The quantum state vector in phase space and Gabor's windowed Fourier transform

TL;DR

This paper explores representing quantum state vectors as phase space amplitudes using Gabor's windowed Fourier transform, connecting to various quantum representations and providing a generalized Born interpretation.

Contribution

It introduces a novel phase space amplitude representation of quantum states linked to Gabor transforms, extending existing quantum phase space methods.

Findings

Amplitudes for simple states illustrated with different windows

Connections established with Bargmann and Husimi representations

Time-dependent Schrödinger equations expressed on phase space amplitudes

Abstract

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.