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American Journal of Physics

WKB energy quantization and first-order perturbation theory

Richard W. Robinett

Published Apr 1, 1997• 4 citations
Publisher

Abstract

The WKB quantization condition can be used to estimate the quantized bound state energies, En, in a potential V(x). If a small perturbation of the form λV(x) is added to this potential, we show that the resulting shift in the quantized energy eigenvalues is given by a form which is quite similar to standard first-order perturbation theory. Several examples are presented in which the large n form of the lowest-order energy shift is more easily evaluated using WKB-perturbation techniques than by using standard first-order perturbation methods.

DOI
10.1119/1.18749

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The WKB quantization condition can be used to estimate the quantized bound state energies, En, in a potential V(x). If a small perturbation of the form λV(x) is added to this potential, we show that the resulting shift in the quantized energy eigenvalues is given by a form which is quite similar to standard first-order perturbation theory. Several examples are presented in which the large n form of the lowest-order energy shift is more easily evaluated using WKB-perturbation techniques than by using standard first-order perturbation methods.
WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
We perform a systematic WKB expansion to all orders for a one--dimensional system with potential V(x)=U0/cos⁡2(αx)V(x)=U_0/\cos^2{(αx)}V(x)=U0​/cos2(αx). We are able to sum the series to the exact energy spectrum. Then we show that any finite order WKB approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.
The partial-wave Schrödinger equation with real boundary conditions is recast as an equation for the probability density. When a small additional potential is included, the changes in the bound-state energy eigenvalues are obtained, up to third order in the perturbation, purely in terms of the perturbing potential and the unperturbed probability density. Although the approach is different, our results are equivalent to those derived by Bender (Bender C M 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p 330). Knowledge of neither the unperturbed energy spectrum nor the wavefunctions of excited states is required. Evaluations of the second-order energy shift are given for some soluble S -wave problems.
Abstract Using both WKB approximation methods and complete eigenstate solutions, we examine four standard, textbook level, bound state problems in quantum mechanics. We show how the action integral from the Bohr–Sommerfeld quantization condition, extended to the WKB energy quantization condition, can be expressed in the language of semi-classical and exact quantum results. Relating classical time averaging and quantum expectation value methods, the action integral can be used to connect the average value of speed squared times period across such systems giving ⟨ v 2 ⟩ n τ n = ( n + C ~ ) ( h / m ) , where n is the bound state quantum number ( n = 0 , 1 , 2 , . . . ) supplemented by the Maslov correction ( C ~ ) . We then extend this analysis using exact results for an infinite hierarchy of potentials, one derived from the repeated supersymmetrization of the familiar infinite well, as a more mathematically sophisticated example. These discussions provide a novel pedagogical twist on h as the ‘quantum of action’, accessible at several levels of the undergraduate quantum mechanics curriculum.