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Physics of Particles and Nuclei

Kinematic Properties of a Quantum System in Phase Space

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, P. V. Afonin, Полякова Римма Васильевна, M. B. Sadovnikova

Published May 25, 2026• 0 citations
Publisher

Abstract

The paper describes the kinematic properties of quantum systems in terms of phase space. Despite the Heisenberg uncertainty principle, the consideration of quantum systems in phase space is widely used with the apparatus of quasi-probability density functions, such as the Wigner function and the gauge-invariant Weyl–Stratonovich function. The term “quasi-density” refers to the presence of phase-space regions with negative values of the probability density function and arouses interest in possible physical interpretations of such values. The kinematic properties of the systems are considered within the Wigner–Vlasov formalism, which enables to link the classical and quantum descriptions through a unified mathematical apparatus using the distribution function in phase space.

DOI
10.1134/s1063779625701564

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The paper describes the kinematic properties of quantum systems in terms of phase space. Despite the Heisenberg uncertainty principle, the consideration of quantum systems in phase space is widely used with the apparatus of quasi-probability density functions, such as the Wigner function and the gauge-invariant Weyl–Stratonovich function. The term “quasi-density” refers to the presence of phase-space regions with negative values of the probability density function and arouses interest in possible physical interpretations of such values. The kinematic properties of the systems are considered within the Wigner–Vlasov formalism, which enables to link the classical and quantum descriptions through a unified mathematical apparatus using the distribution function in phase space.
The paper describes the kinematic properties of quantum systems in terms of phase space. Despite the Heisenberg uncertainty principle, the consideration of quantum systems in phase space is widely used with the apparatus of quasiprobability density functions, such as the Wigner function and the gauge-invariant Weyl–Stratonovich function. The term “quasi-density” refers to the presence of phase regions with negative values of the probability density function and arouses interest in possible physical interpretations of such values. The kinematic properties of the systems are considered within the Wigner–Vlasov formalism, which enables to link the classical and quantum descriptions through a unified mathematical apparatus using the distribution function in phase space.
The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)