Draft:Quantum phase space approach

• Comment: You have a declared conflict of interest it is best NOT to move drafts to main space, but instead submit for review. Theroadislong (talk) 11:09, 1 January 2025 (UTC)

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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 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. Within the quantum phase space approach, the phase space itself is quantum. The QPS concept therefore permits, among other things, to avoid the use of non-positive definite probability distributions.

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 values of the coordinates and momenta for given values of the 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 degree of freedom , for instance, is defined by a coordinate $x$ and the corresponding conjugate momentum $p$. In the framework of quantum mechanics, $x$ and $p$ correspond respectively to the eigenvalues of the coordinate operators $X$ and the momentum operator $P$. The corresponding eigenvalue equations are

${\begin{cases}X|x\rangle =x|x\rangle \\P|p\rangle =p|p\rangle \end{cases}}$

in which $|x\rangle$ and $|p\rangle$ are respectively the coordinates and momentum eigenstates : $|x\rangle$ is the quantum state of the system if the value of its coordinate is equals to $x$ and $|p\rangle$ is its quantum state when the value of its momentum is equal to $p$. The mean values $\langle x\rangle$, $\langle p\rangle$ and statistical variances ${\bigl (}\sigma _{x}{\bigr )}^{2}$, ${\bigl (}\sigma _{p}{\bigr )}^{2}$ of the coordinate and momentum of the system corresponding to a given quantum state $|\psi \rangle$ are defined by the following relations

${\begin{cases}\langle x\rangle =\langle X\rangle =\langle \psi |X|\psi \rangle \\\langle p\rangle =\langle P\rangle =\langle \psi |P|\psi \rangle \\(\sigma _{x})^{2}=\langle (X-\langle x\rangle )^{2}\rangle =\langle \psi |(X-\langle x\rangle )^{2}|\psi \rangle \\(\sigma _{p})^{2}=\langle (P-\langle p\rangle )^{2}\rangle =\langle \psi |(P-\langle p\rangle )^{2}|\psi \rangle \end{cases}}$

As the momentum and coordinate operators $P$ and $X$ satisfy the canonical commutation relation (CCR)

$[P,X]=-i\hbar$

in which $\hbar$ is the reduced Planck constant , it can be shown that one has the following inequality

$(\sigma _{x})$$(\sigma _{p})$$\geq {\frac {\hbar }{2}}$

This formal inequality relating the standard deviation $\sigma _{x}$ of coordinate and the standard deviation $\sigma _{p}$ of momentum is the coordinate-momentum 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 $\{(p,x)\}$ of the possible values of the pairs $(p,x)$. 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.

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 $|\langle z\rangle \rangle$ corresponding to  Gaussian-like wavefunctions . The explicit expression of these wavefunctions in coordinate representation is given by the following relation

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

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

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

As the state $|\langle z\rangle \rangle$ saturate the uncertainty principle, one has the following relation

$A$$B$$=({\frac {\hbar }{2}})^{2}$

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

$Z|\langle z\rangle \rangle =\langle z\rangle |\langle z\rangle \rangle$

with

$\langle z\rangle =\langle p\rangle -{\frac {2i}{\hbar }}B\langle x\rangle$

As will be discussed later, the states $|\langle z\rangle \rangle$ are also the basic quantum states which correspond to wavefunctions that are covariants under the action of the group formed by multidimensional Linear Canonical Transformations . They can also be considered as analogous to what are called coherent states and squeezed states in the literature

Quantum phase space

The quantum phase space can be defined as the set $\{(\langle p\rangle ,\langle x\rangle )\}$ of all possible values of the pair $(\langle p\rangle ,\langle x\rangle )$, or equivalently as the set $\{\langle z\rangle \}$ of all possible values of $\langle z\rangle$, for a given value of the momentum statistical variance $B$ . 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 $B$ can be related to thermodynamics parameters like temperature, pressure and volume shape and size at thermodynamic equilibrium . It follows that the structure of the QPS itself may depend on thermodynamic constraints.

References

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