Simple quantum systems in the momentum representation

TL;DR

This paper introduces quantum mechanics in momentum space, solving standard problems directly in this representation to enhance understanding and address common student misconceptions about the momentum viewpoint.

Contribution

It provides an accessible introduction to momentum space quantum mechanics, solving key problems without relying on coordinate space formulations, highlighting their differences.

Findings

Problems solved entirely in momentum space

Demonstrates differences from coordinate space solutions

Enhances understanding of momentum representation in quantum mechanics

Abstract

The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schr\"odinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.