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Physical Review A

Nonintegral Maslov indices

Hanno Friedrich, Johannes Trost

Published Aug 1, 1996• 71 citations
Publisher

Abstract

The phase loss of a wave reflected by a smooth potential generally varies continuously from \ensuremath{\pi} in the long-wave limit to \ensuremath{\pi}/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of \ensuremath{\pi}/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/${\mathit{x}}^{2}$ potential. The nonintegral Maslov index for a given 1/${\mathit{x}}^{2}$ potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1136

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The phase loss of a wave reflected by a smooth potential generally varies continuously from \ensuremath{\pi} in the long-wave limit to \ensuremath{\pi}/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of \ensuremath{\pi}/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/x2{\mathit{x}}^{2}x2 potential. The nonintegral Maslov index for a given 1/x2{\mathit{x}}^{2}x2 potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.
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.
The notes here presented are of the modifications introduced in the application of WKB method. The problems of two- and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulation of quantization rule respectively. It is found that the energy spectrum of the radial harmonic oscillator, which is reproduced exactly by the standard WKB method with the hanger modification, is also reproduced exactly without the hanger modification via the new quantization rule approach. An alternative way to obtain the non-integral Maslov index for three-dimensional harmonic oscillator is proposed.
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.
The WKB form suitable for the classical region of potential energy less than the total energy, $\ensuremath{\psi}(x)={(\frac{\mathrm{dz}}{\mathrm{dx}})}^{\ensuremath{-}\frac{1}{2}}sinz$, is used to obtain a function z(x)z(x)z(x) that reproduces the time-independent Schr\"odinger equation solutions $\ensuremath{\psi}(x)$, to any desired degree of accuracy through the turning point and into the nonclassical region.