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Journal of Mathematical Physics

On Wentzel’s proof of the quantization condition for a single-well potential

Nanny Fröman, Per Olof Fröman

Published Jan 1, 1977• 23 citations
Publisher

Abstract

Wentzel’s elegant method for deriving the quantal generalization of the Sommerfeld quantization condition, which has been criticized by other authors, is justified and put on an irrefutable basis. Clarifying comments on some questions related to the discussion in the present paper are also made.
DOI
10.1063/1.523122

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arXiv:2006.01434•Jun 2, 2020•0 citations
Extensions of the Bohr–Sommerfeld quantization formula to smooth double wells are established which preserve the formula’s generality and simplicity as well as its accuracy and the ease of use. Moreover, the proposed quantization equations for double wells, while being free from the limitations of the Wentzel–Kramers–Brillouin approximation which is restricted to deep energy levels alone, are equally well suited for determining higher energy levels lying in the vicinity of the barrier’s top as well as above the latter, and they have the correct connection to the conventional Bohr–Sommerfeld formula as the particle’s energy becomes large enough. As a particular application, a simple quantization equation for the pure quartic oscillator is derived which yields correct energy eigenvalues for all eigenstates of the oscillator including the ground one, with relative errors being all less than 1%. Conditions for the validity of the extended quantization equations are discussed and their numerical verification is made.
We present, for the first time ever, with the help of two conjectures, an \textit{exact} quantization condition for two turning-point eigenvalue problems (that recurrently appear in several branches of science) by explicitly summing up the Wentzel-Kramers-Brillouin series to all orders. Surprisingly, the quantization condition looks elegant and resembles exactly the Bohr-Sommerfeld quantization relation with just an extra term having a square of Planck's constant (appearing as a form of repulsive quantum potential). We also find that our exact quantization relation produces Langer-modification for spherically symmetric 3−3-3−dimensional potentials.