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Some Recent Developments in WKB Approximation

Akbar Safari

Published Sep 18, 2013• 3 citations
Publisher

Abstract

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.

DOI
10.71548/12850

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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.
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB0_00​ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave, Ancos⁡(knx+δn)A_n\cos(k_nx+δ_n)An​cos(kn​x+δn​), and are the asymptote of the exact solution.
The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.
In the paper under discussion, it was observed that the accuracy of WKB approximate binding energies for the exponential potential deteriorates as the energy increases, an apparent violation of the correspondence principle. We show that a careful application of the WKB approximation as it applies to the centrifugal barrier potential leads to improved approximate binding energies of the exponential potential. Our analysis supports the Friedrich and Trost “reflection phase” version of the WKB quantization condition.
We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\ensuremath{\pi}/2$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive 1/x2{1/x}^{2}1/x2 potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
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.