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Exact and Proper Quantization Rules and Langer Modification

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Book cover Wave Equations in Higher Dimensions

Abstract

A fundamental interest in quantum mechanics is to obtain the right result without invoking the full mathematics of the Schrödinger equation. Since last decade there has been a great revival of interest in semiclassical methods for obtaining approximate solutions to the Schrödinger equation. Among them, the WKB approximation and its generalization have attracted much attention to many authors since this method is proven to be useful in obtaining an approximate solution to the Schrödinger equation with solvable potentials. Considering the close connection between the WKB method and the exact and proper quantization rules, we first give a brief review of them and then establish the relation between the proper quantization rule, the Maslov index and the Langer modification. Note that the proper quantization rule possesses more symmetry than the original exact quantization rule and greatly simplifies those tedious and complicated integral calculations appearing in the original exact quantization rule. The symmetry and simplicity of the proper quantization rule come from its meaning—whenever the number of the nodes of the logarithmic derivative of wavefunction or the number of the nodes of the wavefunction increases by one, the momentum integral will increase by π. As illustrations we shall study a few solvable potentials via these quantization rules. It should be pointed out that the proper quantization rule can be used to solve all solvable potentials and ends the history of semiclassical quantization rules.

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Notes

  1. 1.

    It should be noted that the validity of the Ma-Xu formula follows from a Barclay’s result. He found that for these potentials the JWKB series can be resumed beyond the lowest-order giving an energy-independent correction which can be absorbed into the Maslov index and written in a closed analytical expression. Moreover, he showed equally that this result is directly correlated to the exactness of the lowest-order SJWKB quantization condition [374, 400]. The starting point is the definition of two classes of potentials, each characterized by a specific change of variable which brings the potential into a quadratic form. It is shown that this two classes coincide with the Barclay-Maxwell classes [401], which are based upon a functional characterization of superpotentials and which cover the whole set of translationally shape invariant potentials.

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  1. Escuela Superior de Física y Matemáticas, Unidad Profesional Adolfo López Mateos, Instituto Politécnico Nacional, Edificio 9, Mexico DF, 07738, Mexico

    Shi-Hai Dong

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Dong, SH. (2011). Exact and Proper Quantization Rules and Langer Modification. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_11

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