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Journal of Physics A Mathematical and General

On the probability density interpretation of smoothed Wigner functions

Marcus A. M. de Aguiar, Alfredo M. Ozorio de Almeida

Published Oct 7, 1990• 10 citations
Publisher

Abstract

It has been conjectured that averages of the Wigner function over phase space volumes, larger than those of minimum uncertainty, are always positive. This is true for Gaussian averaging, so that the Husimi distribution is positive. However, the authors provide a specific counterexample for the averaging with a discontinuous hat function. The analysis of the specific system of a one-dimensional particle in a box also elucidates the respective advantages of the Wigner and the Husimi functions for the study of the semiclassical limit. The falsification of the averaging conjecture is shown not to depend on the discontinuities of the hat function, by considering the latter as the limit of a sequence of analytic functions.

DOI
10.1088/0305-4470/23/19/002

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It has been conjectured that averages of the Wigner function over phase space volumes, larger than those of minimum uncertainty, are always positive. This is true for Gaussian averaging, so that the Husimi distribution is positive. However, the authors provide a specific counterexample for the averaging with a discontinuous hat function. The analysis of the specific system of a one-dimensional particle in a box also elucidates the respective advantages of the Wigner and the Husimi functions for the study of the semiclassical limit. The falsification of the averaging conjecture is shown not to depend on the discontinuities of the hat function, by considering the latter as the limit of a sequence of analytic functions.
For an ergodic system, the time average of a classical observable coincides with that obtained via the Liouville probability density, a delta-function on the energy shell. Reinterpreting this distribution as a Wigner function, that is, the Weyl representation of a density operator, the quantum ergodic conjecture identifies this classical construction as an approximate representation of the quantum eigenstate of the same energy. It is found that this reasonable hypothesis, as far as expectations for observables are concerned, does not satisfy the requirement of positivity, so that the delta-function on the shell cannot be a true Wigner function. This result was first presented by N. Balazs for the special case of a single degree of freedom, such that the system is simultaneously integrable and ergodic. Here, it is shown, for a delta-function on an arbitrary curved energy shell in a -dimensional phase space, that there exists a positive operator for which the conjectured Wigner function predicts a negative expectation.
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.
We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows one to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.
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.
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ?supershifted? so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ?phase space? (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k loc(x)?=?? x arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k loc if L is small enough. These properties are illustrated numerically with two superoscillatory functions.