Perturbation Theory and Stark Effect

Ahmad Amer

This chapter presents approximation methods in quantum mechanics, focusing on both time-independent and time-dependent perturbation theory. The time-independent formulation is developed up to second-order corrections for both nondegenerate and degenerate systems. Applications include the analysis of the Zeeman and Stark effects, where energy level shifts due to magnetic and electric fields are calculated. The time-dependent perturbation theory is introduced, with harmonic perturbations used to derive the first-order transition probabilities and to establish Fermi's golden rule. As an application, the ionization probability of a hydrogen atom under a linearly polarized electromagnetic wave is computed in detail. The treatment is extended to include the case of an unpolarized electromagnetic field. Metadata/Keywords: perturbation theory, Zeeman effect, Stark effect, time-dependent perturbation, harmonic perturbation, Fermi's golden rule, hydrogen ionization

Walter Greiner

George L. Johnston, Garrison Sposito

The Stark effect of the plane rotator is a pedagogically appealing problem for undergraduate courses on quantum mechanics and, accordingly, has appeared in a number of textbooks as an application of bound-state perturbation theory. Despite this fact and the apparent simplicity of the problem, all of the published accounts of it are erroneous in that they employ nondegenerate perturbation theory and present incorrect expressions for the perturbed eigenvalues and eigenstates. In this paper the problem is given a new treatment, based upon a simple analysis of symmetry features and a careful application of degenerate perturbation theory. The major shortcomings of previous discussions are pointed out and correct expressions are derived, to second order, for the perturbed state vectors and energy levels. The lifting of the twofold degeneracy of the mth unperturbed excitation energy in the 2mth order of perturbation theory is demonstrated. Finally, a comparison of the correct perturbation-theoretic results with the exact solution of the problem shows the two to be in identical correspondence.

Jieqing Mo

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.

Willi-Hans Steeb