Double well potential**

symmetric double well potential given by

0.4

0.6

0.8

where *N* is the normalization constant. In this case, the superpotential and its derivative are given by *W*(*x*) = 15/2 − 8 exp(−*x*) and *dW*/*dx* = 8 exp(−*x*), respectively. The minimum SUSY Heisenberg uncertainty product in Eq. (37) is equal to ∆*W*ˆ ∆*p*ˆ*<sup>x</sup>* = 15/4. In addition, the discretized system-specific coherent state basis functions in Eq. (44) are expressed by

$$\Psi\_{\mathfrak{K}\_l}(\mathbf{x}) = \mathrm{Ne}^{i\mathbf{k}\_l(\mathbf{x} - q\_l)} \exp\left[ -8e^{-(\mathbf{x} - q\_l)} - \frac{15}{2}(\mathbf{x} - q\_l) \right]. \tag{50}$$

The phase space grid in Eq. (45) used for the coherent states was *m* = −1, 0, 1 and *n* = −1, 0, 1 for *M* = 9 basis functions and *m* = −1, 0, 1 and *n* = −2, . . . , 2 for *M* = 15 basis functions. The phase space density was set to be *D* = 1. In contrast with the present system-specific coherent states in Eq. (50), different coherent states for the Morse oscillator defined as eigenstates of the charge operator and minimum uncertainty states have been constructed ([8]).

Table 1 presents the computational results for the energy eigenvalues obtained by solving the generalized eigen-equation in Eq. (47) using the discretized system-specific coherent state basis functions with ∆*x* = 0.5. Since the basis includes the exact ground state wave function, the computational result yields the exact ground state energy. As shown in this table, higher accuracy can be achieved when we increase the number of the basis functions from *M* = 9 to *M* = 15. In addition, Table 1 presents the computational results obtained using the harmonic oscillator coherent state basis and the standard harmonic oscillator basis. The discretized harmonic oscillator coherent state basis functions are readily determined by substituting *W*(*x*) = *x* into Eq. (44)

$$\psi\_{\mathfrak{a}\_l}(\mathbf{x}) = N e^{i\mathbf{k}\_l(\mathbf{x} - q\_l)} e^{-(\mathbf{x} - q\_l)^2 / 2}. \tag{51}$$

The standard harmonic oscillator basis is given by

**Figure 2.** The ground state wave function of the double well potential obtained by the imaginary time propagation method is shown with the scaled potential *Vs*(*x*) = *V*(*x*)/20.

$$\phi\_{\rm n}(\mathbf{x}) = \frac{1}{\sqrt{2^{\rm n} n! \sqrt{\pi}}} H\_{\rm n}(\mathbf{x}) e^{-\mathbf{x}^2/2},\tag{52}$$

where *Hn*(*x*) is the Hermite polynomial. Compared with the results obtained from these two basis sets, the computational results from the system-specific coherent states achieve significantly higher accuracy using a small number of basis functions. Thus, the system-specific coherent states provide more accurate approximations of the excited state energies for the Morse oscillator.

Figure 1 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different values for ∆*x* with 15 basis functions. The relative error is defined by

$$Err = \frac{E\_{numerical} - E\_{exact}}{|E\_{exact}|}. \tag{53}$$

As shown in this figure, the system-specific coherent states yield excellent results for the excited state energies. The relative error of the first-excited state energy can even reach 10−<sup>9</sup> for a wide range of ∆*x*. Additionally, compared with the harmonic oscillator coherent states, the system-specific coherent states give much more accurate results for the first three excited state energies. Also, the system-specific coherent states yield stable computational results for a wide range of ∆*x*.

#### **3.2. Double well potential**

12 Quantum Mechanics

**3. Computational results**

The Hamiltonian of the Morse oscillator is given by

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

expression of the ground state wave function is given by

In order to demonstrate features of system-specific coherent states, computational results will be presented for three quantum systems. The first of these concerns the Morse oscillator.

The exact energy eigenvalues are *En* = −(*n* − 15/2)<sup>2</sup> where *n* = 0, . . . , 7, and the analytical

 −8*e*

where *N* is the normalization constant. In this case, the superpotential and its derivative are given by *W*(*x*) = 15/2 − 8 exp(−*x*) and *dW*/*dx* = 8 exp(−*x*), respectively. The minimum SUSY Heisenberg uncertainty product in Eq. (37) is equal to ∆*W*ˆ ∆*p*ˆ*<sup>x</sup>* = 15/4. In addition, the discretized system-specific coherent state basis functions in Eq. (44) are expressed by

> −8*e*

The phase space grid in Eq. (45) used for the coherent states was *m* = −1, 0, 1 and *n* = −1, 0, 1 for *M* = 9 basis functions and *m* = −1, 0, 1 and *n* = −2, . . . , 2 for *M* = 15 basis functions. The phase space density was set to be *D* = 1. In contrast with the present system-specific coherent states in Eq. (50), different coherent states for the Morse oscillator defined as eigenstates of

Table 1 presents the computational results for the energy eigenvalues obtained by solving the generalized eigen-equation in Eq. (47) using the discretized system-specific coherent state basis functions with ∆*x* = 0.5. Since the basis includes the exact ground state wave function, the computational result yields the exact ground state energy. As shown in this table, higher accuracy can be achieved when we increase the number of the basis functions from *M* = 9 to *M* = 15. In addition, Table 1 presents the computational results obtained using the harmonic oscillator coherent state basis and the standard harmonic oscillator basis. The discretized harmonic oscillator coherent state basis functions are readily determined by substituting

the charge operator and minimum uncertainty states have been constructed ([8]).

*ψαi*(*x*) = *Neiki*(*x*−*qi*)

The standard harmonic oscillator basis is given by

*e*

*dx*<sup>2</sup> <sup>+</sup> <sup>64</sup>

 *e* <sup>−</sup>2*<sup>x</sup>* <sup>−</sup> <sup>2</sup>*<sup>e</sup>*

<sup>−</sup>*<sup>x</sup>* − 15 2 *x* 

<sup>−</sup>(*x*−*qi*) − 15

<sup>2</sup> (*<sup>x</sup>* <sup>−</sup> *qi*)

<sup>−</sup>(*x*−*qi*)2/2. (51)

−*x* 

. (48)

. (50)

, (49)

*dx*<sup>2</sup> <sup>+</sup> *<sup>V</sup>*(*x*) = <sup>−</sup> *<sup>d</sup>*<sup>2</sup>

*ψ*0(*x*) = *N* exp

*ψαi*(*x*) = *Neiki*(*x*−*qi*) exp

**3.1. Morse oscillator**

*W*(*x*) = *x* into Eq. (44)

As an example of quantum systems without exact analytical solutions, we consider a symmetric double well potential given by

$$V(\mathbf{x}) = 3\mathbf{x}^4 - 8\mathbf{x}^2. \tag{54}$$


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*E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

**DVR 0.000000 4.751807 6.646349 8.679575** SSCS (*M* = 81) 0 4.754974 6.647358 8.684308 SSCS (*M* = 225) 0 4.751812 6.646353 8.679596 HOCS (*M* = 81) 0.0762 5.3029 6.9378 10.4334 HOCS (*M* = 225) 0.0029 4.7915 6.6554 8.8479 HO (*M* = 81) 0.0870 5.3587 7.0307 10.5626 HO (*M* = 225) 0.0144 4.8953 6.6967 9.2190

**Table 3.** Comparison of the energy eigenvalues for the two-dimensional anharmonic oscillator system obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Grid spacing

**Figure 4.** Logarithm of the relative error, versus grid spacing ∆*x* = ∆*y*, of the first (◦), second (⋄), and third () excited state energies for the two-dimensional anharmonic oscillator system using the system-specific coherent states (—) and the harmonic

using 1000 grid points on the computational domain extending from *x* = −4 to *x* = 4 ([23]). Table 2 presents the computational results for the energy eigenvalues with ∆*x* = 0.5. Again, computational results for the first three excited state energies with significantly higher accuracy were achieved using a small number of the basis functions with *M* = 15. In addition, compared with the harmonic oscillator coherent state basis and the standard harmonic oscillator basis, the system-specific coherent states yields more accurate excited state energies for the double well potential. Moreover, Fig. 2 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different values for ∆*x* with 15 basis functions. As shown in this figure, the system-specific coherent states generally yield much more accurate results for the excited state energies than the harmonic oscillator coherent states except for

functions (HO) with the discrete variable representation (DVR) results.

−6

−5

−4

Log10(Err)

oscillator coherent states (- - -) with 225 basis functions.

small ∆*x*.

−3

−2

−1

**Table 2.** Comparison of the energy eigenvalues for the double well potential obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis functions (HO) with the discrete variable representation (DVR) results.

**Figure 3.** Logarithm of the relative error, versus grid spacing ∆*x*, of the first (◦), second (⋄), and third () excited state energies for the double well potential using the system-specific coherent states (—) and the harmonic oscillator coherent states (- - -) with 15 basis functions.

In order to construct the discretized system-specific coherent state basis functions in Eq. (44), we numerically obtain the ground state wave function. We employed the split-operator method ([35]) to integrate the imaginary time Schrödinger equation from *t* = 0 to *t* = 2 ([36]). The computational grid extends from *x* = −8 to *x* = 8 with 2<sup>13</sup> grid points, and the integration time step was ∆*t* = 0.01. The initial state is a Gaussian wave packet given by

$$
\psi(x) = \left(\frac{2}{\pi}\right)^{1/4} e^{-x^2} \,, \tag{55}
$$

where the wave packet is centered at the origin. Figure 2 presents the resulting ground state wave function of the double well potential with the ground state energy *<sup>E</sup>*<sup>0</sup> = −2.169694.

From the computational result for the ground state, we can construct the approximate discretized system-specific coherent states in Eq. (44) used to determine the excited state energies of the double well potential by solving the generalized eigen-equation in Eq. (47). In order to assess the accuracy of the computational results, accurate results were obtained with a Chebyshev polynomial discrete variable representation (DVR) variational calculation

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14 Quantum Mechanics

variable representation (DVR) results.

with 15 basis functions.

−5

−4

−3

−2

Log10(Err)

−1

0

*E*<sup>0</sup> *E*<sup>1</sup> *E*<sup>2</sup> *E*<sup>3</sup>

**DVR -2.169693 -1.406472 3.102406 7.087930** SSCS (*M* = 9) -2.169697 -1.375254 3.106359 7.807534 SSCS (*M* = 15) -2.169697 -1.406417 3.102440 7.088186 HOCS (*M* = 9) -2.1223 -1.3214 3.3931 7.5166 HOCS (*M* = 15) -2.1688 -1.4048 3.1088 7.0992 HO (*M* = 9) -2.1246 -1.0650 3.5063 8.6640 HO (*M* = 15) -2.1543 -1.3930 3.1555 7.4491

**Table 2.** Comparison of the energy eigenvalues for the double well potential obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis functions (HO) with the discrete

0.4 0.6 0.8 1 1.2

∆ x

**Figure 3.** Logarithm of the relative error, versus grid spacing ∆*x*, of the first (◦), second (⋄), and third () excited state energies for the double well potential using the system-specific coherent states (—) and the harmonic oscillator coherent states (- - -)

In order to construct the discretized system-specific coherent state basis functions in Eq. (44), we numerically obtain the ground state wave function. We employed the split-operator method ([35]) to integrate the imaginary time Schrödinger equation from *t* = 0 to *t* = 2 ([36]). The computational grid extends from *x* = −8 to *x* = 8 with 2<sup>13</sup> grid points, and the integration time step was ∆*t* = 0.01. The initial state is a Gaussian wave packet given by

> 2 *π*

where the wave packet is centered at the origin. Figure 2 presents the resulting ground state wave function of the double well potential with the ground state energy *<sup>E</sup>*<sup>0</sup> = −2.169694.

From the computational result for the ground state, we can construct the approximate discretized system-specific coherent states in Eq. (44) used to determine the excited state energies of the double well potential by solving the generalized eigen-equation in Eq. (47). In order to assess the accuracy of the computational results, accurate results were obtained with a Chebyshev polynomial discrete variable representation (DVR) variational calculation

1/4 *e* −*x*<sup>2</sup>

, (55)

*ψ*(*x*) =

**Table 3.** Comparison of the energy eigenvalues for the two-dimensional anharmonic oscillator system obtained by the system-specific coherent states (SSCS), the harmonic oscillator coherent states (HOCS), and the harmonic oscillator basis functions (HO) with the discrete variable representation (DVR) results.

**Figure 4.** Logarithm of the relative error, versus grid spacing ∆*x* = ∆*y*, of the first (◦), second (⋄), and third () excited state energies for the two-dimensional anharmonic oscillator system using the system-specific coherent states (—) and the harmonic oscillator coherent states (- - -) with 225 basis functions.

using 1000 grid points on the computational domain extending from *x* = −4 to *x* = 4 ([23]). Table 2 presents the computational results for the energy eigenvalues with ∆*x* = 0.5. Again, computational results for the first three excited state energies with significantly higher accuracy were achieved using a small number of the basis functions with *M* = 15. In addition, compared with the harmonic oscillator coherent state basis and the standard harmonic oscillator basis, the system-specific coherent states yields more accurate excited state energies for the double well potential. Moreover, Fig. 2 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different values for ∆*x* with 15 basis functions. As shown in this figure, the system-specific coherent states generally yield much more accurate results for the excited state energies than the harmonic oscillator coherent states except for small ∆*x*.

#### **3.3. Two-dimensional anharmonic oscillator system**

As an example of multidimensional systems, we consider a nonseparable nondegenerate two-dimensional anharmonic oscillator system ([19]). The Hamiltonian is given by

$$\begin{split} H\_1 &= -\nabla^2 + V(\mathbf{x}, \mathbf{y}) \\ &= -\frac{\partial^2}{\partial \mathbf{x}^2} - \frac{\partial^2}{\partial y^2} + \left( 4xy^2 + 2\mathbf{x} \right)^2 \\ &+ \left( 4x^2y + 2\sqrt{2}y \right)^2 - 4\left( x^2 + y^2 \right) - \left( 2 + 2\sqrt{2} \right) . \end{split} \tag{56}$$

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basis of the direct product of the eigenstates of a harmonic oscillator in each dimension

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

calculation with 81 basis functions and a (*Nx*, *Ny*)=(15, 15) basis set calculation with 225 basis functions. Again, compared with the results obtained from the other two basis sets, the computational results from the system-specific coherent states achieve significantly higher accuracy using a small number of basis functions. Furthermore, Figure 4 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different values for the grid spacing with 225 basis functions. As shown in this figure, the system-specific coherent states yield much more accurate results for the excited state energies than the harmonic oscillator coherent states for different grid spacings, and the relative errors reach around 10−<sup>6</sup> for

The application of SUSY-QM to non-relativistic quantum systems generalizes the powerful ladder operator approach used in the treatment of the harmonic oscillator. The lowering operator of the harmonic oscillator annihilates the ground state, while the charge operator annihilates the ground state of the corresponding ground state for other quantum systems. The similarity between the lowering operator of the harmonic oscillator and the SUSY charge operator implies that the superpotential can be regarded as a system-specific generalized displacement variable. Analogous to the ground state of the harmonic oscillator which minimizes the Heisenberg uncertainty product, the ground state of any bound quantum system was identified as the minimizer of the SUSY Heisenberg uncertainty product. Then, system-specific coherent states were constructed by applying shift operators to the ground state of the system, which serves as a fiducial function. In addition, we employed the discretized system-specific coherent states as a dynamically-adapted basis set to determine the excited state energies and wave functions for the Morse oscillator, the double well potential, and the two-dimensional anharmonic oscillator system. Variational calculations in terms of the discretized system-specific coherent states demonstrated that these dynamically-adapted coherent states yield significantly more accurate excited state energies and wave functions than were obtained with the same number of the conventional

As presented in the current study, the ladder operator approach of the harmonic oscillator and the SUSY-QM formulation share strong similarity. This observation suggests that the connection of the SUSY-QM with the Heisenberg minimum uncertainty (*µ*−) wavelets should be explored ([13, 14, 21, 22]). The SUSY-displacement with the SUSY Heisenberg uncertainty product can lead to the construction of the SUSY minimum uncertainty wavelets and the SUSY distributed approximating functionals. These new functions and their potential applications in mathematics and physics are currently under investigation. In addition, this study presents a practical computational approach for discretized system-specific coherent states in calculations of excited states. The issue of completeness of discretized system-specific coherent states should be examined. These relevant studies will be reported

2. These results were obtained by a (*Nx*, *Ny*)=(9, 9) basis set

with frequency *ω* = 2

a wide range of the grid spacings.

**4. Discussion and perspectives**

coherent states and the standard harmonic oscillator basis.

elsewhere in the future.

√

The exact ground state energy of the system is zero and the analytical expression of the ground state wave function is given by

$$
\psi\_0^{(1)}(x,y) = N \exp\left(-2x^2y^2 - x^2 - \sqrt{2}y^2\right),
\tag{57}
$$

where *N* is a normalization constant. Analogous to the one-dimensional case, the discretized system-specific coherent state basis functions are expressed by

$$
\psi\_{\mathfrak{a}\_i}(\mathbf{x}, y) = N e^{i \mathbf{k}\_{\rm ii} (\mathbf{x} - q\_{\rm ii})} e^{i \mathbf{k}\_{\rm yi} (y - q\_{\rm yi})} \psi\_0(\mathbf{x} - q\_{\rm xi}, y - q\_{\rm yi}). \tag{58}
$$

In addition, the two-dimensional separable discretized harmonic oscillator coherent state basis functions are given by

$$\psi\_{\mathfrak{A}\_i}(\mathbf{x}, \mathbf{y}) = N e^{i \mathbf{k}\_{\rm it}(\mathbf{x} - q\_{\rm it})} e^{i \mathbf{k}\_{\rm yi}(y - q\_{\rm yi})} e^{-(\mathbf{x} - q\_{\rm ii})^2 / 2} e^{-(y - q\_{\rm yi})^2 / 2}. \tag{59}$$

The phase space grid points for these two basis sets are defined by

$$\{\{q\_{\rm xi}, q\_{\rm ji}, k\_{\rm xi}, k\_{\rm ji}\}\} = \left\{ \left( m \Delta x \sqrt{\frac{2\pi}{D}}, m \Delta y \sqrt{\frac{2\pi}{D}}, \frac{n}{\Delta x} \sqrt{\frac{2\pi}{D}}, \frac{n}{\Delta y} \sqrt{\frac{2\pi}{D}} \right) \right\} \quad m, n \in \mathbb{Z} \tag{60}$$

where *m* and *n* are integers. For computational results, we chose *m* = −1, 0, 1 and *n* = −1, 0, 1 for *M* = 81 basis functions and *m* = −1, 0, 1 and *n* = −2, . . . , 2 for *M* = 225 basis functions. The phase space density was set to be *D* = 1.

Table 3 presents the computational results for the energy eigenvalues obtained using the discretized system-specific coherent states and the harmonic oscillator coherent states with ∆*x* = ∆*y* = 0.4. Compared with the DVR results using 50 grid points in *x* and in *y* (for a total of 2500 basis functions), the computational results obtained by the system-specific coherent states achieve higher accuracy than the harmonic oscillator coherent states. In addition, Table 3 presents the computational results obtained from the standard harmonic oscillator basis of the direct product of the eigenstates of a harmonic oscillator in each dimension with frequency *ω* = 2 √2. These results were obtained by a (*Nx*, *Ny*)=(9, 9) basis set calculation with 81 basis functions and a (*Nx*, *Ny*)=(15, 15) basis set calculation with 225 basis functions. Again, compared with the results obtained from the other two basis sets, the computational results from the system-specific coherent states achieve significantly higher accuracy using a small number of basis functions. Furthermore, Figure 4 displays the logarithm of the relative error of the excited state energies for the system-specific coherent states and the harmonic oscillator coherent states with different values for the grid spacing with 225 basis functions. As shown in this figure, the system-specific coherent states yield much more accurate results for the excited state energies than the harmonic oscillator coherent states for different grid spacings, and the relative errors reach around 10−<sup>6</sup> for a wide range of the grid spacings.

### **4. Discussion and perspectives**

16 Quantum Mechanics

**3.3. Two-dimensional anharmonic oscillator system**

*<sup>H</sup>*<sup>1</sup> = −∇<sup>2</sup> + *<sup>V</sup>*(*x*, *<sup>y</sup>*)

*<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

4*x*2*y* + 2

<sup>0</sup> (*x*, *y*) = *N* exp

system-specific coherent state basis functions are expressed by

*ψαi*(*x*, *<sup>y</sup>*) = *Neikxi*(*x*−*qxi*)

*ψαi*(*x*, *<sup>y</sup>*) = *Neikxi*(*x*−*qxi*)

 *m*∆*x*

The phase space density was set to be *D* = 1.

The phase space grid points for these two basis sets are defined by

2*π*

*<sup>D</sup>* , *<sup>m</sup>*∆*<sup>y</sup>*

*<sup>∂</sup>y*<sup>2</sup> <sup>+</sup> 

√ 2*y* 2 − 4 *x*<sup>2</sup> + *y*<sup>2</sup> − 2 + 2 √ 2 

<sup>=</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup>

+ 

*<sup>ψ</sup>*(1)

ground state wave function is given by

basis functions are given by

{(*qxi*, *qyi*, *kxi*, *kyi*)} =

As an example of multidimensional systems, we consider a nonseparable nondegenerate

4*xy*<sup>2</sup> + 2*x*

The exact ground state energy of the system is zero and the analytical expression of the

where *N* is a normalization constant. Analogous to the one-dimensional case, the discretized

In addition, the two-dimensional separable discretized harmonic oscillator coherent state

*ikyi*(*y*−*qyi*)

2*π <sup>D</sup>* , *<sup>n</sup>* ∆*x*

where *m* and *n* are integers. For computational results, we chose *m* = −1, 0, 1 and *n* = −1, 0, 1 for *M* = 81 basis functions and *m* = −1, 0, 1 and *n* = −2, . . . , 2 for *M* = 225 basis functions.

Table 3 presents the computational results for the energy eigenvalues obtained using the discretized system-specific coherent states and the harmonic oscillator coherent states with ∆*x* = ∆*y* = 0.4. Compared with the DVR results using 50 grid points in *x* and in *y* (for a total of 2500 basis functions), the computational results obtained by the system-specific coherent states achieve higher accuracy than the harmonic oscillator coherent states. In addition, Table 3 presents the computational results obtained from the standard harmonic oscillator

*e*

<sup>−</sup>(*x*−*qxi*)2/2*<sup>e</sup>*

2*π <sup>D</sup>* , *<sup>n</sup>* ∆*y* 2*π D*

*ikyi*(*y*−*qyi*)

*e*

*e*

2

<sup>−</sup>2*x*2*y*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> <sup>√</sup>

2*y*<sup>2</sup>  . (56)

, (57)

*<sup>ψ</sup>*0(*<sup>x</sup>* − *qxi*, *<sup>y</sup>* − *qyi*). (58)

<sup>−</sup>(*y*−*qyi*)2/2. (59)

*m*, *n* ∈ Z (60)

two-dimensional anharmonic oscillator system ([19]). The Hamiltonian is given by

The application of SUSY-QM to non-relativistic quantum systems generalizes the powerful ladder operator approach used in the treatment of the harmonic oscillator. The lowering operator of the harmonic oscillator annihilates the ground state, while the charge operator annihilates the ground state of the corresponding ground state for other quantum systems. The similarity between the lowering operator of the harmonic oscillator and the SUSY charge operator implies that the superpotential can be regarded as a system-specific generalized displacement variable. Analogous to the ground state of the harmonic oscillator which minimizes the Heisenberg uncertainty product, the ground state of any bound quantum system was identified as the minimizer of the SUSY Heisenberg uncertainty product. Then, system-specific coherent states were constructed by applying shift operators to the ground state of the system, which serves as a fiducial function. In addition, we employed the discretized system-specific coherent states as a dynamically-adapted basis set to determine the excited state energies and wave functions for the Morse oscillator, the double well potential, and the two-dimensional anharmonic oscillator system. Variational calculations in terms of the discretized system-specific coherent states demonstrated that these dynamically-adapted coherent states yield significantly more accurate excited state energies and wave functions than were obtained with the same number of the conventional coherent states and the standard harmonic oscillator basis.

As presented in the current study, the ladder operator approach of the harmonic oscillator and the SUSY-QM formulation share strong similarity. This observation suggests that the connection of the SUSY-QM with the Heisenberg minimum uncertainty (*µ*−) wavelets should be explored ([13, 14, 21, 22]). The SUSY-displacement with the SUSY Heisenberg uncertainty product can lead to the construction of the SUSY minimum uncertainty wavelets and the SUSY distributed approximating functionals. These new functions and their potential applications in mathematics and physics are currently under investigation. In addition, this study presents a practical computational approach for discretized system-specific coherent states in calculations of excited states. The issue of completeness of discretized system-specific coherent states should be examined. These relevant studies will be reported elsewhere in the future.

### **Author details**

Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri

University of Houston, USA

### **References**

[1] Aleixo, A. N. F. & Balantekin, A. B. [2004]. An algebraic construction of generalized coherent states for shape-invariant potentials, *J. Phys. A: Math. Gen.* 37: 8513.

10.5772/54010

517

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[15] Infeld, L. & Hull, T. E. [1951]. The factorization method, *Rev. Mod. Phys.* 23: 21–68.

[16] Jafarpour, M. & Afshar, D. [2002]. Calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential, *J. Phys. A: Math. Gen.* 35: 87.

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

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[19] Kouri, D. J., Maji, K., Markovich, T. & Bittner, E. R. [2010]. New generalization of supersymmetric quantum mechanics to arbitrary dimensionality or number of

[20] Kouri, D. J., Markovich, T., Maxwell, N. & Bittner, E. R. [2009]. Supersymmetric quantum mechanics, excited state energies and wave functions, and the rayleigh–ritz variational principle: A proof of principle study, *J. Phys. Chem. A* 113: 15257–15264.

[21] Kouri, D. J., Papadakis, M., Kakadiaris, I. & Hoffman, D. K. [2003]. Properties of minimum uncertainty wavelets and their relations to the harmonic oscillator and the

[22] Lee, Y., Kouri, D. & Hoffman, D. [2011]. Minimum uncertainty wavelets in non-relativistic super-symmetric quantum mechanics, *J. Math. Chem.* 49: 12–34.

[23] Light, J. C., Hamilton, I. P. & Lill, J. V. [1985]. Generalized discrete variable

[24] Liu, Y., Lei, Y. & Zeng, J. [1997]. Factorization of the radial schrödinger equation and four kinds of raising and lowering operators of hydrogen atoms and isotropic harmonic

[26] Mielnik, B. [1984]. Factorization method and new potentials with the oscillator

[27] Nieto, M. M. & Simmons, L. M. [1978]. Coherent states for general potentials, *Phys. Rev.*

[28] Nieto, M. M. & Simmons, L. M. [1979a]. Coherent states for general potentials. i.

[29] Nieto, M. M. & Simmons, L. M. [1979b]. Coherent states for general potentials. ii.

[30] Nieto, M. M. & Simmons, L. M. [1979c]. Coherent states for general potentials. iii.

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[25] Merzbacher, E. [1997]. *Quantum Mechanics*, Wiley, New York.

oscillators, *Phys. Lett. A* 231: 9–22.

spectrum, *J. Math. Phys.* 25: 3387–3389.

formalism, *Phys. Rev. D* 20: 1321–1331.

*Lett.* 41: 207–210.

41: 2301–2305.


[15] Infeld, L. & Hull, T. E. [1951]. The factorization method, *Rev. Mod. Phys.* 23: 21–68.

18 Quantum Mechanics

**References**

**Author details**

Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri

[1] Aleixo, A. N. F. & Balantekin, A. B. [2004]. An algebraic construction of generalized coherent states for shape-invariant potentials, *J. Phys. A: Math. Gen.* 37: 8513.

[2] Andersson, L. M. [2001]. Quantum dynamics using a discretized coherent state representation: An adaptive phase space method, *J. Chem. Phys.* 115(3): 1158–1165.

[3] Angelova, M. & Hussin, V. [2008]. Generalized and gaussian coherent states for the

[4] Benedict, M. G. & Molnár, B. [1999]. Algebraic construction of the coherent states of the morse potential based on supersymmetric quantum mechanics, *Phys. Rev. A*

[5] Bittner, E. R., Maddox, J. B. & Kouri, D. J. [2009]. Supersymmetric approach to excited

[6] Chou, C.-C., Biamonte, M. T., Bodmann, B. G. & Kouri, D. J. [2012]. New system-specific

[7] Cooper, F., Khare, A. & Sukhatme, U. [2002]. *Supersymmetry in Quantum Mechanics*,

[8] Cooper, I. L. [1992]. A simple algebraic approach to coherent states for the morse

[9] Fernández C, D. J., Hussin, V. & Nieto, L. M. [1994]. Coherent states for isospectral

[10] Fukui, T. & Aizawa, N. [1993]. Shape-invariant potentials and an associated coherent

[11] Gangopadhyaya, A., Mallow, J. V. & Rasinariu, C. [2010]. *Supersymmetric Quantum*

[12] Gerry, C. C. [1986]. Coherent states and a path integral for the morse oscillator, *Phys.*

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[14] Hoffman, D. K. & Kouri, D. J. [2002]. Hierarchy of local minimum solutions of

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[31] Perelomov, A. [1986]. *Generalized Coherent States and Their Applications*, Springer, New York.

**Section 6**

**Quantization and Entanglement**


**Quantization and Entanglement**

20 Quantum Mechanics

518 Advances in Quantum Mechanics

York.

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[31] Perelomov, A. [1986]. *Generalized Coherent States and Their Applications*, Springer, New

[32] Schrödinger, E. [1941a]. Factorization of the hypergeometric equation, *Prod.Roy. Irish*

[33] Schrödinger, E. [1941b]. Further studies on solving eigenvalue problems by

[34] Shreecharan, T., Panigrahi, P. K. & Banerji, J. [2004]. Coherent states for exactly solvable

[35] Tannor, D. J. [2007]. *Introduction to Quantum Mechanics: A Time-Dependent Perspective*,

[36] Varga, K. & Driscoll, J. A. [2011]. *Computational Nanoscience: Applications for Molecules,*

[37] Zhang, W.-M., Feng, D. H. & Gilmore, R. [1990]. Coherent states: Theory and some

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potentials, *Phys. Rev. A* 69: 012102.

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applications, *Rev. Mod. Phys.* 62: 867–927.

**Chapter 22**

**Provisional chapter**

**Quantum Dating Market**

C. M. Arizmendi and O. G. Zabaleta

**Quantum Dating Market**

C. M. Arizmendi and O. G. Zabaleta

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53842

of quantum games models.

**1. Introduction**

Additional information is available at the end of the chapter

Quantum algorithms have proven to be faster than the fastest known classical algorithms. Clearly, such a superiority means counting on a real quantum computer. Although this essential constraint elimination is in development process, many people is working on that and interesting advances are being made [1–3]. Meanwhile, new algorithms and applications of the existing ones are current research topics [4, 5]. One of the main goals of quantum computing is the application of quantum techniques to classical troubleshooting: the Shor algorithm [6], for example, is a purely quantum-mechanical algorithm which comes to solve the classical factoring problem, also the contribution of Lov Grover [7, 8] to speed up the search for items in an N-item database is very important. Both mathematical finds are the cornerstones of quantum computation, so, considerable amount of work on diverse subjects make use of them. Other algorithms which has been very important for quantum computing progress are Simon's and Deutsch-Jozsa's. Through the quantum games, Meyer in [9] and Eisert in [10], among other, showed that quantum techniques are generalizations of classical probability theory, allowing effects which are impossible in a classical setting. These and many other examples, show that there is no contradiction in using quantum techniques to describe non-quantum mechanical problems and solve hard to solve problems with classical tools. Adding, decision theory and game theory, two examples where probabilities theory is applied, deal with decisions made under uncertain conditions by real humans. Basically, the former considers only one agent and her decisions meanwhile the other considers also the conflicts that two or more players cause to each other through the decisions they take. Due to their inherent complexity this kind of problems results convenient to be analyzed by mean

Widely observed phenomena of non-commutativity in patterns of behavior exhibited in experiments on human decisions and choices cannot be obtained with classical decision theory [11] but can be adequately described by putting quantum mechanics and decision theory together. Quantum mechanics and decision theory have been recently combined

> ©2012 Arizmendi and Zabaleta, 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 Arizmendi and Zabaleta; 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 Arizmendi and Zabaleta; 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.

### **Chapter 22**

**Provisional chapter**

## **Quantum Dating Market**

C. M. Arizmendi and O. G. Zabaleta

**Quantum Dating Market**

C. M. Arizmendi and O. G. Zabaleta Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53842

### **1. Introduction**

Quantum algorithms have proven to be faster than the fastest known classical algorithms. Clearly, such a superiority means counting on a real quantum computer. Although this essential constraint elimination is in development process, many people is working on that and interesting advances are being made [1–3]. Meanwhile, new algorithms and applications of the existing ones are current research topics [4, 5]. One of the main goals of quantum computing is the application of quantum techniques to classical troubleshooting: the Shor algorithm [6], for example, is a purely quantum-mechanical algorithm which comes to solve the classical factoring problem, also the contribution of Lov Grover [7, 8] to speed up the search for items in an N-item database is very important. Both mathematical finds are the cornerstones of quantum computation, so, considerable amount of work on diverse subjects make use of them. Other algorithms which has been very important for quantum computing progress are Simon's and Deutsch-Jozsa's. Through the quantum games, Meyer in [9] and Eisert in [10], among other, showed that quantum techniques are generalizations of classical probability theory, allowing effects which are impossible in a classical setting. These and many other examples, show that there is no contradiction in using quantum techniques to describe non-quantum mechanical problems and solve hard to solve problems with classical tools. Adding, decision theory and game theory, two examples where probabilities theory is applied, deal with decisions made under uncertain conditions by real humans. Basically, the former considers only one agent and her decisions meanwhile the other considers also the conflicts that two or more players cause to each other through the decisions they take. Due to their inherent complexity this kind of problems results convenient to be analyzed by mean of quantum games models.

Widely observed phenomena of non-commutativity in patterns of behavior exhibited in experiments on human decisions and choices cannot be obtained with classical decision theory [11] but can be adequately described by putting quantum mechanics and decision theory together. Quantum mechanics and decision theory have been recently combined

©2012 Arizmendi and Zabaleta, 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 Arizmendi and Zabaleta; 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 Arizmendi and Zabaleta; 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.

[11–13] to take into account the indeterminacy of preferences that are determined only when the action takes place. An agent is described by a state that is a superposition of potential preferences to be projected onto one of the possible behaviors at the time of the interaction. In addition to the main goal of modeling uncertainty of preferences that is not due to lack of information, this formalism seems to be adequate to describe widely observed phenomena of non-commutativity in patterns of behavior.

decisions. So, as driven by a no local force, people may make an inconvenient choice in the heat of a competence. In order to formulate in a mathematical way this sort of problem we remodel the quantum dating between men and women with the inclusion of quantum

Quantum Dating Market http://dx.doi.org/10.5772/53842 523

The chapter organization is as follows: First of all, to ease game theory unfamiliar readers comprehension a brief introduction to game basics is presented. In the course of the next sections the quantum dating game is particularly studied. In section 3 the Grover quantum search algorithm as a playing strategy is analyzed. In section 5, the system stability is under study. Finally, section 6 explores entangled strategies performance. At the end of each section the results and the consequent section discussions are set. The chapter ends with a

Game theory [25] is a collection of models (games) designed to study competing agents (players) decisions in some conflict situation. It tries to understand the birth and development of conflicting or cooperative behaviors among a group of individuals who behave rationally and strategically according to their personal interests. Although the theory was conceived in order to analyze and solve social and economy problems, existing applications go beyond [26]. Furthermore, the models reach not only individuals but also governments conflicts,

Before starting to explain quantum games basics, the classic games notation is presented. The game can be set in strategic (or normal) form or in extensive form, in any of them it has three elements: a set of players *i* ∈ J which is taken to be a finite set 0, ..., *N* − 1, the set of pure strategies *Si* = {*s*0,*s*1, ...,*sN*<sup>−</sup>1}, *<sup>i</sup>* = 0, ..., *<sup>N</sup>* − 1 which is the set of all strategies available to the player, and the payoffs function *ui*(*s*0,*s*1, ...,*sN*<sup>−</sup>1), *<sup>i</sup>* = 0, ..., *<sup>N</sup>* − 1, where *si* ∈ *Si*. In the strategic form, the game can be denoted by *<sup>G</sup>*(*N*, *<sup>S</sup>*, *<sup>u</sup>*), where *<sup>S</sup>* = *<sup>S</sup>*<sup>0</sup> × *<sup>S</sup>*<sup>1</sup> × ... × *SN*<sup>−</sup><sup>1</sup> and *<sup>u</sup>* = *<sup>u</sup>*<sup>0</sup> × *<sup>u</sup>*<sup>1</sup> × ... × *uN*−1. Extensive form representation is useful when it is wanted to include not only who makes the move but also when the move is made. Players apply pure strategies when they are certain of what they want, but such condition is not always possible, so mixed strategies must be considered. A mixed strategy is a probability distribution over

As an example, we can mention the well-known *Prisoners Dilemma (PD)* : Two suspected of committing a crime are caught by the police. As there is insufficient evidence to condemn them, the police place the suspects into separate rooms to convince them to confess. If one of the prisoners confesses, and help the police to condene his partner, he gains his freedom and the other prisoner must serve of 10 years. But if both confess, they must serve a sentence of 3 years. In other case, if both refuse to confess, they both will be convicted of a lesser charge and will have to serve a sentence of only one year in prison. In summary, they can choose between two possible strategies "Confess" (C) or "Not Confess" (N). However, observe that the luck of each player depends both on his election as that of the other. As consequence, confessing is a dominant strategy because regardless the other player decision the one who chooses it avoid the worst conviction. The prisoners know that if neither confesses they must serve a minimum sentence. However, as no one knows the other strategy to do not confess is very risky, specially because camaraderie is not a common quality between criminals. It

institutions trades or smart machines (phones, computers) access management.

*S* which corresponds to how frequently each move is chosen.

entanglement between men decision states.

final conclusion.

**2. Quantum games**

Within this framework, we study the dating market decision problem that takes into account progressive mutual learning [14, 15]. This problem is a variation on the Stable Marriage Problem introduced by Gale and Shapley almost four decades ago [16], that has been recently reformulated in a partial information approach [17, 18]. Specifically, perfect information supposition is very far from being a good approximation for the dating market, in which men and women repeatedly go out on dates and learn about each other.

The dating market problem may be included in a more general category of matching problems where the elements of two sets have to be matched by pairs. Matching problems have broad implications in economic and social contexts [19, 20]. As possible applications one could think of job seekers and employers, lodgers and landlords, men and women who want to date, or solitary ciliates *courtship rituals* [21]. In our model players earn an uncertain payoff from being matched with a particular person on the other side of the market in each time period. Players have a list of preferred partners on the other set. Quantum exploration of partners is compared with classical exploration at the dating set. Nevertheless dating is not just finding, but also being accepted by the partner. The preferences of the chosen partner are important in quantum and classic performances.

Recently [22], we introduced a quantum formulation for decision matching problems, specifically for the dating game that takes into account mutual progressive learning. This learning is accomplished by representing women with quantum states whose associated amplitudes must be modified by men's selection strategies, in order to increase a particular state amplitude and to decrease the others, with the final purpose to achieve the best possible choice when the game finishes. Grover quantum search algorithm is used as a playing strategy. Within the same quantum formulation already used in [22], we will concentrate first on the information associated to the dating market problem. Since we deal with mixed strategies, the density matrix formalism is used to describe the system. There exists a strong relationship between game theories, statistical mechanics and information theories. The bonds between these theories are the density operator and entropy. From the density operator we can construct and understand the statistical behavior about our system by using statistical mechanics. The dating problem is analyzed through information theory under a criterion of maximum or minimum entropy. Even though the decisions players make are based on their payoffs, past experiences, believes, etc., we are not interested in that causes but in the consequences of the decision they take, that is, the influence of the strategies they apply on the quantum system stability. In order to identify the conditions of stability we will use the equivalence between maximum entropy states and those states that obey the Collective Welfare Principle that says that a system is stable only if it maximizes the welfare of the collective above the welfare of the individual [23].

Interesting properties merge when entanglement is considered in quantum models of social decision problems [24]. People decisions are usually influenced by other people actions, opinions, or beliefs, to the extent that they may proceed in ways that they would rarely or never do if moved by their own benefit. Love, hate, envy, or a close friendship, which encase a bit of everything, are examples of relationships between people that may correlate their decisions. So, as driven by a no local force, people may make an inconvenient choice in the heat of a competence. In order to formulate in a mathematical way this sort of problem we remodel the quantum dating between men and women with the inclusion of quantum entanglement between men decision states.

The chapter organization is as follows: First of all, to ease game theory unfamiliar readers comprehension a brief introduction to game basics is presented. In the course of the next sections the quantum dating game is particularly studied. In section 3 the Grover quantum search algorithm as a playing strategy is analyzed. In section 5, the system stability is under study. Finally, section 6 explores entangled strategies performance. At the end of each section the results and the consequent section discussions are set. The chapter ends with a final conclusion.

### **2. Quantum games**

2 Quantum Mechanics

of non-commutativity in patterns of behavior.

are important in quantum and classic performances.

of the collective above the welfare of the individual [23].

[11–13] to take into account the indeterminacy of preferences that are determined only when the action takes place. An agent is described by a state that is a superposition of potential preferences to be projected onto one of the possible behaviors at the time of the interaction. In addition to the main goal of modeling uncertainty of preferences that is not due to lack of information, this formalism seems to be adequate to describe widely observed phenomena

Within this framework, we study the dating market decision problem that takes into account progressive mutual learning [14, 15]. This problem is a variation on the Stable Marriage Problem introduced by Gale and Shapley almost four decades ago [16], that has been recently reformulated in a partial information approach [17, 18]. Specifically, perfect information supposition is very far from being a good approximation for the dating market, in which

The dating market problem may be included in a more general category of matching problems where the elements of two sets have to be matched by pairs. Matching problems have broad implications in economic and social contexts [19, 20]. As possible applications one could think of job seekers and employers, lodgers and landlords, men and women who want to date, or solitary ciliates *courtship rituals* [21]. In our model players earn an uncertain payoff from being matched with a particular person on the other side of the market in each time period. Players have a list of preferred partners on the other set. Quantum exploration of partners is compared with classical exploration at the dating set. Nevertheless dating is not just finding, but also being accepted by the partner. The preferences of the chosen partner

Recently [22], we introduced a quantum formulation for decision matching problems, specifically for the dating game that takes into account mutual progressive learning. This learning is accomplished by representing women with quantum states whose associated amplitudes must be modified by men's selection strategies, in order to increase a particular state amplitude and to decrease the others, with the final purpose to achieve the best possible choice when the game finishes. Grover quantum search algorithm is used as a playing strategy. Within the same quantum formulation already used in [22], we will concentrate first on the information associated to the dating market problem. Since we deal with mixed strategies, the density matrix formalism is used to describe the system. There exists a strong relationship between game theories, statistical mechanics and information theories. The bonds between these theories are the density operator and entropy. From the density operator we can construct and understand the statistical behavior about our system by using statistical mechanics. The dating problem is analyzed through information theory under a criterion of maximum or minimum entropy. Even though the decisions players make are based on their payoffs, past experiences, believes, etc., we are not interested in that causes but in the consequences of the decision they take, that is, the influence of the strategies they apply on the quantum system stability. In order to identify the conditions of stability we will use the equivalence between maximum entropy states and those states that obey the Collective Welfare Principle that says that a system is stable only if it maximizes the welfare

Interesting properties merge when entanglement is considered in quantum models of social decision problems [24]. People decisions are usually influenced by other people actions, opinions, or beliefs, to the extent that they may proceed in ways that they would rarely or never do if moved by their own benefit. Love, hate, envy, or a close friendship, which encase a bit of everything, are examples of relationships between people that may correlate their

men and women repeatedly go out on dates and learn about each other.

Game theory [25] is a collection of models (games) designed to study competing agents (players) decisions in some conflict situation. It tries to understand the birth and development of conflicting or cooperative behaviors among a group of individuals who behave rationally and strategically according to their personal interests. Although the theory was conceived in order to analyze and solve social and economy problems, existing applications go beyond [26]. Furthermore, the models reach not only individuals but also governments conflicts, institutions trades or smart machines (phones, computers) access management.

Before starting to explain quantum games basics, the classic games notation is presented. The game can be set in strategic (or normal) form or in extensive form, in any of them it has three elements: a set of players *i* ∈ J which is taken to be a finite set 0, ..., *N* − 1, the set of pure strategies *Si* = {*s*0,*s*1, ...,*sN*<sup>−</sup>1}, *<sup>i</sup>* = 0, ..., *<sup>N</sup>* − 1 which is the set of all strategies available to the player, and the payoffs function *ui*(*s*0,*s*1, ...,*sN*<sup>−</sup>1), *<sup>i</sup>* = 0, ..., *<sup>N</sup>* − 1, where *si* ∈ *Si*. In the strategic form, the game can be denoted by *<sup>G</sup>*(*N*, *<sup>S</sup>*, *<sup>u</sup>*), where *<sup>S</sup>* = *<sup>S</sup>*<sup>0</sup> × *<sup>S</sup>*<sup>1</sup> × ... × *SN*<sup>−</sup><sup>1</sup> and *<sup>u</sup>* = *<sup>u</sup>*<sup>0</sup> × *<sup>u</sup>*<sup>1</sup> × ... × *uN*−1. Extensive form representation is useful when it is wanted to include not only who makes the move but also when the move is made. Players apply pure strategies when they are certain of what they want, but such condition is not always possible, so mixed strategies must be considered. A mixed strategy is a probability distribution over *S* which corresponds to how frequently each move is chosen.

As an example, we can mention the well-known *Prisoners Dilemma (PD)* : Two suspected of committing a crime are caught by the police. As there is insufficient evidence to condemn them, the police place the suspects into separate rooms to convince them to confess. If one of the prisoners confesses, and help the police to condene his partner, he gains his freedom and the other prisoner must serve of 10 years. But if both confess, they must serve a sentence of 3 years. In other case, if both refuse to confess, they both will be convicted of a lesser charge and will have to serve a sentence of only one year in prison. In summary, they can choose between two possible strategies "Confess" (C) or "Not Confess" (N). However, observe that the luck of each player depends both on his election as that of the other. As consequence, confessing is a dominant strategy because regardless the other player decision the one who chooses it avoid the worst conviction. The prisoners know that if neither confesses they must serve a minimum sentence. However, as no one knows the other strategy to do not confess is very risky, specially because camaraderie is not a common quality between criminals. It is very common to represent in a bimatrix the possible strategies combinations with their respective reward. The corresponding bimatrix for the prisoners game is 1.

**woman feature** |0� a |1� b |2� c |3� d

**Table 2.** Sample woman database. Left column contains women states and right column displays a letter representing some

chosen woman state is: |3� in this example. The procedure is very simple if the table has just a few rows, but when the database gets bigger, the table in the best case would have to be entered *Nw*/2 times [30]. Under this framework we propose to use Grover algorithm in order to achieve man's decision in less time. Without losing generality let *Nw* = 2*<sup>n</sup>* being *n* the qubits needed to code *Nw* women. Quantum states transformation are made by applying Hilbert space operators *U* to them, following Ψ<sup>1</sup> = *U*1Ψ<sup>0</sup> is a new system state starting from Ψ0. As a consequence any quantum algorithm can be thought as a set of suitable linear transformations. Grover algorithm starts with *n* qubits in |0�, resulting

product. Initially, the woman identified by state |0� is chosen with probability one. The next step is to create superposition states and like many other quantum algorithms Grover uses Hadamard transform to do this task since it maps *n* qubits initialized with |0� to a superposition of all *n* orthogonal states in the |0�, |1�,.. |*n* − 1� basis with equal weight,

*<sup>n</sup>*, see [27],

*<sup>H</sup>* <sup>=</sup> <sup>1</sup> √2  1 1 1 −1 

Another quantum search algorithms characteristic, is the "Oracle", which is basically a black box capable of marking the problem solution. We call *Uf* the operator which implement the

where *f*(*w*) is the oracle function which takes the value 1 if *w* correspond to the searched woman, *f*(*w*) = 1, and if it is not the case it takes the value 0, *f*(*w*) = 0. The value of *f*(*w*) on a superposition of every possible input *w* may be obtained [27]. The algorithm sets the

<sup>2</sup> ) �−→*Uf* (−1)*f*(*w*)

Observe that the second register is in an eigenstate, so we can ignore it, considering only the

*<sup>n</sup>* the system initial state, where symbol denotes Kronecker tensor

*<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�. As an example, when *Nw* <sup>=</sup> 4, the state results *<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>H</sup>*|00� <sup>=</sup>

<sup>2</sup> . One-qubit Hadamard transform matrix representation is (1),

*Uf*(|*w*�|*q*�) = |*w*�|*<sup>q</sup>* ⊕ *<sup>f</sup>*(*w*)�, (2)


<sup>2</sup> ) (3)

(|0�−|1�). As a result, the corresponding mathematical expression is:


(1)

Quantum Dating Market http://dx.doi.org/10.5772/53842 525

feature or a feature set that characterizes each woman on the left.

*<sup>ψ</sup>ini* = |00..00�≡|0�

*ψ*<sup>1</sup> = *Hψini* = <sup>√</sup>

1 <sup>2</sup> <sup>∑</sup><sup>3</sup>

oracle

target qubit |*q*� to <sup>√</sup>

effect on the first register.

1 2



1 *Nw* <sup>∑</sup>*Nw*−<sup>1</sup>

*<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*� <sup>=</sup> <sup>|</sup>00�+|01�+|10�+|11�

and n-qubits extension is *H*


**Table 1.** Prisoners Dilemma: C ≡ confess; N ≡ do not confess. The number on the left is for the years the prisoner *S*<sup>1</sup> prisoner must serve.

Quantum game theory is a classic game theory generalization. That is, quantum game strategies and outcomes include the classical as particularities, but also quantum features let the application of new strategies which leads to solutions classically imposible. The *N* players quantum game si denoted by *G*(*N*, H, *ρ*, *S*(H), *u*), where H is the Hilbert space of the physical system and *ρ* ∈ *S*(H) is the system initial condition, being *S*(H) the associated space state. In quantum games, players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the system's Hamiltonian [27]. If we call *Ui* the operator corresponding to player *i* strategy, the N-players strategies operator results *U* = *U*<sup>0</sup> *U*1...*Ui* ... *UN*−1. Starting from the initial pure state |Ψ0� of the system, players apply their strategies *<sup>U</sup>* in order to modify it according to their preferences, that is modifying the probability amplitudes associated with each base state. As a consequence, evolution from the initial system state to some state |Ψ1� is given by |Ψ1� = *<sup>U</sup>*|Ψ0�. Quantum games provide new ways to cooperate, to eliminate dilemmas, and as a consequence new equilibriums arise. As can be seen in [10], for example, the dilemma is avoided in the quantum Prisoner's game. That is, the system equilibrium is not longer (C,C) to be (N,N).

#### **3. Quantum search strategy**

In the classic dating market game [28, 29], men choose women simultaneously from *N* options, looking for those women who would have some "property" they want. Unlike the traditional game, in the quantum version of the dating game, players get the chance to use quantum techniques, for example they can explore their possibilities using a quantum search algorithm. Grover algorithm capitalizes quantum states superposition characteristic to find some "marked" state from a group of possible solutions in considerably less time than a classical algorithm can do [8]. That state space must be capable of being translatable, say to a graph *G* where to find some particular state which has a searched feature or distinctive mark, throughout the execution of the algorithm. By "distinctive mark" we mean problems whose algorithmic solution are inspired by physical processes. Furthermore it is possible to guarantee that the searched node is marked by a minimum (maximum) value of a physical property included in the algorithm.

Let agents be coded as Hilbert space base states. As a result, men are able to choose from *Nw* women set *W* = {|0�, |1�, ..., |*Nw* − 1�}. Table 1 displays four women states in the first column and some feature that makes them unique in the second column which we will code with a letter for simplicity.

If a player is looking for a woman with a feature "d", the table must be searched on its second column and when the desired "d" is found, look at the first column where the corresponding


4 Quantum Mechanics

must serve.

to be (N,N).

strategies operator results *U* = *U*<sup>0</sup>

**3. Quantum search strategy**

property included in the algorithm.

with a letter for simplicity.

is very common to represent in a bimatrix the possible strategies combinations with their

**S1** \ **S2 C N** *C* 3,3 0,10 *N* 10,0 1,1

**Table 1.** Prisoners Dilemma: C ≡ confess; N ≡ do not confess. The number on the left is for the years the prisoner *S*<sup>1</sup> prisoner

Quantum game theory is a classic game theory generalization. That is, quantum game strategies and outcomes include the classical as particularities, but also quantum features let the application of new strategies which leads to solutions classically imposible. The *N* players quantum game si denoted by *G*(*N*, H, *ρ*, *S*(H), *u*), where H is the Hilbert space of the physical system and *ρ* ∈ *S*(H) is the system initial condition, being *S*(H) the associated space state. In quantum games, players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the system's Hamiltonian [27]. If we call *Ui* the operator corresponding to player *i* strategy, the N-players


In the classic dating market game [28, 29], men choose women simultaneously from *N* options, looking for those women who would have some "property" they want. Unlike the traditional game, in the quantum version of the dating game, players get the chance to use quantum techniques, for example they can explore their possibilities using a quantum search algorithm. Grover algorithm capitalizes quantum states superposition characteristic to find some "marked" state from a group of possible solutions in considerably less time than a classical algorithm can do [8]. That state space must be capable of being translatable, say to a graph *G* where to find some particular state which has a searched feature or distinctive mark, throughout the execution of the algorithm. By "distinctive mark" we mean problems whose algorithmic solution are inspired by physical processes. Furthermore it is possible to guarantee that the searched node is marked by a minimum (maximum) value of a physical

Let agents be coded as Hilbert space base states. As a result, men are able to choose from *Nw* women set *W* = {|0�, |1�, ..., |*Nw* − 1�}. Table 1 displays four women states in the first column and some feature that makes them unique in the second column which we will code

If a player is looking for a woman with a feature "d", the table must be searched on its second column and when the desired "d" is found, look at the first column where the corresponding

... *UN*−1. Starting from the initial pure state

respective reward. The corresponding bimatrix for the prisoners game is 1.

*U*1...*Ui*

**Table 2.** Sample woman database. Left column contains women states and right column displays a letter representing some feature or a feature set that characterizes each woman on the left.

chosen woman state is: |3� in this example. The procedure is very simple if the table has just a few rows, but when the database gets bigger, the table in the best case would have to be entered *Nw*/2 times [30]. Under this framework we propose to use Grover algorithm in order to achieve man's decision in less time. Without losing generality let *Nw* = 2*<sup>n</sup>* being *n* the qubits needed to code *Nw* women. Quantum states transformation are made by applying Hilbert space operators *U* to them, following Ψ<sup>1</sup> = *U*1Ψ<sup>0</sup> is a new system state starting from Ψ0. As a consequence any quantum algorithm can be thought as a set of suitable linear transformations. Grover algorithm starts with *n* qubits in |0�, resulting *<sup>ψ</sup>ini* = |00..00�≡|0� *<sup>n</sup>* the system initial state, where symbol denotes Kronecker tensor product. Initially, the woman identified by state |0� is chosen with probability one. The next step is to create superposition states and like many other quantum algorithms Grover uses Hadamard transform to do this task since it maps *n* qubits initialized with |0� to a superposition of all *n* orthogonal states in the |0�, |1�,.. |*n* − 1� basis with equal weight, *ψ*<sup>1</sup> = *Hψini* = <sup>√</sup> 1 *Nw* <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�. As an example, when *Nw* <sup>=</sup> 4, the state results *<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> *<sup>H</sup>*|00� <sup>=</sup> 1 <sup>2</sup> <sup>∑</sup><sup>3</sup> *<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*� <sup>=</sup> <sup>|</sup>00�+|01�+|10�+|11� <sup>2</sup> . One-qubit Hadamard transform matrix representation is (1), and n-qubits extension is *H <sup>n</sup>*, see [27],

$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \tag{1}$$

Another quantum search algorithms characteristic, is the "Oracle", which is basically a black box capable of marking the problem solution. We call *Uf* the operator which implement the oracle

$$\mathcal{U}\_f(|w\rangle|q\rangle) = |w\rangle|q \oplus f(w)\rangle,\tag{2}$$

where *f*(*w*) is the oracle function which takes the value 1 if *w* correspond to the searched woman, *f*(*w*) = 1, and if it is not the case it takes the value 0, *f*(*w*) = 0. The value of *f*(*w*) on a superposition of every possible input *w* may be obtained [27]. The algorithm sets the target qubit |*q*� to <sup>√</sup> 1 2 (|0�−|1�). As a result, the corresponding mathematical expression is:

$$(|w\rangle(\frac{|0\rangle - |1\rangle}{\sqrt{2}}) \longmapsto ^{U\_f}(-1)^{f(w)}|w\rangle(\frac{|0\rangle - |1\rangle}{\sqrt{2}})\tag{3}$$

Observe that the second register is in an eigenstate, so we can ignore it, considering only the effect on the first register.

**Figure 2.** Grover Quantum searching algorithm

$$|w\rangle \longleftarrow \, ^{\mathrm{II}}\!/ \ (-1)^{f(w)} |w\rangle \tag{4}$$

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>0</sup>**

Quantum Dating Market http://dx.doi.org/10.5772/53842 527

**Searched State Other**

**Iteration Number**

**Figure 3.** Evolution of the probability to find the chosen woman and the probability to find other woman as a function of the

As the number of iterations the algorithm makes depends on the size of the options set, this must be known at the beginning of simulations. Every operator has its matrix representation to be used in simulations. We suppose the player chooses a woman who has some specific particularity that would distinguish her from any other of the group, so we construct matrix *Uf* and other matrixes for that purpose. The evolution of the squared amplitude with the iteration number is shown in Figure 3. The searched state amplitude is initially the same for all possible states |*i* > in the Ψ<sup>1</sup> expression. The fast increasing of the probability to find the preferred state on each iteration contrasts with the decreasing of the probability to find every other state. The example displayed is for *Nw* = 1024 women and as the can be seen in Figure 3, the number of iterations needed to get certainty to find the preferred woman are 25. Classically, a statistical algorithm would need approximately *Nw* = 1024 iterations.

Thus when a given man who wants to date a *Nw* size set selected woman, he must set his own *Uf* operator out, according to his preferences, and then let the algorithm do the job. The case of *Nm* men may be obtained generalizing the single man case: every one of them must follow the same steps. Nevertheless, achieving top choice is hard because of competition from other players and your dream partner may not share your feelings. If all players play quantum, the time to find woman is not an issue and the *N* stable solutions will be the same

To compare the quantum approach efficiency with the classical one we will consider some players playing quantum and others playing classic. Let us follow the evolution of agents

**0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1**

iteration number with Grover's algorithm.

as for the classic formulation [32].

representative from each group, *Q* and *C* respectively.

**4. Quantum vs classic**

**Probability**

Consequently, if *f*(*w*) = 1 a phase shift is produced, otherwise nothing happens. As we already stated our algorithm is based on the classical Gale-Shapley (GS) algorithm which assigns the role of proposers to the elements of one set, the men say, and of judges to the elements of the other.

Actually, for a more symmetric formulation of the algorithm where both sets are, at the same time, proposers and judges, it would be necessary another oracle which evaluates women features matching by means of another function *g*(*x*) [31], but we will not go into that. As far as we are concerned up to now the Oracle is a device capable of recognizing and "mark" a woman who has some special feature, said hair color, money, good manners, etc. Oracle operator *Uf* makes one of two central operations comprising of a whole operation named Grover iterate *G* (Fig.1), and a rotation operator *UR*, or conditional phase shift operator represented by equation (5).

*UR* and *Uf* , together with Hadamard transformations represented by *H* blocks (1), in the order depicted by (Fig. 1), make the initial state vector asymptotically going to reach the solution state vector amplitudes. The symbol *I* in *UR* equation is the identity operator.

$$
\hbar L\_R = 2|0\rangle\langle 0| - I \tag{5}
$$

Furthermore, after applying Grover iterate, *G*, *O*( *Nw*) times, the man finds the woman he is looking for. In Figure 1 Grover iterate is shown and Grover quantum algorithm scheme is depicted in 2.

√

**Figure 3.** Evolution of the probability to find the chosen woman and the probability to find other woman as a function of the iteration number with Grover's algorithm.

As the number of iterations the algorithm makes depends on the size of the options set, this must be known at the beginning of simulations. Every operator has its matrix representation to be used in simulations. We suppose the player chooses a woman who has some specific particularity that would distinguish her from any other of the group, so we construct matrix *Uf* and other matrixes for that purpose. The evolution of the squared amplitude with the iteration number is shown in Figure 3. The searched state amplitude is initially the same for all possible states |*i* > in the Ψ<sup>1</sup> expression. The fast increasing of the probability to find the preferred state on each iteration contrasts with the decreasing of the probability to find every other state. The example displayed is for *Nw* = 1024 women and as the can be seen in Figure 3, the number of iterations needed to get certainty to find the preferred woman are 25. Classically, a statistical algorithm would need approximately *Nw* = 1024 iterations.

Thus when a given man who wants to date a *Nw* size set selected woman, he must set his own *Uf* operator out, according to his preferences, and then let the algorithm do the job. The case of *Nm* men may be obtained generalizing the single man case: every one of them must follow the same steps. Nevertheless, achieving top choice is hard because of competition from other players and your dream partner may not share your feelings. If all players play quantum, the time to find woman is not an issue and the *N* stable solutions will be the same as for the classic formulation [32].

### **4. Quantum vs classic**

6 Quantum Mechanics

**Figure 1.** Grover Iteration

elements of the other.

represented by equation (5).

depicted in 2.

Furthermore, after applying Grover iterate, *G*, *O*(


**Figure 2.** Grover Quantum searching algorithm

*Uf H UR H*


Consequently, if *f*(*w*) = 1 a phase shift is produced, otherwise nothing happens. As we already stated our algorithm is based on the classical Gale-Shapley (GS) algorithm which assigns the role of proposers to the elements of one set, the men say, and of judges to the

Actually, for a more symmetric formulation of the algorithm where both sets are, at the same time, proposers and judges, it would be necessary another oracle which evaluates women features matching by means of another function *g*(*x*) [31], but we will not go into that. As far as we are concerned up to now the Oracle is a device capable of recognizing and "mark" a woman who has some special feature, said hair color, money, good manners, etc. Oracle operator *Uf* makes one of two central operations comprising of a whole operation named Grover iterate *G* (Fig.1), and a rotation operator *UR*, or conditional phase shift operator

*UR* and *Uf* , together with Hadamard transformations represented by *H* blocks (1), in the order depicted by (Fig. 1), make the initial state vector asymptotically going to reach the solution state vector amplitudes. The symbol *I* in *UR* equation is the identity operator.

is looking for. In Figure 1 Grover iterate is shown and Grover quantum algorithm scheme is

√

{

O( ) *<sup>N</sup>1/2*


*UR* = <sup>2</sup>|0��0| − *<sup>I</sup>* (5)

*Nw*) times, the man finds the woman he

*Hon G G G*

{

*G*

To compare the quantum approach efficiency with the classical one we will consider some players playing quantum and others playing classic. Let us follow the evolution of agents representative from each group, *Q* and *C* respectively.

*Q*, that plays quantum can keep his state as a linear combination of all the prospective results when unitary transforms such as the described above for Grover's algorithm are applied, provided no measurement producing collapse to any of them is done. On the other hand, the only way *C* has to search such a database is to test the elements sequentially against the condition until the target is found. For a database of size *Nw*, this brute force search requires an average of *O*(*Nw*/2) comparisons [7].

Two different games where both men want to date with the same woman are presented: In the first one player *Q* gives player *C* the chance to play first and both have only one attempt per turn, which means only one question to the oracle. The second game, in order that *Q* plays handicapped, is set out in the way that *C* can play *Nw*/2 times while *Q* only once, and player *C* plays first again. For the last case we analyzed two alternatives for the classic player: in the first one he plays without memory of his previous result and therefore, in every try he has 1/*Nw* probability to find the chosen woman to date, the other alternative permits the classic player to discard previous unfavorable outcomes at any try in order to avoid choosing them again and diminish the selection universe.

The player who invites the chosen woman first has more chances to succeed, as well as that who asks the same woman more times. Nevertheless the woman has the last word, and therefore the dating success for each player depends on that woman preferences. So, let us define *P<sup>i</sup> <sup>c</sup>* as the probability that woman *i* accepts dating the classic player *C* and *P<sup>i</sup> q* as the probability that she accepts the quantum player *Q* proposal. In order to compare performances, we consider *T* = 1000 playing times on turns and count the dating success times, then calculate the mean relative difference between *Q* and *C* success total number as *<sup>D</sup>*/*<sup>T</sup>* <sup>=</sup> *Qsuccess*−*Csuccess T* , for different woman acceptation probabilities.

0

**Figure 4.** First game: One attempt for both players. Mean relative difference between *Q* and *C* success total number as

his chances to win, reaching 0.78 as the probability to find his preferred woman in only one

*<sup>q</sup>* values used in the simulation, which means that even at extremes where *P<sup>i</sup>*

*C* attempt the system is forced to collapse to one base state, so a third party, that could be the oracle, arrange the states again and mark the solution. As we explained above, to mark a state means to change its phase but nothing happens to the state amplitude, consequently, for the classic player *C*, the probability that state results the one the Oracle have signaled is,

dating success chances increase considerably with respect to the first case. Figure 5 shows the corresponding results, where it is possible to see that classic player *C* begins to outperform

*Q* has only one, *C*'s odds of success in dating increases, and there are zones on the graph where *D*/*T* < 0. This implies that player *C* outperforms player *Q*. Nevertheless, to achieve that, the chosen woman preferences must be considerably greater for the classic player, that

*<sup>q</sup>*, that is, when woman has a marked preference for player *C*.

the quantum player performs better. However there is a very small region where *P<sup>i</sup>*

marked or not, 1/*Nw* = 1/8, even though, due to his "insistence", he tries *Nw*

algorithm like "Brute-Force algorithm". As shown in figure 5, when *<sup>C</sup>* has *Nw*

PCvec PQvec

0

0.2

*<sup>T</sup>* , for different woman acceptation probabilities *Pi*

iteration. Figure 4 shows that situation outcomes for different *P<sup>i</sup>*

*<sup>q</sup>* ≈ 0 not shown in the figure that corresponds to a prevailing *C*.

Player *C* probability to find the chosen woman can increase to <sup>1</sup>

0.4

vertical axis depicts *D*/*T* values as a function of *P<sup>i</sup>*

Under the second game conditions player *<sup>C</sup>* have *Nw*

0.6

0.8

small region where *C* prevails is not shown.

1 0

*<sup>D</sup>*/*<sup>T</sup>* = *Qsuccess*−*Csuccess*

*Pi <sup>c</sup>* and *P<sup>i</sup>*

*Pi*

*Q* when *P<sup>i</sup>*

is *P<sup>i</sup>*

*<sup>c</sup>* > 2*P<sup>i</sup> q*.

*<sup>c</sup>* >> *P<sup>i</sup>*

0.2

0.4

D/T

0.6

0.8

0.2

*<sup>c</sup>* and *P<sup>i</sup>*

0.4

*<sup>c</sup>* and *Pi*

0.6

*<sup>c</sup>* and *P<sup>i</sup>*

0.8

*<sup>q</sup>*. *Q* outperforms *C* in all shown cases. The

*<sup>q</sup>* respectively. *D*/*T* is positive for all

<sup>2</sup> <sup>=</sup> 4 attempts before *<sup>Q</sup>* plays. After each

1

*q* combinations. The

*<sup>c</sup>* >> *P<sup>i</sup> q*,

*<sup>c</sup>* ≈ 1 and

<sup>2</sup> = 4 times, his

<sup>2</sup> = 4 tries while

<sup>2</sup> when using a classical

0.1

0.2

0.3

0.4

0.5

0.6

0.7

529

Quantum Dating Market http://dx.doi.org/10.5772/53842

Initially, both players begin with the system in the initial state *ψ*<sup>1</sup> = <sup>√</sup> 1 *Nw* <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�, therefore the probability to select any woman is the same for both, *p*(*wi*) = 1/*N*. In the next step the Oracle marks one of the prospective women state according men preferences.

The results are highly dependent on the women set size *Nw* because, as mentioned above, Grover algorithm needs *O*( (*Nw*)) steps to find the quantum player's chosen partner while the classic player must use *O*(*Nw*) for the same task. In the case of only one woman and one man, for example, classic and quantum will not have any advantage on searching and the dating success difference for the first game will depend only on that woman preferences, that is, if *Pc* > *Pq* then *D*/*T* < 0 and the quantum player will do better when *Pq* > *Pc* . Similar chances for both players is not usual in most quantum games, such as, for example the coin flip game introduced by Meyer [9] where the quantum player always beats the classic player in a *"mano a mano"* game. For a two women set *Q* uses only one step, but *C* needs two steps to find the right partner. In this case *Q* does better when *Pq* > *Pc*/4. Winning conditions improve for the quantum player for increasing *Nw*, but not in a monotonous way, because the number of steps used by Grover algorithm in *Q* search is an integer that increases in discrete steps.

In order to facilitate comprehension the set size in the simulations results shown is *Nw* = 8, that is the biggest *Nw* (taken as 2*n*) in which *Q* uses only one step in Grover algorithm.

Under the first game conditions both players have only one attempt by turn. Since *C* cannot modify state *ψ*<sup>1</sup> amplitudes, he has 1/8 chance to be right. On the other hand player *Q*, using Grover algorithm as his strategy, can modify states amplitudes in order to increase

8 Quantum Mechanics

let us define *P<sup>i</sup>*

*<sup>D</sup>*/*<sup>T</sup>* <sup>=</sup> *Qsuccess*−*Csuccess*

Grover algorithm needs *O*(

steps.

an average of *O*(*Nw*/2) comparisons [7].

them again and diminish the selection universe.

*Q*, that plays quantum can keep his state as a linear combination of all the prospective results when unitary transforms such as the described above for Grover's algorithm are applied, provided no measurement producing collapse to any of them is done. On the other hand, the only way *C* has to search such a database is to test the elements sequentially against the condition until the target is found. For a database of size *Nw*, this brute force search requires

Two different games where both men want to date with the same woman are presented: In the first one player *Q* gives player *C* the chance to play first and both have only one attempt per turn, which means only one question to the oracle. The second game, in order that *Q* plays handicapped, is set out in the way that *C* can play *Nw*/2 times while *Q* only once, and player *C* plays first again. For the last case we analyzed two alternatives for the classic player: in the first one he plays without memory of his previous result and therefore, in every try he has 1/*Nw* probability to find the chosen woman to date, the other alternative permits the classic player to discard previous unfavorable outcomes at any try in order to avoid choosing

The player who invites the chosen woman first has more chances to succeed, as well as that who asks the same woman more times. Nevertheless the woman has the last word, and therefore the dating success for each player depends on that woman preferences. So,

as the probability that she accepts the quantum player *Q* proposal. In order to compare performances, we consider *T* = 1000 playing times on turns and count the dating success times, then calculate the mean relative difference between *Q* and *C* success total number as

the probability to select any woman is the same for both, *p*(*wi*) = 1/*N*. In the next step the

The results are highly dependent on the women set size *Nw* because, as mentioned above,

the classic player must use *O*(*Nw*) for the same task. In the case of only one woman and one man, for example, classic and quantum will not have any advantage on searching and the dating success difference for the first game will depend only on that woman preferences, that is, if *Pc* > *Pq* then *D*/*T* < 0 and the quantum player will do better when *Pq* > *Pc* . Similar chances for both players is not usual in most quantum games, such as, for example the coin flip game introduced by Meyer [9] where the quantum player always beats the classic player in a *"mano a mano"* game. For a two women set *Q* uses only one step, but *C* needs two steps to find the right partner. In this case *Q* does better when *Pq* > *Pc*/4. Winning conditions improve for the quantum player for increasing *Nw*, but not in a monotonous way, because the number of steps used by Grover algorithm in *Q* search is an integer that increases in discrete

In order to facilitate comprehension the set size in the simulations results shown is *Nw* = 8, that is the biggest *Nw* (taken as 2*n*) in which *Q* uses only one step in Grover algorithm.

Under the first game conditions both players have only one attempt by turn. Since *C* cannot modify state *ψ*<sup>1</sup> amplitudes, he has 1/8 chance to be right. On the other hand player *Q*, using Grover algorithm as his strategy, can modify states amplitudes in order to increase

*T* , for different woman acceptation probabilities.

Initially, both players begin with the system in the initial state *ψ*<sup>1</sup> = <sup>√</sup>

Oracle marks one of the prospective women state according men preferences.

*<sup>c</sup>* as the probability that woman *i* accepts dating the classic player *C* and *P<sup>i</sup>*

(*Nw*)) steps to find the quantum player's chosen partner while

1 *Nw* <sup>∑</sup>*Nw*−<sup>1</sup> *q*

*<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�, therefore

**Figure 4.** First game: One attempt for both players. Mean relative difference between *Q* and *C* success total number as *<sup>D</sup>*/*<sup>T</sup>* = *Qsuccess*−*Csuccess <sup>T</sup>* , for different woman acceptation probabilities *Pi <sup>c</sup>* and *Pi <sup>q</sup>*. *Q* outperforms *C* in all shown cases. The small region where *C* prevails is not shown.

his chances to win, reaching 0.78 as the probability to find his preferred woman in only one iteration. Figure 4 shows that situation outcomes for different *P<sup>i</sup> <sup>c</sup>* and *P<sup>i</sup> q* combinations. The vertical axis depicts *D*/*T* values as a function of *P<sup>i</sup> <sup>c</sup>* and *P<sup>i</sup> <sup>q</sup>* respectively. *D*/*T* is positive for all *Pi <sup>c</sup>* and *P<sup>i</sup> <sup>q</sup>* values used in the simulation, which means that even at extremes where *P<sup>i</sup> <sup>c</sup>* >> *P<sup>i</sup> q*, the quantum player performs better. However there is a very small region where *P<sup>i</sup> <sup>c</sup>* ≈ 1 and *Pi <sup>q</sup>* ≈ 0 not shown in the figure that corresponds to a prevailing *C*.

Under the second game conditions player *<sup>C</sup>* have *Nw* <sup>2</sup> <sup>=</sup> 4 attempts before *<sup>Q</sup>* plays. After each *C* attempt the system is forced to collapse to one base state, so a third party, that could be the oracle, arrange the states again and mark the solution. As we explained above, to mark a state means to change its phase but nothing happens to the state amplitude, consequently, for the classic player *C*, the probability that state results the one the Oracle have signaled is, marked or not, 1/*Nw* = 1/8, even though, due to his "insistence", he tries *Nw* <sup>2</sup> = 4 times, his dating success chances increase considerably with respect to the first case. Figure 5 shows the corresponding results, where it is possible to see that classic player *C* begins to outperform *Q* when *P<sup>i</sup> <sup>c</sup>* >> *P<sup>i</sup> <sup>q</sup>*, that is, when woman has a marked preference for player *C*.

Player *C* probability to find the chosen woman can increase to <sup>1</sup> <sup>2</sup> when using a classical algorithm like "Brute-Force algorithm". As shown in figure 5, when *<sup>C</sup>* has *Nw* <sup>2</sup> = 4 tries while *Q* has only one, *C*'s odds of success in dating increases, and there are zones on the graph where *D*/*T* < 0. This implies that player *C* outperforms player *Q*. Nevertheless, to achieve that, the chosen woman preferences must be considerably greater for the classic player, that is *P<sup>i</sup> <sup>c</sup>* > 2*P<sup>i</sup> q*.

**5. Stability of couples**

normalization condition <sup>∑</sup>*Nw*−<sup>1</sup>

women spaces,i.e. *SC* = *SM*

strategies operator results *U* = *U*<sup>0</sup>

identify men states is given by *UW*.

**5.2. Density matrix and system entropy**

of all pure states and is defined by the equation

<sup>Ψ</sup>*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*M*−<sup>1</sup>

**5.1. Strategies**

There is a group of *Nm* men and *Nw* women playing the game. Be *Si* = {|0�, |1�, ..., |*Nw* − <sup>1</sup>�} the states in a Hilbert space of man *i* decisions, where {0, 1, ..., *Nw* − 1} are indexes in decimal notation identifying all the women he may choose. As a result each man has been assigned *log*2(*Nw*) qubits in order to identify each woman. Generally, the state vector of one man

is the probability that man *i* selects woman *j* when system state is Ψ*<sup>i</sup>* so must satisfied the

where is the Kronecker product. Note that the *SM* extends to any possible combination of men elections. On the other side there are the women who receive men proposals and must decide whether to accept or not one of them. With greater or lesser probability they will receive the all men's proposals, so following the same argument used with the men, be

the women acceptances space state. Finally, to close the circle, we define the couples possible states which must include so all possible men's elections as all possible women's acceptances. Accordingly, state space of the couples emerge from the Kronecker product of the men and

In quantum games, players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the system's Hamiltonian [27]. If we call *Ui* the operator corresponding to player *i* strategy, the N-players


to the qubits that identify the women states, meanwhile the women action on the qubits that

Often, as in life, players are not completely sure about which strategy to apply, that is, by the way of example, the case where someone chooses between the strategy *Ua* with probability *pa* and *Ub* with probability *pb* = <sup>1</sup> − *pa*, that situation is referred in a mixed strategies game. Despite the complete system can be represented by its state vector, when it comes to mixed states the density matrix is more suitable. It was introduced by von Neumann to describe a mixed ensemble in which each member has assigned a probability of being in a determined state. The density operator, as it is also commonly called, represents the statistical mixture

*U*1...*Ui*

*<sup>j</sup>*=<sup>0</sup> *<sup>α</sup>j*|*j*�, where <sup>|</sup>*αj*<sup>|</sup>

Quantum Dating Market http://dx.doi.org/10.5772/53842

... *SM*−<sup>2</sup>

... *SNw*<sup>−</sup><sup>2</sup>

<sup>2</sup> = 1. If there is no correlation between players, the state

*S*<sup>1</sup>

*S*<sup>1</sup>

... *UN*−1. Starting from the initial pure state

*UW*. That is, *UM* is applied by men

2

531

*SM*−1,

*SNw*<sup>−</sup><sup>1</sup>

decisions will be in quantum superposition of the base states, <sup>Ψ</sup>*<sup>i</sup>* <sup>=</sup> <sup>∑</sup>*Nw*−<sup>1</sup>

*<sup>j</sup>*=<sup>0</sup> <sup>|</sup>*αj*<sup>|</sup>

*SW*.

strategies *UM* and *UW* respectively through *U* = *UM*

space of all men decision system is represented through *SM* = *S*<sup>0</sup>

*<sup>i</sup>*=<sup>0</sup> *<sup>α</sup>i*|*i*� the woman *<sup>j</sup>* acceptation state and be *SW* <sup>=</sup> *<sup>S</sup>*<sup>0</sup>

**Figure 5.** Second game: Classic player *C* has four tries while *Q* has only one. Mean relative difference between *Q* and *C* success total number as *<sup>D</sup>*/*<sup>T</sup>* = *Qsuccess*−*Csuccess <sup>T</sup>* , for different woman acceptation probabilities *Pi <sup>c</sup>* and *Pi <sup>q</sup>*. *C* outperforms *Q* when *Pi <sup>c</sup>* >> *Pi q*

#### **4.1. Section discussion**

In this section we have introduced a quantum formulation for decision matching problems, specifically for the dating game. In that framework women are represented with quantum states whose associated amplitudes must be modified by men's selection strategies, in order to increase a particular state amplitude and to decrease the others, with the final purpose to achieve the best possible choice when the game finishes. This is a highly time consuming task that takes a *O*(*N*) runtime for a classical probabilistic algorithm, being *N* the women database size. Grover quantum search algorithm is used as a playing strategy that takes the man *O*( <sup>√</sup>*N*) runtime to find his chosen partner. As a consequence, if every man uses quantum strategy, no one does better than the others, and stability is quickly obtained.

The performances of quantum vs. classic players depend on the number of players *N*. In a "one on one" game there is no advantage from any of them and the woman preferences rule. Similar chances for quantum and classic players in "one on one" situation is not usual in most quantum games. Winning conditions improve for the quantum player for increasing *N* and the same number of attempts, but not in a monotonous way. The comparison between quantum and classic performances shows that for the same numbers of attempts, the quantum approach outperforms the classical approach. If the game is set in order that the classic player has *<sup>N</sup>* <sup>2</sup> opportunities and the quantum player only one, the former player begins to have an advantage over the quantum one when his probability to be accepted by the chosen woman is much higher than the probability for the quantum player.

### **5. Stability of couples**

10 Quantum Mechanics

0

**Figure 5.** Second game: Classic player *C* has four tries while *Q* has only one. Mean relative difference between *Q* and *C*

In this section we have introduced a quantum formulation for decision matching problems, specifically for the dating game. In that framework women are represented with quantum states whose associated amplitudes must be modified by men's selection strategies, in order to increase a particular state amplitude and to decrease the others, with the final purpose to achieve the best possible choice when the game finishes. This is a highly time consuming task that takes a *O*(*N*) runtime for a classical probabilistic algorithm, being *N* the women database size. Grover quantum search algorithm is used as a playing strategy that takes

quantum strategy, no one does better than the others, and stability is quickly obtained.

The performances of quantum vs. classic players depend on the number of players *N*. In a "one on one" game there is no advantage from any of them and the woman preferences rule. Similar chances for quantum and classic players in "one on one" situation is not usual in most quantum games. Winning conditions improve for the quantum player for increasing *N* and the same number of attempts, but not in a monotonous way. The comparison between quantum and classic performances shows that for the same numbers of attempts, the quantum approach outperforms the classical approach. If the game is set in order that

begins to have an advantage over the quantum one when his probability to be accepted by

the chosen woman is much higher than the probability for the quantum player.

PCvec PQvec

0

0.2

0.4

0.6

0.8

success total number as *<sup>D</sup>*/*<sup>T</sup>* = *Qsuccess*−*Csuccess*

1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

D/T

when *Pi*

*<sup>c</sup>* >> *Pi q* **4.1. Section discussion**

the man *O*(

the classic player has *<sup>N</sup>*

0.2

*<sup>T</sup>* , for different woman acceptation probabilities *Pi*

<sup>√</sup>*N*) runtime to find his chosen partner. As a consequence, if every man uses

<sup>2</sup> opportunities and the quantum player only one, the former player

0.4

0.6

0.8

*<sup>c</sup>* and *Pi*

1

*<sup>q</sup>*. *C* outperforms *Q*

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

There is a group of *Nm* men and *Nw* women playing the game. Be *Si* = {|0�, |1�, ..., |*Nw* − <sup>1</sup>�} the states in a Hilbert space of man *i* decisions, where {0, 1, ..., *Nw* − 1} are indexes in decimal notation identifying all the women he may choose. As a result each man has been assigned *log*2(*Nw*) qubits in order to identify each woman. Generally, the state vector of one man decisions will be in quantum superposition of the base states, <sup>Ψ</sup>*<sup>i</sup>* <sup>=</sup> <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> *<sup>α</sup>j*|*j*�, where <sup>|</sup>*αj*<sup>|</sup> 2 is the probability that man *i* selects woman *j* when system state is Ψ*<sup>i</sup>* so must satisfied the normalization condition <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> <sup>|</sup>*αj*<sup>|</sup> <sup>2</sup> = 1. If there is no correlation between players, the state space of all men decision system is represented through *SM* = *S*<sup>0</sup> *S*<sup>1</sup> ... *SM*−<sup>2</sup> *SM*−1, where is the Kronecker product. Note that the *SM* extends to any possible combination of men elections. On the other side there are the women who receive men proposals and must decide whether to accept or not one of them. With greater or lesser probability they will receive the all men's proposals, so following the same argument used with the men, be <sup>Ψ</sup>*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*M*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> *<sup>α</sup>i*|*i*� the woman *<sup>j</sup>* acceptation state and be *SW* <sup>=</sup> *<sup>S</sup>*<sup>0</sup> *S*<sup>1</sup> ... *SNw*<sup>−</sup><sup>2</sup> *SNw*<sup>−</sup><sup>1</sup> the women acceptances space state. Finally, to close the circle, we define the couples possible states which must include so all possible men's elections as all possible women's acceptances. Accordingly, state space of the couples emerge from the Kronecker product of the men and women spaces,i.e. *SC* = *SM SW*.

#### **5.1. Strategies**

In quantum games, players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the system's Hamiltonian [27]. If we call *Ui* the operator corresponding to player *i* strategy, the N-players strategies operator results *U* = *U*<sup>0</sup> *U*1...*Ui* ... *UN*−1. Starting from the initial pure state |Ψ0� of the system, players apply their strategies *<sup>U</sup>* in order to modify it according to their preferences, that is modifying the probability amplitudes associated with each base state. As a consequence, evolution from the initial system state to some state |Ψ1� is given by |Ψ1� = *<sup>U</sup>*|Ψ0�. Note that, following the reasoning of the preceding paragraph, when <sup>Ψ</sup><sup>0</sup> is the initial state and Ψ<sup>1</sup> is the final state of the couples system, *U* arises from men and women strategies *UM* and *UW* respectively through *U* = *UM UW*. That is, *UM* is applied by men to the qubits that identify the women states, meanwhile the women action on the qubits that identify men states is given by *UW*.

#### **5.2. Density matrix and system entropy**

Often, as in life, players are not completely sure about which strategy to apply, that is, by the way of example, the case where someone chooses between the strategy *Ua* with probability *pa* and *Ub* with probability *pb* = <sup>1</sup> − *pa*, that situation is referred in a mixed strategies game. Despite the complete system can be represented by its state vector, when it comes to mixed states the density matrix is more suitable. It was introduced by von Neumann to describe a mixed ensemble in which each member has assigned a probability of being in a determined state. The density operator, as it is also commonly called, represents the statistical mixture of all pure states and is defined by the equation

$$\rho = \sum\_{i} p\_{i} |\Psi\_{i}\rangle\langle\Psi\_{i}|\_{\prime} \tag{6}$$

where the coefficients *pi* are non-negative and add up to one. From the density operator we can construct and understand the statistical behavior about our system by using statistical mechanics and a criterion of maximum or minimum entropy. Continuing the example, if it is supposed that the system starts in the pure state *<sup>ρ</sup>*<sup>0</sup> = |Ψ0��Ψ0|, after players mixed actions density matrix evolution is

$$
\rho\_1 = p\_a \mathcal{U}\_a \rho\_0 \mathcal{U}\_a^\dagger + p\_b \mathcal{U}\_b \rho\_0 \mathcal{U}\_b^\dagger. \tag{7}
$$

Entropy is the central concept of information theories, [33]. The quantum analogue of entropy was introduced in quantum mechanics by von Neumann,[34] and it is defined by the formula

$$S(\rho) = -Tr\{\rho \log\_2 \rho\}.\tag{8}$$

**Figure 6.** Quantum entropy corresponding to the situation where player 0 varies the probability *p* to apply strategy *U*<sup>0</sup>

because it is considered that people prefer to be coupled.

follows *U*(*θ*, 0) is always replaced by the simplest notation *U*(*θ*).

*<sup>U</sup>*(*θ*, *<sup>γ</sup>*) =

<sup>0</sup>, while *<sup>U</sup>*<sup>1</sup>

Let *p*<sup>0</sup> be the probability of player 0 to apply strategy *U*<sup>0</sup>

<sup>1</sup> to the initial state <sup>Ψ</sup>*<sup>i</sup>*

strategy *U*<sup>0</sup>

other two are the two women possible selections, then 16 are the possible couples states. For example, the state |0101� corresponds to the case that man 0 chooses woman 0 and she accepts him and the same occurs with man 1 and woman 1. Note that not all states corresponds to possible dates, some of them are considering the cases where there are no date, or the ones where only one couple is formed, the state |0001� is an example of the last case where the man man 0 chooses woman 0 and she accepts but on the other hand man 1 also chooses woman but she doesn't and woman 1 does not receive any proposition. As the game progress, probability amplitudes associated with mismatches must decrease, that

Single players moves or strategies are associated with unitary operators *Ui*(*θ*), with 0 ≤ *<sup>θ</sup>* ≤ *π*, applied on each one of their qubits, that in the general case where players have 2*<sup>n</sup>* options, each pure strategy *U* is composed by *n* different *Ui*(*θk*), being *k* each state qubit. The general formula of *Ui* is 10, that are rotation operators, as explained in [27] any qubit operation can be decomposed as a product of rotations. In this work we consider *γ* = 0, therefore in what

> *ei<sup>γ</sup>* · *cos*(*θ*/2) *sin*(*θ*/2) <sup>−</sup>*sin*(*θ*/2) *<sup>e</sup>*−*i<sup>γ</sup>* · *cos*(*θ*/2)

> > <sup>0</sup> and *<sup>U</sup>*<sup>1</sup>

probability *<sup>p</sup>*<sup>1</sup> and 1 − *<sup>p</sup>*<sup>1</sup> respectively. The strategies operators used in the examples are defined below, equations 11 and 12 are applied by man 0. Both of them transform the initial

<sup>0</sup> and 1 <sup>−</sup> *<sup>p</sup>*<sup>0</sup> the probability to apply

<sup>1</sup> are the strategies the man 1 applies with

0

Quantum Dating Market http://dx.doi.org/10.5772/53842 533

(10)

#### **5.3.** *N* = 2 **Model**

In order to set up the notation let us look at the following example of two men and two women that interact for *T* times periods. Let define Ψ*<sup>i</sup>* <sup>0</sup> <sup>=</sup> *<sup>α</sup>*|0� <sup>+</sup> *<sup>β</sup>*|1� as the initial decision state of men *i* which is a linear superposition of the two possible options he has, they are woman 0 or 1. Without losing generalization consider *α* = 1 and *β* = 0 which is consistent with thinking that they both have preference for the most popular, the most beautiful, the richest, or any superficial feature that most of the time makes men desire a woman at first glance. Consequently, the men's initial state vector is Ψ*<sup>M</sup>* <sup>0</sup> <sup>=</sup> <sup>Ψ</sup><sup>0</sup> 0 Ψ<sup>1</sup> <sup>0</sup> <sup>=</sup> <sup>|</sup>00�, where the first qubit represent man's 0 choice and the second is man's 1 choice. As we explain above, the initial quantum pure state is not stable, so during the game the state will change to the general form Ψ*<sup>M</sup> <sup>a</sup>* = <sup>∑</sup><sup>1</sup> *<sup>i</sup>*=0,*j*=<sup>0</sup> *<sup>α</sup>ij*|*ij*� with probability *pa* and <sup>Ψ</sup>*<sup>M</sup> <sup>b</sup>* <sup>=</sup> <sup>∑</sup><sup>1</sup> *<sup>i</sup>*=0,*j*=<sup>0</sup> *<sup>β</sup>ij*|*ij*� with probability *pb*. As women have the last decision, they must evaluate men proposals and

decide to accept one of them or reject all. We consider, just for the example that woman 0 chooses man 0 with *<sup>p</sup>*0*<sup>m</sup>* and man 1 with probability 1 − *<sup>p</sup>*0*m*, similar condition for woman 1 but in this case being *p*1*<sup>m</sup>* the probability to choose man 0. That condition doesn't affect system stability but depending on the probabilities values does affect the maximum and minimum of the couple system's entropy. Equation 9 shows the women density matrix which has no off diagonal elements.

$$p\_{100} = p\_{0m} p\_{1m} |00\rangle\langle 00| + p\_{0m} (1 - p\_{1m}) |01\rangle\langle 01| + (1 - p\_{0m}) p\_{1m} |10\rangle\langle 10| + (1 - p\_{0m}) (1 - p\_{1m}) |11\rangle\langle 11|.\tag{9}$$

The direct product of all possible men proposals with all possible women decisions generates a possible partners state vector which in decimal notation is Ψ*<sup>P</sup>* <sup>0</sup> <sup>=</sup> <sup>∑</sup><sup>15</sup> *<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�. Index *<sup>i</sup>* is a four qubits number, the first two qubits represent men 0 and 1 choices respectively and the

**Figure 6.** Quantum entropy corresponding to the situation where player 0 varies the probability *p* to apply strategy *U*<sup>0</sup> 0

12 Quantum Mechanics

density matrix evolution is

**5.3.** *N* = 2 **Model**

the general form Ψ*<sup>M</sup>*

*<sup>ρ</sup>* <sup>=</sup> ∑ *i*

*ρ*<sup>1</sup> = *paUaρ*0*U*†

women that interact for *T* times periods. Let define Ψ*<sup>i</sup>*

glance. Consequently, the men's initial state vector is Ψ*<sup>M</sup>*

a possible partners state vector which in decimal notation is Ψ*<sup>P</sup>*

*<sup>a</sup>* = <sup>∑</sup><sup>1</sup>

which has no off diagonal elements.

where the coefficients *pi* are non-negative and add up to one. From the density operator we can construct and understand the statistical behavior about our system by using statistical mechanics and a criterion of maximum or minimum entropy. Continuing the example, if it is supposed that the system starts in the pure state *<sup>ρ</sup>*<sup>0</sup> = |Ψ0��Ψ0|, after players mixed actions

Entropy is the central concept of information theories, [33]. The quantum analogue of entropy was introduced in quantum mechanics by von Neumann,[34] and it is defined by the formula

In order to set up the notation let us look at the following example of two men and two

state of men *i* which is a linear superposition of the two possible options he has, they are woman 0 or 1. Without losing generalization consider *α* = 1 and *β* = 0 which is consistent with thinking that they both have preference for the most popular, the most beautiful, the richest, or any superficial feature that most of the time makes men desire a woman at first

first qubit represent man's 0 choice and the second is man's 1 choice. As we explain above, the initial quantum pure state is not stable, so during the game the state will change to

probability *pb*. As women have the last decision, they must evaluate men proposals and decide to accept one of them or reject all. We consider, just for the example that woman 0 chooses man 0 with *<sup>p</sup>*0*<sup>m</sup>* and man 1 with probability 1 − *<sup>p</sup>*0*m*, similar condition for woman 1 but in this case being *p*1*<sup>m</sup>* the probability to choose man 0. That condition doesn't affect system stability but depending on the probabilities values does affect the maximum and minimum of the couple system's entropy. Equation 9 shows the women density matrix

*<sup>i</sup>*=0,*j*=<sup>0</sup> *<sup>α</sup>ij*|*ij*� with probability *pa* and <sup>Ψ</sup>*<sup>M</sup>*

*ρw*0=*p*0*<sup>m</sup> p*1*<sup>m</sup>*|00��00|+*p*0*<sup>m</sup>*(1−*p*1*<sup>m</sup>*)|01��01|+(1−*p*0*<sup>m</sup>*)*p*1*<sup>m</sup>*|10��10|+(1−*p*0*<sup>m</sup>*)(1−*p*1*<sup>m</sup>*)|11��11|. (9)

The direct product of all possible men proposals with all possible women decisions generates

four qubits number, the first two qubits represent men 0 and 1 choices respectively and the

*<sup>a</sup>* + *pbUbρ*0*U*†

*pi*|Ψ*i*��Ψ*i*|, (6)

*<sup>S</sup>*(*ρ*) = <sup>−</sup>*Tr*{*<sup>ρ</sup>* log2 *<sup>ρ</sup>*}. (8)

<sup>0</sup> <sup>=</sup> <sup>Ψ</sup><sup>0</sup> 0 Ψ<sup>1</sup>

*<sup>b</sup>* . (7)

<sup>0</sup> <sup>=</sup> *<sup>α</sup>*|0� <sup>+</sup> *<sup>β</sup>*|1� as the initial decision

*<sup>b</sup>* <sup>=</sup> <sup>∑</sup><sup>1</sup>

<sup>0</sup> <sup>=</sup> <sup>∑</sup><sup>15</sup>

<sup>0</sup> <sup>=</sup> <sup>|</sup>00�, where the

*<sup>i</sup>*=0,*j*=<sup>0</sup> *<sup>β</sup>ij*|*ij*� with

*<sup>i</sup>*=<sup>0</sup> <sup>|</sup>*i*�. Index *<sup>i</sup>* is a

other two are the two women possible selections, then 16 are the possible couples states. For example, the state |0101� corresponds to the case that man 0 chooses woman 0 and she accepts him and the same occurs with man 1 and woman 1. Note that not all states corresponds to possible dates, some of them are considering the cases where there are no date, or the ones where only one couple is formed, the state |0001� is an example of the last case where the man man 0 chooses woman 0 and she accepts but on the other hand man 1 also chooses woman but she doesn't and woman 1 does not receive any proposition. As the game progress, probability amplitudes associated with mismatches must decrease, that because it is considered that people prefer to be coupled.

Single players moves or strategies are associated with unitary operators *Ui*(*θ*), with 0 ≤ *<sup>θ</sup>* ≤ *π*, applied on each one of their qubits, that in the general case where players have 2*<sup>n</sup>* options, each pure strategy *U* is composed by *n* different *Ui*(*θk*), being *k* each state qubit. The general formula of *Ui* is 10, that are rotation operators, as explained in [27] any qubit operation can be decomposed as a product of rotations. In this work we consider *γ* = 0, therefore in what follows *U*(*θ*, 0) is always replaced by the simplest notation *U*(*θ*).

$$\mathcal{U}(\theta,\gamma) = \begin{pmatrix} e^{i\gamma} \cdot \cos(\theta/2) & \sin(\theta/2) \\ -\sin(\theta/2) & e^{-i\gamma} \cdot \cos(\theta/2) \end{pmatrix} \tag{10}$$

Let *p*<sup>0</sup> be the probability of player 0 to apply strategy *U*<sup>0</sup> <sup>0</sup> and 1 <sup>−</sup> *<sup>p</sup>*<sup>0</sup> the probability to apply strategy *U*<sup>0</sup> <sup>1</sup> to the initial state <sup>Ψ</sup>*<sup>i</sup>* <sup>0</sup>, while *<sup>U</sup>*<sup>1</sup> <sup>0</sup> and *<sup>U</sup>*<sup>1</sup> <sup>1</sup> are the strategies the man 1 applies with probability *<sup>p</sup>*<sup>1</sup> and 1 − *<sup>p</sup>*<sup>1</sup> respectively. The strategies operators used in the examples are defined below, equations 11 and 12 are applied by man 0. Both of them transform the initial

**Figure 7.** Quantum entropy corresponding to the situation where player 1 varies the probability *p* to apply strategy *U*<sup>1</sup> 0

state |0� into states that are linear superpositions of 0 and 1, representing states with different probabilities of choosing one woman or the other. On the other hand, strategies applied by man 1 are presented in 13 and 14.

Figures 6 and 7 show two situations where the system entropy varies considerably as a function of the strategies the players use. Figure 2, for example, shows the case where the man 1 applies his strategies with fixed probability, just varying the angle *θ* while the other man (0) varies both strategies angle and the probability *p*. In all the cases we present here, in order to simplify the outcomes display, women density matrix doesn't change as explained above.

$$\mathcal{U}\_0^0 = \mathcal{U}(\theta) \tag{11}$$

woman to choose both of them at the same time (we assume). This correspond to minimum entropy as can be easily seen in Fig. 1. Depending on the strategies applied by men, the whole system entropy, that is the couple system entropy changes reaching maximum and minimum limits. As *p* increases, the mixing of the strategies also increases producing an increase in entropy that indicates a tendency to stability. The mixing of the strategies means that the men proposals are less focused on one woman. Fig. 2 shows the case where men's role change, that is man 0 fixes his strategies probabilities while 1 varies his own. Although for fixed *θ* angle, as expected, the minimum entropy points are located where player is applying a pure strategy (*p* = 0), for *θ* = *π*/2 the entropy value does not change regardless

strategy. A result not shown in the figures is that entropy maxima increase when women

In this way, maxima and minima entropy points may be used to find stable states. Nevertheless, these stable states may not correspond to equilibria states of the game, because the players utilities has not been considered. In order to find Nash equilibria states, these

As a continuation of the analysis of a quantum formulation for the dating game that takes into account mutual progressive learning by representing women with quantum states whose associated amplitudes must be modified by men's selection strategies. we concentrate on the information associated to the problem. Since we deal with mixed strategies, the density matrix formalism is used to describe the system. Even though the decisions players make are based on their payoffs, past experiences, believes, etc., we are not interested in that causes but in the consequences of the decision they take, that is, the influence of the strategies they apply on the quantum system stability by means of the equivalence between maximum entropy states and those states that obey the Collective Welfare Principle that says that a system is stable only if it maximizes the welfare of the collective above the welfare of the individual. Maxima and minima entropy points are used to find characteristic strategies that lead to stable and unstable states. Nevertheless, in order to find Nash equilibria states, the

Maxima and minima entropy do not depend only on the strategies of men but also on women preferences, reaching the highest value when they have no preferences, that is when they choose every man with equal probability. On the other hand, minimum entropy correspond to men betting all chips to a single woman, without giving a chance to other woman.

The quantum dating market problem has been formulated as a two-sided bandit model [28], where in one side there are the men who must choose one "item" from the other side, which

The quantum formulation, which was presented in previous section, proceeds by assigning one basis of a Hilbert state space to each woman. As a consequence, if *Nw* is number of women playing, *Si* = {|0�, |1�, ..., |*Nw* − <sup>1</sup>�} are the states in the Hilbert space representing a man *i* decisions, therefore every man needs at least *log*2(*Nw*) qubits to identify each woman.

unlike the one side bandit, is composed by women able to reject the invitations.

<sup>1</sup> are equivalent and therefore player 1 is applying a pure

Quantum Dating Market http://dx.doi.org/10.5772/53842 535

of the value of *p*, because *U*<sup>0</sup>

**5.4. Section discussion**

players utilities must be considered.

**6. Entangled strategies**

preferences are the same for every men.

<sup>0</sup> and *<sup>U</sup>*<sup>0</sup>

utilities must be considered. This is beyond this chapter goals.

$$\mathcal{U}^{0}\_{1} = \mathcal{U}(\theta)\mathcal{U}(\pi) \tag{12}$$

$$\mathcal{U}^1\_0 = \mathcal{U}(\theta) \tag{13}$$

$$
\mathcal{U}\_1^1 = \mathcal{U}(-\theta) \tag{14}
$$

For example, if both men choose the same woman with probability one, this is represented in Fig. 1 with *p* = 0. This situation is completely unstable because it is impossible for the woman to choose both of them at the same time (we assume). This correspond to minimum entropy as can be easily seen in Fig. 1. Depending on the strategies applied by men, the whole system entropy, that is the couple system entropy changes reaching maximum and minimum limits. As *p* increases, the mixing of the strategies also increases producing an increase in entropy that indicates a tendency to stability. The mixing of the strategies means that the men proposals are less focused on one woman. Fig. 2 shows the case where men's role change, that is man 0 fixes his strategies probabilities while 1 varies his own. Although for fixed *θ* angle, as expected, the minimum entropy points are located where player is applying a pure strategy (*p* = 0), for *θ* = *π*/2 the entropy value does not change regardless of the value of *p*, because *U*<sup>0</sup> <sup>0</sup> and *<sup>U</sup>*<sup>0</sup> <sup>1</sup> are equivalent and therefore player 1 is applying a pure strategy. A result not shown in the figures is that entropy maxima increase when women preferences are the same for every men.

In this way, maxima and minima entropy points may be used to find stable states. Nevertheless, these stable states may not correspond to equilibria states of the game, because the players utilities has not been considered. In order to find Nash equilibria states, these utilities must be considered. This is beyond this chapter goals.

### **5.4. Section discussion**

14 Quantum Mechanics

man 1 are presented in 13 and 14.

above.

**Figure 7.** Quantum entropy corresponding to the situation where player 1 varies the probability *p* to apply strategy *U*<sup>1</sup>

state |0� into states that are linear superpositions of 0 and 1, representing states with different probabilities of choosing one woman or the other. On the other hand, strategies applied by

Figures 6 and 7 show two situations where the system entropy varies considerably as a function of the strategies the players use. Figure 2, for example, shows the case where the man 1 applies his strategies with fixed probability, just varying the angle *θ* while the other man (0) varies both strategies angle and the probability *p*. In all the cases we present here, in order to simplify the outcomes display, women density matrix doesn't change as explained

<sup>0</sup> <sup>=</sup> *<sup>U</sup>*(*θ*) (11)

<sup>1</sup> <sup>=</sup> *<sup>U</sup>*(*θ*)*U*(*π*) (12)

<sup>0</sup> <sup>=</sup> *<sup>U</sup>*(*θ*) (13)

<sup>1</sup> <sup>=</sup> *<sup>U</sup>*(−*θ*) (14)

*U*0

*U*0

*U*1

*U*1

For example, if both men choose the same woman with probability one, this is represented in Fig. 1 with *p* = 0. This situation is completely unstable because it is impossible for the

0

As a continuation of the analysis of a quantum formulation for the dating game that takes into account mutual progressive learning by representing women with quantum states whose associated amplitudes must be modified by men's selection strategies. we concentrate on the information associated to the problem. Since we deal with mixed strategies, the density matrix formalism is used to describe the system. Even though the decisions players make are based on their payoffs, past experiences, believes, etc., we are not interested in that causes but in the consequences of the decision they take, that is, the influence of the strategies they apply on the quantum system stability by means of the equivalence between maximum entropy states and those states that obey the Collective Welfare Principle that says that a system is stable only if it maximizes the welfare of the collective above the welfare of the individual. Maxima and minima entropy points are used to find characteristic strategies that lead to stable and unstable states. Nevertheless, in order to find Nash equilibria states, the players utilities must be considered.

Maxima and minima entropy do not depend only on the strategies of men but also on women preferences, reaching the highest value when they have no preferences, that is when they choose every man with equal probability. On the other hand, minimum entropy correspond to men betting all chips to a single woman, without giving a chance to other woman.

### **6. Entangled strategies**

The quantum dating market problem has been formulated as a two-sided bandit model [28], where in one side there are the men who must choose one "item" from the other side, which unlike the one side bandit, is composed by women able to reject the invitations.

The quantum formulation, which was presented in previous section, proceeds by assigning one basis of a Hilbert state space to each woman. As a consequence, if *Nw* is number of women playing, *Si* = {|0�, |1�, ..., |*Nw* − <sup>1</sup>�} are the states in the Hilbert space representing a man *i* decisions, therefore every man needs at least *log*2(*Nw*) qubits to identify each woman. The state of man *<sup>i</sup>* decisions is a quantum superposition of the base states, <sup>Ψ</sup>*<sup>i</sup>* <sup>=</sup> <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> *<sup>α</sup>j*|*j*�, where |*αj*| <sup>2</sup> is the probability that man *<sup>i</sup>* selects woman *<sup>j</sup>* when system state is <sup>Ψ</sup>*<sup>i</sup>* and |·� is known as Dirac's notation. The normalization condition is <sup>∑</sup>*Nw*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> <sup>|</sup>*αj*<sup>|</sup> <sup>2</sup> = 1. On the other side of the market, women receive men proposals and must decide which is the best, accept it and reject the others. Thus, <sup>Ψ</sup>*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*Nm*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> *<sup>α</sup>i*|*i*� is woman *<sup>j</sup>* acceptation state. Finally, combining proposals and acceptances the couples space which is the Kronecker product of all men's and all women's spaces is defined, i.e. *SC* = *SM SW*.

single actions. Therefore judging on the probability amplitudes, there is 50% probability that man 0 chooses woman 0 while man 1 chooses woman 1 and the other 50% for the other case. Since there is no way that men choose the same woman it is a state of mutual cooperation. Women acceptation state is initialized to *ψ<sup>w</sup>* = 0.5(|00� + |01� + |10� + |11�), implying that there is initially 25% chance that they choose the same man. In order to analyze the effect of woman behavior on men payoffs, for this and the following two cases, we consider that men decision state *ψ*<sup>0</sup> is invariant, while women strategies and their acceptation state *ψ<sup>w</sup>* change. Figure 8 shows the payoff for man 0 as a function of women strategies which are set as *Ui* = (*t* · *π*, *t* · *π*/2) for *t* ∈ [−1, 1] and *i* = 0, 1. Different strategies imply changes on women preferences, so some change in *Ui*(*θ*, *γ*) implies that woman *i* acceptation probability distribution is modified. Following [9], equation 16 represents man 0 payoff, where *P*<sup>00</sup> and

In the second example we introduce competition between men. The initial men state is

function of women strategies. Finally a third case is considered where men decision state is *ψ<sup>m</sup>* = 0.5(|00� + |01� + |10� + |11�). In this case men make independent choices choosing one of four possible options with equal probability. Figure 10 show the resulting payoffs as

As the figures show, the different scenarios present significant differences on payoff topology

The cooperative situation presents the highest payoff compared with the competitive and the independent ones as shown in figure 8. Figures 9 and 10 show that a better payoff may be obtained in the competitive setup compared to the independent one, but on the other hand, also a much lower payoff for other women strategies may be available. The independent decision scenario is thus characterized by lowest maxima and less variation on payoffs.

We have considered the dating market decision problem under the quantum mechanics point of view with the addition of entanglement between players states. Women and men are represented with quantum states and strategies are represented by means of unitary operator on a complex Hilbert space. Men final payoff, considering payoff as a measure of satisfaction, depends on the woman he is paired with. If men decision states are entangled, their actions are non-locally correlated modeling competition or cooperation scenarios. Three examples are shown in order to illustrate the more usual scenarios. In two of them the men strategies are correlated in a cooperative and a competitive way respectively. In the other example men strategies are independent. Although cooperative and competitive strategies can drive to higher payoffs, changing of women preferences on those scenarios can lead to very low payoffs. The independent decision scenario is characterized by less variation on payoffs.

<sup>2</sup> (|00� <sup>+</sup> <sup>|</sup>11�). Figure 9 depicts again the resulting payoffs for man 0 as a

\$*m*<sup>0</sup> = <sup>2</sup> · *<sup>P</sup>*<sup>00</sup> + <sup>5</sup> · *<sup>P</sup>*<sup>01</sup> (16)

Quantum Dating Market http://dx.doi.org/10.5772/53842 537

*P*<sup>01</sup> are his chances to be accepted for a date with woman 0 and 1 respectively.

given by *ψ*<sup>0</sup> =

√2

a function of woman strategies.

and maximum payoff values.

**6.2. Section discussion**

Men decision states are separable when there is no connection among players, that is, for instance, no man has any emotional bond with some other that could condition their actions, thus all men decision state, *<sup>ψ</sup>M*, is defined as *<sup>ψ</sup><sup>M</sup>* <sup>=</sup> *<sup>M</sup>*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *ψi*. The same reasoning corresponds to women states. On the other hand, if there is some relationship between two or more men, their actions are non-locally correlated, that is, their decisions are far from being independent. John Stuart Bell shown in 1966 that systems in entangled states exhibit correlations beyond those explainable by local "hidden" properties, or in other words, a non-local connection appears when two quantum particles are entangled, [35]. Therefore, we will study the case with correlation between agents by means of quantum entanglement, in other words, how harmful or beneficial can be for players knowing each other in advance.

As we mention in the previous section, players strategies are represented by unitary operators in quantum games. Starting the system in some state |Ψ0� at time *<sup>t</sup>*0, players apply their strategies *U* in order to modify it according to their preferences, that is modifying the probability amplitudes associated with each base state. Thus, evolution from the initial system state to some state |Ψ1� in time *<sup>t</sup>*<sup>1</sup> is given by |Ψ1� = *<sup>U</sup>*|Ψ0�. The strategy operator *U* arises from men and women preferences operators *UM* and *UW* respectively through *U* = *UM UW*, where *UM* is applied by men to the qubits that identify the women states, meanwhile the women action on the qubits that identify men states is given by *UW*.

In order to understand the problem we analyze here a simple example of two men and two women. Single players moves or strategies are associated with 2 × 2 unitary rotation operators *Ui*(*θ*, *<sup>γ</sup>*) applied on each one of their qubits (15), where 0 ≤ *<sup>θ</sup>* ≤ *<sup>π</sup>* and 0 ≤ *<sup>γ</sup>* ≤ *<sup>π</sup>*/2. Men choices are coded by states |*w*0� = |0� and |*w*1� = |1�, meanwhile women must decide between men |*m*0� = |0� and |*m*1� = |1�. Since any qubit operation can be decomposed as a product of rotations, strategies combinations and possible outcomes are infinite. As a consequence, focusing on men relationship, we study three relevant cases. We suppose, as a measure of satisfaction, that men receive some payoff *pwi* if accepted by woman *wi*, so for the example we have considered that *pw*<sup>0</sup> = 2 and *pw*<sup>1</sup> = 5.

$$\mathcal{U}(\theta,\gamma) = \begin{pmatrix} e^{i\gamma}\cos(\theta/2) & \sin(\theta/2) \\ -\sin(\theta/2) & e^{-i\gamma}\cos(\theta/2) \end{pmatrix} \tag{15}$$

#### **6.1. Results**

For the first case, let us consider *ψ*<sup>0</sup> = 2 <sup>2</sup> (|01� <sup>+</sup> <sup>|</sup>10�) as the initial state of men decisions system, where the left qubit of *ψ*<sup>0</sup> is representing man 0 election while the right one represents man 1 choice. As men states are entangled, it is not possible to uncouple their

√

single actions. Therefore judging on the probability amplitudes, there is 50% probability that man 0 chooses woman 0 while man 1 chooses woman 1 and the other 50% for the other case. Since there is no way that men choose the same woman it is a state of mutual cooperation. Women acceptation state is initialized to *ψ<sup>w</sup>* = 0.5(|00� + |01� + |10� + |11�), implying that there is initially 25% chance that they choose the same man. In order to analyze the effect of woman behavior on men payoffs, for this and the following two cases, we consider that men decision state *ψ*<sup>0</sup> is invariant, while women strategies and their acceptation state *ψ<sup>w</sup>* change. Figure 8 shows the payoff for man 0 as a function of women strategies which are set as *Ui* = (*t* · *π*, *t* · *π*/2) for *t* ∈ [−1, 1] and *i* = 0, 1. Different strategies imply changes on women preferences, so some change in *Ui*(*θ*, *γ*) implies that woman *i* acceptation probability distribution is modified. Following [9], equation 16 represents man 0 payoff, where *P*<sup>00</sup> and *P*<sup>01</sup> are his chances to be accepted for a date with woman 0 and 1 respectively.

$$\pounds\_{m0} = \text{2} \cdot P\_{00} + \text{5} \cdot P\_{01} \tag{16}$$

In the second example we introduce competition between men. The initial men state is given by *ψ*<sup>0</sup> = √2 <sup>2</sup> (|00� <sup>+</sup> <sup>|</sup>11�). Figure 9 depicts again the resulting payoffs for man 0 as a function of women strategies. Finally a third case is considered where men decision state is *ψ<sup>m</sup>* = 0.5(|00� + |01� + |10� + |11�). In this case men make independent choices choosing one of four possible options with equal probability. Figure 10 show the resulting payoffs as a function of woman strategies.

As the figures show, the different scenarios present significant differences on payoff topology and maximum payoff values.

The cooperative situation presents the highest payoff compared with the competitive and the independent ones as shown in figure 8. Figures 9 and 10 show that a better payoff may be obtained in the competitive setup compared to the independent one, but on the other hand, also a much lower payoff for other women strategies may be available. The independent decision scenario is thus characterized by lowest maxima and less variation on payoffs.

#### **6.2. Section discussion**

16 Quantum Mechanics

where |*αj*|

*U* = *UM*

**6.1. Results**

For the first case, let us consider *ψ*<sup>0</sup> =

The state of man *<sup>i</sup>* decisions is a quantum superposition of the base states, <sup>Ψ</sup>*<sup>i</sup>* <sup>=</sup> <sup>∑</sup>*Nw*−<sup>1</sup>

of the market, women receive men proposals and must decide which is the best, accept it and

proposals and acceptances the couples space which is the Kronecker product of all men's

Men decision states are separable when there is no connection among players, that is, for instance, no man has any emotional bond with some other that could condition their

corresponds to women states. On the other hand, if there is some relationship between two or more men, their actions are non-locally correlated, that is, their decisions are far from being independent. John Stuart Bell shown in 1966 that systems in entangled states exhibit correlations beyond those explainable by local "hidden" properties, or in other words, a non-local connection appears when two quantum particles are entangled, [35]. Therefore, we will study the case with correlation between agents by means of quantum entanglement, in other words, how harmful or beneficial can be for players knowing each other in advance. As we mention in the previous section, players strategies are represented by unitary operators in quantum games. Starting the system in some state |Ψ0� at time *<sup>t</sup>*0, players apply their strategies *U* in order to modify it according to their preferences, that is modifying the probability amplitudes associated with each base state. Thus, evolution from the initial system state to some state |Ψ1� in time *<sup>t</sup>*<sup>1</sup> is given by |Ψ1� = *<sup>U</sup>*|Ψ0�. The strategy operator *U* arises from men and women preferences operators *UM* and *UW* respectively through

known as Dirac's notation. The normalization condition is <sup>∑</sup>*Nw*−<sup>1</sup>

actions, thus all men decision state, *<sup>ψ</sup>M*, is defined as *<sup>ψ</sup><sup>M</sup>* <sup>=</sup> *<sup>M</sup>*−<sup>1</sup>

reject the others. Thus, <sup>Ψ</sup>*<sup>j</sup>* <sup>=</sup> <sup>∑</sup>*Nm*−<sup>1</sup>

and all women's spaces is defined, i.e. *SC* = *SM*

<sup>2</sup> is the probability that man *<sup>i</sup>* selects woman *<sup>j</sup>* when system state is <sup>Ψ</sup>*<sup>i</sup>* and |·� is

*SW*.

*UW*, where *UM* is applied by men to the qubits that identify the women states,

 *eiγcos*(*θ*/2) *sin*(*θ*/2) <sup>−</sup>*sin*(*θ*/2) *<sup>e</sup>*−*iγcos*(*θ*/2)

<sup>2</sup> (|01� <sup>+</sup> <sup>|</sup>10�) as the initial state of men decisions

meanwhile the women action on the qubits that identify men states is given by *UW*.

√2

system, where the left qubit of *ψ*<sup>0</sup> is representing man 0 election while the right one represents man 1 choice. As men states are entangled, it is not possible to uncouple their

*wi*, so for the example we have considered that *pw*<sup>0</sup> = 2 and *pw*<sup>1</sup> = 5.

*U*(*θ*, *γ*) =

In order to understand the problem we analyze here a simple example of two men and two women. Single players moves or strategies are associated with 2 × 2 unitary rotation operators *Ui*(*θ*, *<sup>γ</sup>*) applied on each one of their qubits (15), where 0 ≤ *<sup>θ</sup>* ≤ *<sup>π</sup>* and 0 ≤ *<sup>γ</sup>* ≤ *<sup>π</sup>*/2. Men choices are coded by states |*w*0� = |0� and |*w*1� = |1�, meanwhile women must decide between men |*m*0� = |0� and |*m*1� = |1�. Since any qubit operation can be decomposed as a product of rotations, strategies combinations and possible outcomes are infinite. As a consequence, focusing on men relationship, we study three relevant cases. We suppose, as a measure of satisfaction, that men receive some payoff *pwi* if accepted by woman

*<sup>j</sup>*=<sup>0</sup> <sup>|</sup>*αj*<sup>|</sup>

*<sup>i</sup>*=<sup>0</sup> *<sup>α</sup>i*|*i*� is woman *<sup>j</sup>* acceptation state. Finally, combining

*<sup>j</sup>*=<sup>0</sup> *<sup>α</sup>j*|*j*�,

(15)

<sup>2</sup> = 1. On the other side

*<sup>i</sup>*=<sup>1</sup> *ψi*. The same reasoning

We have considered the dating market decision problem under the quantum mechanics point of view with the addition of entanglement between players states. Women and men are represented with quantum states and strategies are represented by means of unitary operator on a complex Hilbert space. Men final payoff, considering payoff as a measure of satisfaction, depends on the woman he is paired with. If men decision states are entangled, their actions are non-locally correlated modeling competition or cooperation scenarios. Three examples are shown in order to illustrate the more usual scenarios. In two of them the men strategies are correlated in a cooperative and a competitive way respectively. In the other example men strategies are independent. Although cooperative and competitive strategies can drive to higher payoffs, changing of women preferences on those scenarios can lead to very low payoffs. The independent decision scenario is characterized by less variation on payoffs.

**Figure 8.** Payoff for man 0 if men never choose the same woman, as function of women acceptation strategies. For the example *γ* varies as *θ*/2.

**Figure 10.** Payoff for man 0 if men choose without restrictions, states are not entangled, as function of women acceptation

Quantum Dating Market http://dx.doi.org/10.5772/53842 539

The dating market problem may be included in a more general category of matching problems where the elements of two sets have to be matched by pairs. Matching problems have broad implications not only in economic and social contexts but in other very different research fields such as communications engineering or molecular biology, for example. The main goal of this chapter is to introduce and analyze a quantum formulation for the dating market game, whose nearest classical antecedent is the Stable Marriage Problem. Players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the Hamiltonian of the system. Significant outcomes arise when classic players play against quantum ones. For instance, when a quantum player uses Grover search algorithm as her strategy, her winning probabilities grow with increasing number of players, but none leads in a "one on one" game. Besides, from stability point of view, maxima and minima entropy points are used to find characteristic strategies that lead to unstable and stable states, resulting the highest entropy values when women have no preferences, that is, when they choose every man with equal probability. On the other hand, minimum entropy correspond to men betting all chips to a single woman, without giving a chance to other woman. Finally, to model relationships between people that may correlate their decisions, our model consider the situation when men decision states are entangled and their actions are non-locally correlated modeling competition or cooperation scenarios. One of the main outcomes is for example that, although cooperative and competitive strategies can drive to higher payoffs, changing of women preferences on those scenarios can lead to

strategies. For the example *γ* varies as *θ*/2.

**7. Final remarks**

very low payoffs.

**Figure 9.** Payoff for man 0 if men always choose the same woman, as function of women acceptation strategies. For the example *γ* varies as *θ*/2.

**Figure 10.** Payoff for man 0 if men choose without restrictions, states are not entangled, as function of women acceptation strategies. For the example *γ* varies as *θ*/2.

### **7. Final remarks**

18 Quantum Mechanics

example *γ* varies as *θ*/2.

example *γ* varies as *θ*/2.

**Figure 8.** Payoff for man 0 if men never choose the same woman, as function of women acceptation strategies. For the

**Figure 9.** Payoff for man 0 if men always choose the same woman, as function of women acceptation strategies. For the

The dating market problem may be included in a more general category of matching problems where the elements of two sets have to be matched by pairs. Matching problems have broad implications not only in economic and social contexts but in other very different research fields such as communications engineering or molecular biology, for example. The main goal of this chapter is to introduce and analyze a quantum formulation for the dating market game, whose nearest classical antecedent is the Stable Marriage Problem. Players strategies are represented by unitary operators, which in quantum mechanics are also known as evolution operators related to the Hamiltonian of the system. Significant outcomes arise when classic players play against quantum ones. For instance, when a quantum player uses Grover search algorithm as her strategy, her winning probabilities grow with increasing number of players, but none leads in a "one on one" game. Besides, from stability point of view, maxima and minima entropy points are used to find characteristic strategies that lead to unstable and stable states, resulting the highest entropy values when women have no preferences, that is, when they choose every man with equal probability. On the other hand, minimum entropy correspond to men betting all chips to a single woman, without giving a chance to other woman. Finally, to model relationships between people that may correlate their decisions, our model consider the situation when men decision states are entangled and their actions are non-locally correlated modeling competition or cooperation scenarios. One of the main outcomes is for example that, although cooperative and competitive strategies can drive to higher payoffs, changing of women preferences on those scenarios can lead to very low payoffs.

### **Author details**

C. M. Arizmendi and O. G. Zabaleta

Facultad de Ingeniería, Universidad Nacional de Mar del Plata, Argentina

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**Chapter 23**

**Quantization as Selection**

Peter Enders

**1. Introduction**

http://dx.doi.org/10.5772/53847

Planck's theory." (p. 7)

**Rather than Eigenvalue Problem**

Additional information is available at the end of the chapter

following "principal assumptions" for a theory of atoms.

stationary states cannot be treated on that basis.

The experimental, in particular, spectroscopic results about atoms brought Bohr [3] to the

**1.** That the dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different

**2.** That the latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by

Assumption (1) states a strange contraposition of the conservation and of the changes of stationary states. Indeed, the conservation of energy - a generalization of Newton's Law 1 holds true quite general, while the change of stationary states along Newton's Law 2 is restricted to classical mechanics (CM). A smooth transition from representations of CM, which axiomatically fix not only the conditions of conservation, but also the manner of change of stationary states and the equation(s) of motion, respectively (Newton, Lagrange, Hamilton), is impossible, as observed also by Heisenberg [30] and Schrödinger [43]. However, in Leonhard Euler's representation of CM [24][25], only the conditions of conservation of stationary states are fixed, while their change has to be described according to the situation under consideration. This makes it suitable to serve as starting point for a smooth transition from classical to quantum mechanics (QM). Euler's principles of the change of stationary states of bodies will be generalized to classical conservative systems as well as to quantum systems. The latter will be used for deriving the time-dependent from the time-independent Schrödinger equation.

> © 2013 Enders; 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 Enders; 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.


### **Chapter 23**
