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Modern Physics Letters B

THE QUANTIZATION RULE AND MASLOV INDEX

Xiaoyan Gu

Published Aug 27, 2010• 1 citation
Publisher

Abstract

Within extensions of the new quantization rule approach in arbitrary dimensions, the Maslov indices and energy spectra of some exactly solvable potentials are presented. We find that the Maslov index for the harmonic oscillator in three dimensions agrees well with those obtained by other methods.
DOI
10.1142/s021798491002464x

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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.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
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.