Canonical Quantization and Impenetrable Barriers

TL;DR

This paper explores how to reconcile canonical quantization with boundary-imposed spatial restrictions, using the infinite well as a case study to analyze spectral properties of confined and global observables.

Contribution

It demonstrates a method to interpret operators for trapped particles without neglecting the unoccupied space, resolving conflicts between quantization and boundary conditions.

Findings

Operators for trapped particles can be consistently defined considering the entire real line.

Spectral analysis reveals how boundary conditions affect quantum observables.

The approach clarifies the role of the unoccupied space in quantum confinement.

Abstract

We address an apparent conflict between the traditional canonical quantization framework of quantum theory and the spatially restricted quantum dynamics, when the translation invariance of the otherwise free quantum system is broken by boundary conditions. By invoking an exemplary case of a particle in an infinite well, we analyze spectral problems for related, confined and global, observables. In particular, we show how one can make sense of various operators pertaining to trapped particles by not ignoring the rest of the real line (e.g., that space which is never occupied by the particle in question).