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1983

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April

  

Reviews of topical problems


The Wigner representation of quantum mechanics

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.

PACS: 03.65.Db
DOI: 10.1070/PU1983v026n04ABEH004345
URL: https://ufn.ru/en/articles/1983/4/b/
Citation: Tatarskii V I "The Wigner representation of quantum mechanics" Sov. Phys. Usp. 26 311–327 (1983)
BibTexBibNote ® (generic)BibNote ® (RIS) MedlineRefWorks
PT Journal Article
TI The Wigner representation of quantum mechanics
AU Tatarskii V I
FAU Tatarskii VI
DP 10 Apr, 1983
TA Phys. Usp.
VI 26
IP 4
PG 311-327
RX 10.1070/PU1983v026n04ABEH004345
URL https://ufn.ru/en/articles/1983/4/b/
SO Phys. Usp. 1983 Apr 10;26(4):311-327

Оригинал: Татарский В И «Вигнеровское представление квантовой механики» УФН 139 587–619 (1983); DOI: 10.3367/UFNr.0139.198304b.0587

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