A New Concept for the Momentum of a Quantum Mechanical Particle in a Box

TL;DR

This paper introduces a novel momentum operator for a quantum particle in a box, decomposing it into Hermitian and anti-Hermitian parts, enabling a consistent description of momentum measurements within the confined space.

Contribution

It proposes an alternative momentum operator with a Hermitian part that can be extended to a self-adjoint operator, addressing limitations of the traditional operator in a box.

Findings

The new operator has a Hermitian component suitable for physical interpretation.

It allows for a consistent description of momentum measurements inside a box.

The operator's structure clarifies the measurement process in confined quantum systems.

Abstract

For a particle in a box, the operator $- i \partial_x$ is not Hermitean. We provide an alternative construction of a momentum operator $p = p_R + i p_I$, which has a Hermitean component $p_R$ that can be extended to a self-adjoint operator, as well as an anti-Hermitean component $i p_I$. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box.