Quantum phase space with a basis of Wannier functions

TL;DR

This paper introduces a quantum phase space framework using Wannier functions that localize wave functions at Planck cells, offering a smoother alternative to Wigner functions for quantum-classical analysis and signal processing.

Contribution

The paper develops a novel method to construct Wannier functions localized at Planck cells, enabling a unitary mapping of wave functions onto phase space with improved smoothing capabilities.

Findings

Method effectively smooths wave function oscillations

Provides a better tool for quantum-classical correspondence

Applicable to time-frequency analysis of signals

Abstract

A quantum phase space with Wannier basis is constructed: (i) classical phase space is divided into Planck cells; (ii) a complete set of Wannier functions are constructed with the combination of Kohn's method and L\"owdin method such that each Wannier function is localized at a Planck cell. With these Wannier functions one can map a wave function unitarily onto phase space. Various examples are used to illustrate our method and compare it to Wigner function. The advantage of our method is that it can smooth out the oscillations in wave functions without losing any information and is potentially a better tool in studying quantum-classical correspondence. In addition, we point out that our method can be used for time-frequency analysis of signals.