On the hydrogen wave function in Momentum-space, Clifford algebra and the Generating function of Gegenbauer polynomial

TL;DR

This paper derives the hydrogen atom's wave function in momentum space using Fourier transforms, explores Clifford algebra connections, and provides generating functions for Gegenbauer polynomials.

Contribution

It introduces a novel analytic expression for the hydrogen wave function in momentum space and links Clifford algebra with Gegenbauer polynomial generating functions.

Findings

Analytic momentum-space wave function derived

Matrix elements between basis states calculated

Relationship established between Clifford algebra and Gegenbauer polynomials

Abstract

Using the quadratic transformation and the generating function method we Perform the Fourier transformation of the wave function of coordinates of hydrogen atom and we find the analytic expression of the wave function in momentum space. We derive the matrix elements between the basis to 4-dimensions and integral representation of the generating functions of Gegenbauer polynomials. We find a relationship between a class of Clifford algebra and the generating functions of these polynomials.