On the energy density in quantum mechanics

TL;DR

This paper investigates different definitions of energy density in quantum mechanics and proposes a method to experimentally distinguish them by relating the mean work done during a system's size variation to a specific energy density definition.

Contribution

It introduces a practical approach to identify the physically relevant energy density in quantum systems through work measurements during boundary manipulations.

Findings

The mean work done is related to a specific energy density definition.

The proposed method extends to multi-dimensional quantum systems.

The energy density at the boundary corresponds to the local force exerted by the particle.

Abstract

There are several definitions of energy density in quantum mechanics. These yield expressions that differ locally, but all satisfy a continuity equation and integrate to the value of the expected energy of the system under consideration. Thus, the question of whether there are physical grounds to choose one definition over another arises naturally. In this work, we propose a way to probe a system by varying the size of a well containing a quantum particle. We show that the mean work done by moving the wall is closely related to one of the definitions for energy density. Specifically, the appropriate energy density, evaluated at the wall corresponds to the force exerted by the particle locally, against which the work is done. We show that this identification extends to two and three dimensional systems.