Ensemble in phase space: statistical formalism of quantum mechanics

TL;DR

This paper introduces a novel statistical phase space formalism for quantum mechanics, deriving core equations and demonstrating its ability to reproduce key results, including relativistic wave equations, without relying on traditional formalisms.

Contribution

It develops an autonomous quantum mechanics formalism in phase space based on statistical ensemble methods, deriving the master equation and connecting it to known quantum equations.

Findings

Derivation of a master quantum-mechanical equation from phase space ensembles.

Approximation of the Schrödinger equation within this formalism.

Successful derivation of the relativistic wave equation using the formalism.

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 formalisms of quantum mechanics, and gives the idea of operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space. We have explained with the help of this equation the structure of quantum mechanics in phase space and the approximation to the Schrodinger equation. Furthermore, we have shown that this formalism provides reasonable results of quantization by dealing with some simple cases, 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.