Wave functions of the Hydrogen atom in the momentum representation

TL;DR

This paper derives explicit momentum-space wave functions for the hydrogen atom using integral transforms, expressing them via Gegenbauer functions and demonstrating their symmetry under the SO(4) group.

Contribution

It provides a new explicit integral transform approach to obtain hydrogen atom wave functions in momentum space, linking them to Gegenbauer functions and symmetry properties.

Findings

Explicit momentum-space wave functions derived

Wave functions expressed in Gegenbauer and trigonometric functions

Demonstrated SO(4) symmetry of the wave functions

Abstract

We construct the integral transform passing from the space representation to the momentum representation for the Hydrogen atom using polar spherical coordinates. The resulting radial wave functions are explicitly given in terms of complex finite expansions of Gegenbauer functions of the first and second kind, or in terms of (elementary) trigonometric functions. We show their symmetry under the $SO(4)$ group, and their equivalence with those of Lombardi and Oglivie.