Stark Effect of Electron-Proton System with Computer-generated Visualizations

Abstract

The energy level and wave function obtained by solving the Schrödinger equation are extremely important in quantum mechanics. In this study, the solution to the Schrödinger equation of the electron-proton system by separation of variables was briefly described and the visualizations of the various solution were generated by Mathematica. The perturbation theory was discussed with the exclusion of the correction terms coming from the degenerate subspace. Applying the perturbation theory to the wavefunctions of the electron-proton system to study the Stark effect, the perturbation Hamiltonian is a constant electrostatic field. For the principal quantum numbers n=1, n=2 and n=3, this paper obtains the energy corrections to the first and second degree, the corrections of the wavefunction to the first degree. The visualizations of the perturbed wavefunctions were obtained by setting the summation limits k=20 and other parameters to 1. The visualizations of various unperturbed-states and perturbed-states were generated by Mathematica version 13.2.