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Journal of Physics A Mathematical and Theoretical

Probability in the formalism of quantum mechanics on phase space

Franklin E. Schroeck

Published Jan 20, 2012• 3 citations
Publisher

Abstract

The methods of Born and Einstein are used to obtain the probability density in the formalism of quantum mechanics on phase space. The resulting probability leads to a contextual measurement scheme. The Wigner representation, the Husimi representation and the mass shell representation are discussed from the point of view of quantum mechanics on phase space. We also give ramifications for paradoxes in standard quantum mechanics.

DOI
10.1088/1751-8113/45/6/065303

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The methods of Born and Einstein are used to obtain the probability density in the formalism of quantum mechanics on phase space. The resulting probability leads to a contextual measurement scheme. The Wigner representation, the Husimi representation and the mass shell representation are discussed from the point of view of quantum mechanics on phase space. We also give ramifications for paradoxes in standard quantum mechanics.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.
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.
Standard formulations of quantum mechanics successfully describe physical systems through isolated representations, most notably in position or momentum spaces. Attempts to unify these descriptions into a single joint phase space have historically faced severe mathematical dilemmas, such as the appearance of negative values in the Wigner quasiprobability distribution. This paper addresses this challenge by extending the recently developed Analytic Probability Framework and the Primacy of the Whole Principle into a geometric domain. We demonstrate that the separate, valid representations of standard quantum mechanics can be consistently unified into a joint hybrid representation without relying on classical phase space assumptions. By defining a strictly positive and normalized joint probability distribution, we construct a genuine quantum phase space where individual values of position and momentum do not exist as autonomous points, but rather function as interconnected structural elements of a single, indivisible quantum state. This approach provides a rigorous, non-paradoxical geometric foundation that naturally preserves quantum logic while extending the standard formalism.
The concept of quantum phase space offers a view on quantum mechanics, which is different from the standard Hilbert space approach, but which more closely resembles the classical phase space. Due to the properties of quantum mechanics there are several equivalent quantum phase space descriptions, and one cannot always prefer one or another as they all have certain merits and drawbacks. For example, the Husimi-Kano Q function is a probability distribution and thus gives rise to entropic quantities, namely the Renyi-Wehrl entropies, of which several properties are known. The Wigner function, on the other hand, has an easier physical explanation, but may take negative values. In this article, we investigate entropic measures of localisation for a state in quantum phase space by using the Beckner-Brascamp-Lieb inequality to relate different phase-space functions.