The hydrogen atom according to wave mechanics in cartesian coordinates

TL;DR

This paper demonstrates a method to solve the Schrödinger equation for the hydrogen atom using Cartesian coordinates, providing algebraic solutions and visualizations similar to spherical coordinates.

Contribution

It introduces a partial separation of variables in Cartesian coordinates for the hydrogen atom, simplifying the algebraic form of the wave functions without using angular variables.

Findings

Algebraic formulas for wave functions in Cartesian coordinates

Visualizations of constant wave function surfaces

Correspondence with spherical coordinate solutions

Abstract

A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.