Gaussian wave packets in phase space: The Fermi g_F function

TL;DR

This paper explores representing Gaussian quantum states in phase space using the Fermi g_F function, visualizing their evolution, and proposing a thought experiment for measuring their properties.

Contribution

It introduces a phase space visualization method for Gaussian wave packets via the g_F function and analyzes their dynamics and measurement techniques.

Findings

Gaussian wave packets can be visualized as curves g_F=0 in phase space.

The evolution of Gaussian packets under various forces can be easily interpreted.

A thought experiment is proposed to fully determine the state of a Gaussian packet.

Abstract

Any pure quantum state can be equivalently represented by means of its wave function psi(q) or of the Fermi function g_F(q,p), with q and p coordinates and conjugate momenta of the system under investigation.We show that a Gaussian wave packet can be conveniently visualized in phase space by means of the curve g_F(q,p)=0. The evolution in time of the g_F=0 curve is then computed for a Gaussian packet evolving freely or under a constant or a harmonic force. As a result, the spreading or shrinking of the packet is easily interpreted in phase space. Finally, we discuss a gedanken prism microscope experiment for measuring the position-momentum correlation. This gedanken experiment, together with the well-known Heisenberg microscope and von Neumann velocimeter, is sufficient to fully determine the state of a Gaussian packet.