Self-adjoint Momentum Operator for a Particle Confined in a Multi-Dimensional Cavity

TL;DR

This paper extends the construction of a self-adjoint momentum operator to particles in multi-dimensional cavities, analyzing its properties, measurement implications, and related quantum principles.

Contribution

It introduces a generalized self-adjoint momentum operator for multi-dimensional regions and explores its non-commuting components and measurement considerations.

Findings

Components of the momentum operator generally do not commute.

Momentum measurements should be performed one direction at a time.

Extended Ehrenfest theorem and uncertainty relations to higher dimensions.

Abstract

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.