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arXiv:1506.08880 · SIAM Journal on Mathematical Analysis

A New Phase Space Density for Quantum Expectations

Johannes Keller, Caroline Lasser, Tomoki Ohsawa

Published Jan 1, 2016• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1137/15m1028388

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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.
We discuss a new phase space method for the computation of quantum expectation values in the high frequency regime. Instead of representing a wavefunction by its Wigner function, which typically attains negative values, we define a new phase space density by adding a first-order Hermite spectrogram term as a correction to the Husimi function. The new phase space density yields accurate approximations of the quantum expectation values as well as allows numerical sampling from non-negative densities. We illustrate the new method by numerical experiments in up to 128128128 dimensions.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.
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 present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.
This paper proposes an approach to interpreting quantum expectation values that may help address the quantum measurement problem. Quantum expectation values are usually calculated via Hilbert space inner products and, thereby, differently from expectation values in classical mechanics, which are weighted phase-space integrals. It is shown that, by using Anti-Wick quantization to associate dynamical variables with self-adjoint linear operators, quantum expectation values can be interpreted as genuine weighted averages over phase space, paralleling their classical counterparts. This interpretation arises naturally in the Segal-Bargmann space, where creation and annihilation operators act as simple multiplication and differentiation operators. In this setting, the Husimi Q-function - the coherent-state representation of the quantum state - can be seen as a true probability density in phase space. Unlike Bohmian mechanics, the present approach retains the standard correspondence between dynamical variables and self-adjoint operators while paving the way for a classical-like probabilistic interpretation of quantum statistics.
The Husimi distribution, introduced by Kodi Husimi in 1940 [1], is a quasi-probability distribution commonly used to study the correspondence between quantum and classical dynamics [2]. Also, it is employed to describe systems in different areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory [3–8]. Additionally, in nanotechnology it is possible to obtain a clear description of localization –which corresponds to classicality– and is crucial to determine correctly the size of systems when the particle dynamics takes into account mobility boundaries [9]. Among its properties, it is always positive definite and unique, conversely it cannot be considered as a true probability distribution over the quantum-mechanical phase space, reason why it is often considered as a quasi probability distribution. Although it possesses no correct marginal properties, its usefulness is to allow the assessment of the expectation values in quantum mechanics in a way similar to the classical case [10]. The semiclassical Husimi probability distribution refers to a special type of probability, this is for simultaneous but approximate location of position and momentum in phase space.