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Journal of Modern Optics

Wigner distribution for operators at different times

Leon W. Cohen

Published Oct 25, 2006• 0 citations
Publisher

Abstract

The paper considers the problem of finding the joint probability distribution of position, P(q 0,q 1,…, qN ), at the different times t 0,t 1, …, tN . The problem is addressed both classically and quantum mechanically. The characteristic function approach is used and it is shown that for certain cases the quantum mechanical answer is the same as the classical answer with replacement of the initial classical probability distribution by the Wigner distribution of the initial wave function. This is the case for taking a certain form of the quantum characteristic function. Other choices are also discussed.

DOI
10.1080/09500340600952051

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The paper considers the problem of finding the joint probability distribution of position, P(q 0,q 1,…, qN ), at the different times t 0,t 1, …, tN . The problem is addressed both classically and quantum mechanically. The characteristic function approach is used and it is shown that for certain cases the quantum mechanical answer is the same as the classical answer with replacement of the initial classical probability distribution by the Wigner distribution of the initial wave function. This is the case for taking a certain form of the quantum characteristic function. Other choices are also discussed.
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 fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint “distribution” derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists of some observations concerning the axiomatic foundations of quantum mechanics.
The Wigner function has been studied for more than eight decades. Historically, it was developed from the Schrodinger wave equation, although it could have been developed on its own right. The use of the Wigner function is particularly important in scattering problems and clearly shows the transition to the classical world. In general, the Wigner function is not a positive definite function. One of the interesting aspects of the Wigner function is the ability to transfer many of the results of classical transport theory into quantum approaches merely by replacing the Boltzmann distribution with the Wigner function. The Wigner function for either of the two oscillators can be found by integrating out the coordinates of the other oscillator. Modern approaches with the Wigner function provide a distinct formulation that is recognized as equivalent to, but is a different alternative to normal operator-based quantum mechanics.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.
Quantum phase‐space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook‐level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” . Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section contains his answer. In Appendix D , we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.