# The ambiguity function and the displacement operator basis in quantum mechanics

## TL;DR

This paper introduces a novel method for calculating quantum operator expectation values using a displacement operator basis, providing an alternative to the Wigner-Weyl phase-space formalism.

## Contribution

It presents a new c-function formalism based on the displacement operator basis, expanding the tools for quantum expectation value calculations beyond traditional phase-space methods.

## Key findings

- Displacement operator forms a complete orthogonal basis for quantum operators.
- The method connects to Wigner distribution and Weyl procedures.
- Examples demonstrate the practical application of the formalism.

## 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.