A Method for Treating Large Perturbations

Abstract

A method is developed for treating large perturbations to which the ordinary Schr\"odinger perturbation theory is not applicable. The method can be used for all Hamiltonian matrices which are sufficiently regular. The condition is that the matrix elements in any given diagonal should change slowly from one row to the next (cf. \textsection{}2). It is not necessary that the Hamiltonian be expressible as an explicit function of coordinates and momenta. The wave function is expanded in the usual way in "unperturbed eigenfunctions," $\ensuremath{\psi}=\ensuremath{\Sigma}{n}^{}c(n){\ensuremath{\psi}}_{n}$. The expansion coefficients $c(n)$ are then calculated by a method similar to the wellknown WKB method; only we deal with a difference rather than a differential equation (\textsection{}3). It is found that $c(n)$ behaves, as a function of $n$, either exponentially or oscillatory or exponentially with alternating sign (\textsection{}4), the first two types of behavior being similar to the ordinary WKB, while the last type is peculiar to difference equations. Several examples are given to illustrate the method, viz. some perturbed oscillator problems (\textsection{}3, $5A$, $B$) and the Matthieu functions ($\textsection{}5C$). After some generalizations (\textsection{}6, 7), the method is applied to fast electrons in metals (\textsection{}8) whose wave functions are determined in momentum space. It is shown that a plane wave is an insufficient approximation even for high energy.