View source for Draft:Quantum phase space approach

{{AFC submission|d|nn|u=Rshon13|ns=118|decliner=Ldm1954|declinets=20250108132102|ts=20250103072446}} <!-- Do not remove this line! -->

{{AFC comment|1=Two issues:
1. This article currently fails as [[WP:TOOSOON]]. Wikipedia is very conservative, and only includes topics which are well established and (for science) have been cited/discussed by many groups. While the formalism here might be correct, it will only become notable as far as Wikipedia is concerned in a few years when/if it is widely accepted by the community. Just being published is not enough.
2. A quick Google search finds the term "quantum phase space". One I found quickly is https://doi.org/10.1016/0370-1573(95)00007-4 but there are many other well cited papers which use the term. It is unclear whether these are different. If similar then older precedents should be cited; if different the name here is problematic.
I strongly suggest waiting for a few years unless you can find significant secondary sources now. [[User:Ldm1954|Ldm1954]] ([[User talk:Ldm1954|talk]]) 13:21, 8 January 2025 (UTC)}}

{{AFC comment|1=You have a declared conflict of interest it is best NOT to move drafts to main space, but instead submit for review. [[User:Theroadislong|Theroadislong]] ([[User talk:Theroadislong|talk]]) 11:09, 1 January 2025 (UTC)}}

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{{Short description|Concept in quantum physics}}
{{Draft topics|physics}}
{{AfC topic|stem}}

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Quantum phase space (QPS) is an extension of the classical [[phase space]] concept to the realm of [[quantum physics]]. It corresponds to one of the approaches that has been considered to tackle the challenge of integrating the phase space concept into quantum mechanics, given the constraints imposed by [[uncertainty principle]]. This approach is distinct from the [[Phase-space formulation]]<ref>R. P Rundle, M. J Everitt (2021): "''Overview of the phase space formulation of quantum mechanics with application to quantum technologies''", https://onlinelibrary.wiley.com/doi/10.1002/qute.202100016, [[arxiv:2102.11095|arXiv:2102.11095 [quant-ph]]]</ref> which offers an alternative quantum mechanical framework that continues to employ a classical phase space. This use of a classical phase space for the formulation of quantum mechanics requires the introduction of non-positive definite probability distributions<ref>A. Kenfack, K. Zyczkowski (2004):"''Negativity of the Wigner function as an indicator of non-classicality", [https://iopscience.iop.org/article/10.1088/1464-4266/6/10/003 J. Opt. B: Quantum Semiclass. Opt. 6 396]''</ref>. Within the '''quantum phase space approach''', the phase space itself is quantum<ref name=":0">R.T. Ranaivoson et al (2022) : "''Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates''" [https://iopscience.iop.org/article/10.1088/2399-6528/ac8520/pdf J. Phys. Commun. 6 095010], [[arxiv:2008.10602|arXiv:2008.10602 [quant-ph]]]</ref><ref name=":1">R.T. Ranaivoson et al (2024) : "Quantum phase space symmetry and sterile neutrinos". [https://www.researchgate.net/publication/378907651_Quantum_Phase_Space_symmetry_and_Sterile_Neutrinos Read on ResearchGate]</ref>. The QPS concept therefore permits, among other things, to avoid the use of non-positive definite probability distributions<ref name=":4" />. 

In [[classical mechanics]], phase space is defined as the set of all possible exact values of the coordinates and momenta associated with the mechanical description of a system. Quantum mechanics, however, introduces the [[uncertainty principle]], limiting the precision with which these quantities can be simultaneously determined. This poses a challenge in directly applying the classical phase space concept to quantum systems. The main idea behind the concept of QPS is to define it as the set of all possible [[Mean|mean values]] of the coordinates and momenta for given values of the [[Standard deviation|uncertainties]]. When the uncertainties are taken to be zero (in the [[classical limit]]), the QPS reduces to its classical counterpart. 

Connections between the concept of QPS, [[statistical mechanics]] and [[thermodynamics]] have been explored. It was shown, for instance, that at thermodynamic equilibrium, quantum uncertainties can be related to thermodynamic parameters. Other works have also considered the [[relativistic]] generalization of the QPS concept and its relation to multidimensional [[linear canonical transformation]] for application in [[particle physics]]. 

== Mathematical definition of the quantum phase space ==

=== Uncertainty relation and the problem of phase space in quantum mechanics ===
In [[Hamiltonian mechanics]], a system is described with its [[canonical coordinates]] . The mechanical state of a  system with one [[Degrees of freedom (mechanics)|degree of freedom]] , for instance, is defined by a [[Canonical coordinates|coordinate]] <math>x</math> and the corresponding [[Canonical coordinates|conjugate momentum]] '''<math>p</math>'''. In the framework of [[quantum mechanics]], <math>x</math> and '''<math>p</math>''' correspond respectively to the eigenvalues of the [[Position operator|coordinate operators]] <math>X</math> and the [[momentum operator]] <math>P</math>. The corresponding eigenvalue equations are<ref>{{Cite web |title=Recap- Position and Momentum States |url=https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/01%3A_Scattering_Theory/1.02%3A_Recap-_Position_and_Momentum_States |access-date=29 December 2024 |website=https://phys.libretexts.org/|date=14 April 2021 }}</ref><ref>{{Cite web |title=Quantum Theory I, Lecture 5 Notes |url=https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |access-date=29 December 2024 |website=https://ocw.mit.edu/courses}}</ref>

<math>\begin{cases} X|x\rangle =x|x\rangle\\ P|p\rangle =p|p\rangle \end{cases}</math>

in which <math>|x\rangle</math> and  <math>|p\rangle</math> are respectively the coordinates and momentum [[eigenstates]] : <math>|x\rangle</math> is the quantum state of the system if the value of its coordinate is equals to  <math>x</math> and <math>|p\rangle</math> is its quantum state when the value of its momentum is equal to  '''<math>p</math>'''. The mean values <math>\langle x\rangle</math>, <math>\langle p\rangle</math>  and statistical variances <math>\bigl( \sigma_x\bigr)^2</math>, <math>\bigl( \sigma_p\bigr)^2</math> of the coordinate and momentum of the system corresponding to a given quantum state <math>|\psi\rangle</math> are defined by the following relations 

<math>\begin{cases} \langle x\rangle =\langle X\rangle = \langle \psi|X|\psi\rangle \\ \langle p\rangle =\langle P\rangle = \langle \psi|P|\psi\rangle \\ (\sigma_x)^2 =\langle (X-\langle x\rangle)^2\rangle = \langle \psi|(X-\langle x\rangle)^2|\psi\rangle \\(\sigma_p)^2 =\langle (P-\langle p\rangle)^2\rangle = \langle \psi|(P-\langle p\rangle)^2|\psi\rangle \end{cases}</math>  

As the momentum and coordinate operators <math>P</math> and <math>X</math> satisfy the [[canonical commutation relation]] (CCR) 

<math>[P,X]=-i\hbar</math> 

in which <math>\hbar</math> is the reduced [[Planck constant]] , [[Uncertainty principle|it can be shown]] that one has the following inequality 

<math>(\sigma_x)</math><math>(\sigma_p)</math><math>\geq \frac{\hbar}{2}</math> 

This formal inequality relating the [[standard deviation]] <math>\sigma_x</math> of coordinate  and the standard deviation <math>\sigma_p</math> of momentum is the coordinate-momentum [[Uncertainty principle|uncertainy relation]]. According to this relation, there is a limit to the precision with which the values of the coordinate and momentum can be simultaneously known. However, in classical physics, phase space is defined as the set <math>\{(p,x)\}</math>  of the possible values of the pairs <math>(p,x)</math>. It follows that it is not trivial to extend the definition of phase space from classical to quantum physics. The concept of QPS gives a rigorous solution to this problem.<ref name=":0" />

=== Joint momentum-coordinate quantum states ===
The introduction of the concept of Quantum Phase Space (QPS) needs the search for some kind of joint momentum-coordinate quantum states that are compatible with the CCR and the [[uncertainty principle]]. It can be shown that the kind of states satisfying these criterion and which saturate the uncertainty relation are the states denoted <math>|\langle z \rangle\rangle</math>  corresponding to  Gaussian-like wavefunctions <ref name=":0" /><ref name=":2">R. T. Ranaivoson et al (2021): "''Linear Canonical Transformations in Relativistic Quantum Physics''" [https://iopscience.iop.org/article/10.1088/1402-4896/abeba5 Phys. Scr. 96, 065204], [[arxiv:1804.10053|arXiv:1804.10053 [quant-ph]]]</ref><ref name=":1" />. The explicit expression of these wavefunctions in coordinate representation is given by the following relation

<math>\varphi(x)=\langle x|\langle z\rangle\rangle =\frac{e^{- \frac{B}{(\hbar)^2}(x-\langle x\rangle)^2+\frac{i}{\hbar}\langle p \rangle x}}{(2\pi A)^\tfrac{1}{4}}</math>

in which <math>\langle x\rangle</math>, <math>\langle p\rangle</math> ,  <math>B</math> and <math>A</math> are respectively the mean values  and statistical variances of the coordinate and momentum corresponding to the quantum state <math>|\langle z \rangle\rangle</math> itself

<math>\begin{cases} \langle x\rangle =\langle X\rangle = \langle \langle z \rangle|X|\langle z\rangle\rangle \\ \langle p\rangle =\langle P\rangle = \langle \langle z \rangle|P|\langle z \rangle \rangle \\ A=\langle (X-\langle x\rangle)^2\rangle = \langle \langle z\rangle|(X-\langle x\rangle)^2|\langle z \rangle \rangle \\ B =\langle (P-\langle p\rangle)^2\rangle = \langle \langle z \rangle|(P-\langle p\rangle)^2|\langle z \rangle\rangle \end{cases}</math>

As the state  <math>|\langle z \rangle\rangle</math> saturate the uncertainty principle, one has the following relation

<math>A</math><math>B</math><math>=(\frac{\hbar}{2})^2</math>

It can be shown that a state <math>|\langle z \rangle\rangle</math> is an eigenstate of the operator <math>Z=P - \frac{2i}{\hbar}BX </math>. The corresponding eigenvalue equation is

<math>Z|\langle z \rangle\rangle =\langle z \rangle|\langle z \rangle\rangle</math>

with

<math>\langle z \rangle = \langle p \rangle - \frac{2i}{\hbar} B\langle x\rangle </math>

As will be discussed later, the states  <math>|\langle z \rangle\rangle</math> are also the basic quantum states which correspond to wavefunctions that are covariants under the action of the [[Group action|group]] formed by multidimensional [[Linear canonical transformation|Linear Canonical Transformations]]<ref name=":2" /> . They can also be considered as analogous to what are called [[coherent state]]s and [[Squeezed coherent state|squeezed states]] in the literature<ref>T.G. Philbin (2014) : "''Generalized coherent states"''  , [https://pubs.aip.org/aapt/ajp/article-abstract/82/8/742/1057585/Generalized-coherent-states?redirectedFrom=fulltext Am. J. Phys. 82, 742] , [[arxiv:1311.1920|arXiv:1311.1920 [quant-ph]]]</ref><ref>B. Bagchi, R. Ghosh, A. Khare (2020) : "''A pedestrian introduction to coherent and squeezed states''", [https://www.worldscientific.com/doi/abs/10.1142/S0217751X20300112 Int. J. Mod. Phys. A35, 2030011], [[arxiv:2004.08829|arXiv:2004.08829 [quant-ph]]]</ref>

=== Quantum phase space ===
The quantum phase space can be defined as the set <math>\{(\langle p \rangle ,\langle x\rangle)\}</math> of all possible values of the pair <math>(\langle p \rangle ,\langle x\rangle)</math>, or equivalently as the set  <math>\{\langle z\rangle\}</math> of all possible values of <math>\langle z\rangle</math>, for a given value of the momentum statistical variance <math>B</math> <ref name=":0" /><ref name=":4">R.T Ranaivoson et al (2023) : "''Highlighting relations between Wave-particle duality, Uncertainty principle, Phase space and Microstates''", [[arxiv:2205.08538|arXiv:2205.08538 [quant-ph]]]</ref>. It follows from this definition that the structure of the quantum phase space depends explicitly on the value of the momentum statistical variance. It is this explicit dependence that makes this definition naturally compatible with the uncertainty principle. It can also be remarked here that, according to some works, the momentum statistical variance <math>B</math> can be related to thermodynamics parameters like temperature, pressure and volume shape and size at thermodynamic equilibrium <ref name=":3">R. H. M. Ravelonjato et al (2023) [https://link.springer.com/article/10.1007/s10701-023-00727-5 ''Found Phys'' '''53''', 88], [[arxiv:2302.13973|arXiv:2302.13973 [cond-mat.stat-mech]]]</ref> . It follows that the structure of the QPS itself may depend on thermodynamic constraints.

== References ==

{{reflist}}

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[[:Category:Quantum mechanics]]
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