Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Relationship between the Wigner function and the probability density function in quantum phase space representation

Qian Shu Li, Gong Min Wei, Li Qiang Lü

Published Aug 20, 2004• 26 citations
Publisher

Abstract

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

DOI
10.1103/physreva.70.022105

Related papers

On quantum mechanical phase-space wave functions

Joachim J. Wl odarz

An approach to quantum mechanics based on the notion of a phase-space wave function is proposed within the Weyl–Wigner–Moyal representation. It is shown that the Schrödinger equation for the phase-space wave function is equivalent to the quantum Liouville equation for the Wigner distribution function. The relationship to the recent results by Torres-Vega and Frederick [J. Chem. Phys. 98, 3103 (1993)] is also discussed.

The Journal of Chemical Physics•May 15, 1994•23 citations

The Wigner representation of quantum mechanics

Valerian I. Tatarskii

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.

Soviet Physics Uspekhi•Apr 30, 1983•483 citations

Rigorous solutions of a particle in δ potential fields in phase space

Jun Lu, Qian‐Shu Li, Zheng Sun

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrödinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.

Physical Chemistry Chemical Physics•Jan 1, 2001•6 citations

Schrödinger like equation for wavelets

Arturo Zúñiga-Segundo, H. M. Moya-Cessa, Francisco Soto

An explicit phase space representation of the wave function is build based on a wavelet transformation. The wavelet transformation allows us to understand the relationship between s − ordered Wigner function, (or Wigner function when s = 0), and the Torres-Vega-Frederick’s wave functions. This relationship is necessary to find a general solution of the Schrödinger equation in phase-space.

AIP Advances•Jan 1, 2016•4 citations

- Phase Space Picture and Canonical Transformations

S. Rajasekar, R. Velusamy

Given the wave function ψ(x, t) we compute the probability density function ρ(x, t) = |ψ(x, t)|2 and easily visualize the distribution. Knowing ψ(x, t) the distribution in momentum px is given by σ(px, t) = |χ(px, t)|2 = ∣∣∣ 1√ 2π ψ(x, t)e−ikx dx ∣∣∣2 , (6.1) where χ(px, t) is the momentum wave function. The momentum distribution is difficult to visualize from the given ψ(x, t). Thus, we wish to have a function that can display the probability distribution in the variables x and px simultaneously. Recall that, in classical mechanics phase space is used to visualize the solutions of systems. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture [1-8]. The idea is that every state of a quantum system is describable by a distribution (or function) on the classical phase space. In the phase space picture, starting from the Schrödinger wave function, we are able to construct the Wigner distribution function in terms of x and px. It does not represent a joint probability distribution for x and px because the uncertainty principle disallows the simultaneous determination of these variables with desired accuracy.

CRC Press eBooks•Dec 10, 2014•0 citations

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

Chol Jong, Jin-Sim Kim, Shin-Hyok Jon +1 more

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

World Journal of Mechanics•Jan 1, 2023•1 citation
This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.
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.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrödinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.
Given the wave function ψ(x, t) we compute the probability density function ρ(x, t) = |ψ(x, t)|2 and easily visualize the distribution. Knowing ψ(x, t) the distribution in momentum px is given by σ(px, t) = |χ(px, t)|2 = ∣∣∣ 1√ 2π ψ(x, t)e−ikx dx ∣∣∣2 , (6.1) where χ(px, t) is the momentum wave function. The momentum distribution is difficult to visualize from the given ψ(x, t). Thus, we wish to have a function that can display the probability distribution in the variables x and px simultaneously. Recall that, in classical mechanics phase space is used to visualize the solutions of systems. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture [1-8]. The idea is that every state of a quantum system is describable by a distribution (or function) on the classical phase space. In the phase space picture, starting from the Schrödinger wave function, we are able to construct the Wigner distribution function in terms of x and px. It does not represent a joint probability distribution for x and px because the uncertainty principle disallows the simultaneous determination of these variables with desired accuracy.
The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.