The ambiguity function and the displacement operator basis in quantum mechanics

J S Ben-Benjamin,W G Unruh

doi:10.1088/1402-4896/ab3376

Abstract

We present a method for calculating expectation values of operators in terms of a corresponding c-function formalism which is not the Wigner–Weyl position-momentum phase-space, but another space. Here, the quantity representing the quantum system is the expectation value of the displacement operator, parametrized by the position and momentum displacements, and expectation values are evaluated as classical integrals over these parameters. The displacement operator is found to offer a complete orthogonal basis for operators, and some of its other properties are investigated. Connection to the Wigner distribution and Weyl procedure are discussed and examples are given.

Highlights

In the early 1930’s, Wigner pioneered the phase-space formulation of quantum mechanics, introducing the Wigner distribution [1]
Besides to construct a quantum theory of statistical mechanics, the drive to create phase-space distributions was due to several aspects: One was a fundamental aspect – trying to create new formulations of qunatum mechanics and studying the uncertainty principle; another, not too distant motivation was the study of the classical–quantum interface; still other reasons are for mathematical, as well as conceptual simplicity
We present the theory behind one such distribution, the ambiguity distribution, which is seldom used in quantum mechanics, and present many of its properties and the properties of its accompanying classical operator, including the A ↔ A

Summary

Introduction

In the early 1930’s, Wigner pioneered the phase-space formulation of quantum mechanics, introducing the Wigner distribution [1]. This formalism could be used for mathematical manipulations of operators, and it was used in [12] for studying the decoherence of a harmonic oscillator in a heat bath, and in [13, 14], for studying fundamental issues in relativistic quantum field theory. In our formalism, calculation of the expectation value of (polynomial) operators does not involve any integration – just derivatives and multiplication (as in Eq (38) This is a clear advantage over the Wigner distribution. We present a method for obtaining the c-number distribution (ambiguity function) A(η, ξ) for any arbitrary operator A, and show how it can be used for calculating quantum mechanical expectation values. Starting with an arbitrary operator A, we define the c-number

Time evolution

Examples

Conclusions

B Proof of our expectation value method