Canonical quantization on the half-line and in an interval based upon a new concept for the momentum in a space with boundaries

TL;DR

This paper introduces a new self-adjoint momentum operator for particles confined to a half-line or interval, enabling proper canonical quantization and accurate description of boundary reflection measurements.

Contribution

It proposes a novel concept for a self-adjoint momentum operator that allows canonical quantization in bounded spaces, addressing a key limitation of traditional momentum operators.

Findings

Self-adjoint momentum operator $ ilde{p}_R$ constructed for bounded domains.

Proper domain considerations lead to meaningful quantum measurements.

Quantization framework applicable to particles reflected at boundaries.

Abstract

For a particle moving on a half-line or in an interval the operator $\hat p = - i \partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\hat p$ fails. Based upon a new concept for a self-adjoint momentum operator $\hat p_R$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\hat H$ and the momentum operator $\hat p_R$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\hat H)$ and $D(\hat p_R)$ in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The new concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.