The Momentum Representation of the Hydrogen Atom in Paraboloidal Coordinates

TL;DR

This paper develops a method for constructing momentum space wave functions of the hydrogen atom in paraboloidal coordinates, highlighting the benefits and limitations of using momentum representation in quantum mechanics.

Contribution

It introduces a systematic approach to derive momentum space wave functions for the hydrogen atom in paraboloidal coordinates, ensuring proper conjugate variable relationships.

Findings

Derived explicit momentum space wave functions in paraboloidal coordinates.

Analyzed the advantages and disadvantages of momentum representation.

Ensured valid uncertainty relationships in the constructed variables.

Abstract

We examine the procedure to construct the variables of use for the momentum representation in quantum mechanics. The momentum variables must be chosen properly conjugate to the corresponding position space variables, such that valid uncertainty relationships are maintained. We then apply such considerations to the hydrogen atom to obtain the momentum space wave functions corresponding to the position space functions in paraboloidal coordinates. The advantages and disadvantages of employing the momentum representation are explored.