Saul T. Epstein

Note on Perturbation Theory

Perturbation theory is derived in such a way that the wave function renormalization terms appear automatically. The derivation is based on the Feynman theorem for the rate of change of an eigenvalue with respect to a parameter, and a corresponding theorem for the rate of change of an eigenfunction.

A differential equation for the energy eigenvalues of relativistic hydrogenic atoms, and its solution

By using various general theorems, differential equations for the energy are written down for Klein–Gordon and Dirac hydrogenic atoms. When coupled with the fact that in the nonrelativistic limit one must get the Bohr formula, these equations are shown to determine the exact energy eigenvalues. Also, in the case of the Klein–Gordon equation, a recursion relation is derived from which the average values of powers of the radius can be found.

Application of the Rayleigh-Schrödinger Perturbation Theory to the Delta Function Potential

E. P. Wigner has shown that although there are reasons why one might hope to succeed, one cannot in fact derive the binding energy of the hydrogen atom by second-order perturbation theory applied to a free particle in a box. To further clarify the reasons for this failure we here consider the one-dimensional attractive delta function potential.

Harmonic Oscillator Wave Packets

Time-dependent operators are used to find harmonic oscillator wave packets which oscillate without change in shape. The results are in agreement with those of Senitzky and others. Some useful averages are computed for these packets and some comments are made on the classical limit.

The Parity Operator and the Fourier Transform Operator

Explicit formulas, in terms of coordinate and momentum operators, are given for the parity operator and the Fourier-transform operator.

General Solutions of the Hamilton-Jacobi Equation

The importance of finding general solutions to the Hamilton-Jacobi equation is emphasized. The procedure for doing so is presented together with examples of its use.

Spin Without Commutation Rules

The Pauli spin matrices are derived in a way which makes no appeal to the usual apparatus of abstract operators, commutation rules, etc.