The polarizability of a particle in power-law potentials: a WKB analysis

Abstract

We consider the problem of a particle confined to a one-dimensional power-law potential of the form , but also subject to a constant force F. This might represent, for example, the application of an external electric field, , via the perturbing potential . The electric polarizability of the system in a given state is defined via the energy shift due to the external field. Exact expressions for are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well . For the harmonic oscillator (infinite well) the polarizability scales as and is positive (negative) for large values of n indicating a qualitatively different response of the system to an external field for large quantum numbers. In order to examine this problem for a more general power-law potential, we apply WKB techniques to evaluate the energy shifts for large n and we find that (i) the n-dependence of the polarizability scales as as a function of k and (ii) the approximate value of k at which the crossover from to (for large n) occurs is . This study provides a useful example of numerical WKB techniques, the use of scaling ideas in simple one-dimensional quantum mechanical systems, and a better `feel' for the meaning of the polarizability which is an important physical quantity in more realistic atomic and molecular systems.