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Quantization

Mark Sheard Child

Published Jan 3, 1991• 0 citations

Abstract

Abstract Questions of quantization turn on a proper treatment of normally isolated regions where the JWKB description, expressed by eqn (2.13), breaks down. The most obvious of such special regions are those around the classical turning points, at each of which the analysis in Sections 2.2 and 2.3 yields an associated so-called Maslov (1972) phase correction of π/4; Delos (1986) and Littlejohn and Robbins (1987) discuss this Maslov correction from a more fundamental standpoint and Appendix A and parts of Section 11.2 bear on its choice. Other special regions around potential singularities are most conveniently handled by introducing a Langer (1932) correction term to the physical potential function (see Section 2.3). It is shown in Section 3.1 that systems with no other complications may be quantized by the Bohr-Sommerfeld rule (3.1)∮p(q)dq=(n+δ)h, n=0,1,2,3⋯ where n is the quantum number and b is a Maslov index, which takes the values ½, 0, ½ for vibrational, plane rotational, and orbital motion respectively.

DOI: 10.1093/oso/9780198556541.003.0003

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Abstract Questions of quantization turn on a proper treatment of normally isolated regions where the JWKB description, expressed by eqn (2.13), breaks down. The most obvious of such special regions are those around the classical turning points, at each of which the analysis in Sections 2.2 and 2.3 yields an associated so-called Maslov (1972) phase correction of π/4; Delos (1986) and Littlejohn and Robbins (1987) discuss this Maslov correction from a more fundamental standpoint and Appendix A and parts of Section 11.2 bear on its choice. Other special regions around potential singularities are most conveniently handled by introducing a Langer (1932) correction term to the physical potential function (see Section 2.3). It is shown in Section 3.1 that systems with no other complications may be quantized by the Bohr-Sommerfeld rule (3.1)∮p(q)dq=(n+δ)h, n=0,1,2,3⋯ where n is the quantum number and b is a Maslov index, which takes the values ½, 0, ½ for vibrational, plane rotational, and orbital motion respectively.

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.