Observables, evolution equation,and stationary states equation in the joint probability representation of quantum mechanics

TL;DR

This paper develops a formalism for symplectic and optical joint probability representations in quantum mechanics, deriving evolution and stationary state equations using star product quantization and Gaussian distributions.

Contribution

It introduces a comprehensive formalism for joint probability representations in quantum mechanics, including quantizers, dequantizers, and operator symbols, with new evolution and stationary state equations.

Findings

Derived evolution equations for joint probability distributions.

Established correspondence rules for physical operators.

Obtained explicit expressions for operator symbols in Gaussian cases.

Abstract

Symplectic and optical joint probability representations of quantum mechanics are considered, in which the functions describing the states are the probability distributions with all random arguments (except the argument of time ). The general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations is given. Taking the Gaussian functions as the distributions of the tomographic parameters the correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators in the form of singular and regular generalized functions are derived. Evolution equations and stationary states equations for symplectic and optical joint probability distributions are obtained.