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

We consider the problem of a particle confined to a one-dimensional power-law potential of the form V^(k)(x) = V₀|x/a|^k, but also subject to a constant force F. This might represent, for example, the application of an external electric field, ε, via the perturbing potential Ṽ(x) = Fx = qεx. The electric polarizability of the system in a given state is defined via the energy shift ΔE_n = α_nε²/2 due to the external field. Exact expressions for α_n are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well (k → ∞). For the harmonic oscillator (infinite well) the polarizability scales as n⁰ (n_-2) 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 (2 < k < ∞) 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 α_n ∝ n^2(2-k)/(2+k) as a function of k and (ii) the approximate value of k at which the crossover from α_n > 0 to α_n < 0 (for large n) occurs is k ≈ 6. 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.