Cartesian operator factorization method for Hydrogen

TL;DR

This paper introduces a Cartesian operator factorization method for the Hydrogen atom, extending Schrödinger's traditional approach by representing the Hamiltonian as a sum over coupled operators in Cartesian coordinates.

Contribution

It generalizes the factorization method to Cartesian coordinates, providing a new framework for solving the Hydrogen atom and potentially other Hamiltonians.

Findings

Eigenstates and energies are determined.

Wavefunctions in coordinate and momentum space are obtained.

The method links to the confluent hypergeometric equation approach.

Abstract

We generalize Schroedinger's factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach, is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schroedinger factorizations.