Ensemble in phase space: Statistical formalism of quantum mechanics

Abstract

We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the master quantum-mechanical equation without recourse to the other formulations of quantum mechanics, and gives the idea for operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space and, as a result, reproduces Liouville's theorem and virial theorem for quantum mechanics. We have explained with the help of this equation the structure of quantum mechanics in phase space and the approximation to the Schrödinger equation. Furthermore, we have shown that this formalism provides reasonable results of quantization by dealing with some simple cases such as the quantization of harmonic oscillation, the two-slit interference and the uncertainty relation, which confirm the validity of this formalism. In particular, we have demonstrated that this formalism can easily give the relativistic wave equation without treating the problem of linearizing the Hamiltonian operator by making the most of the point that the master equation is a first-order partial differential equation with respect to time, position and momentum variables, and makes use of the phase velocity. The ultimate outcome this formalism produces is that primary and general matters of quantum mechanics can be studied reasonably within the framework of statistical mechanics. phase space functions of classical mechanics by mathematical representations in configuration space.On the other hand, since the advent of quantum mechanics, some attempts have been made to modify the standard interpretations and mathematical formalism of quantum mechanics or to replace them by any other theories. From the point of view of interpretation, the causal theory of quantum mechanics should be noted. The causal theory of quantum mechanics aims to clarify the dynamical causes of quantum-mechanical movements. It furnishes the methods of analysis and interpretation for solving quantum dynamical problems, reproducing the concepts of classical mechanics even for quantum mechanics [6,7,8,9,10]. De Broglie, Madelung, Bohm and others representative of Bohmian mechanics, and Groenewold, Moyal, Takabayasi and others representative of quantum mechanics in phase space (abbreviated as QMPS) had established the foundations of the causal theory of quantum mechanics

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

“…In this paper, we referred to some parts of our previous papers [73] to systematically explain a new theme distinct from the previous subjects: Born's probabilistic interpretation. Nevertheless, we look upon this paper as having an independent significance, since it treats the most important open question of quantum mechanics as a particular subject.…”

“…In Refs. [73] and as its extension [73], we explained with the help of mechanical picture the reason why the wave function should be represented as a complex number function. This assumption implies that the wave field has the phase determined by the action, and h is the quantum of the action.…”

“…D and E as the citations of Ref. [73], we describe how our formalism offers the generalized proof of the uncertainty relations in terms of commutation relations of operators. Thus, the uncertainty relation not only for position and momentum but also for energy and time has been proved in a general way in terms of commutation relation of operators.…”

“…However, it is necessary to review the problem of whether the Schrödinger equation is mathematically rigorous, since it essentially was assumed. Starting with the definition of mean value, we can explicitly demonstrate that the Schrödinger equation makes some approximations besides non-relativistic one [73]. For convenience, we consider the Schrödinger equation for one particle.…”