Obtaining Hydrogen energy wave functions using the Runge-Lenz vector

TL;DR

This paper explores deriving hydrogen atom wave functions using the conserved Runge-Lenz vector, providing a group-theoretic approach that complements traditional Schrödinger equation methods.

Contribution

It introduces a recursive, symmetry-based method for obtaining hydrogen wave functions via the Runge-Lenz vector, enhancing understanding of degeneracies.

Findings

Recursive relations for wave functions derived

Method aligns with Schrödinger equation results

Group theory underpins the analysis

Abstract

The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schr\"{o}dinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate $|n,l,m\rangle$ with other degenerate states $|n, l\pm 1,m'\rangle$. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schr\"{o}dinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.