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American Journal of Physics

Extensions of the Bohr–Sommerfeld formula to double-well potentials

L. V. Chebotarev

Published Dec 1, 1998• 19 citations
Publisher

Abstract

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.

DOI
10.1119/1.19049

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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.
The generalized Bohr-Sommerfeld quantization condition up to the ninth-order approximation is given explicitly, in terms of complete elliptic integrals, for two kinds of quartic oscillators V(z)=v2{\mathit{v}}_{2}v2​z2{\mathit{z}}^{2}z2+v4{\mathit{v}}_{4}v4​z4{\mathit{z}}^{4}z4: (i) for the quartic double barrier, where v2{\mathit{v}}_{2}v2​>0 and ${\mathit{v}}_{4}$0, and the energy is below the potential maximum; and (ii) for the quartic double well, where ${\mathit{v}}_{2}$0 and v4{\mathit{v}}_{4}v4​>0. Resonance energies for some quartic double barriers and energy levels for some quartic double wells were evaluated by solving the quantization condition numerically. In some cases the results are very good.
For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).
For a general smooth one-dimensional potential well with N minima (containing N−1 barrier humps of arbitrary shape), a phase-integral quantization condition (including σ terms) is obtained, from which the energy eigenvalues can be calculated with an accuracy relative to the chosen order of the Fröman phase-integral approximation. The quantization equation is derived from a condition for total transmission through a general system of potential barriers of arbitrary shape, by means of a limiting procedure, in which the two outermost barriers are made to grow large in a particular way. Finally, the quantization condition is expressed by means of a concise recursion formula. As an illustration, the special case N=2 is treated in some detail. For N=2, the quantization condition turns out to be identical to a previously well-known phase-integral quantization condition for the double-minimum potential, the accuracy of which has been examined in several papers by other authors. Their results concerning a symmetric double-minimum model potential show that when the σ term is included, the third- and fifth-order energy eigenvalues are accurate enough for all practical purposes, even close to the barrier maximum, but if the σ term is neglected, the quantization condition breaks down near the barrier maximum, this breakdown becoming more dramatic with increasing order. A table displaying phase-integral eigenvalues for the model potential as compared with exact (numerical) quantum mechanical results is given here; the figures are quoted from the papers mentioned. Applying the results to the special case N=3, it will be given explicitly the quantization condition for an arbitrary triple-minimum potential.
This investigation concerns some model potentials with very steep walls, for which the quantal Bohr–Sommerfeld half-integer quantization condition, if necessary generalized to correspond to modified phase-integral approximations of arbitrary order, and used without or with higher-order corrections included, can be used for obtaining, very accurately, the energy eigenvalues of the bound states, apart possibly from the lowest ones. One of the cases treated is the potential proportional to cot2x, for which a modified Bohr–Sommerfeld half-integer quantization condition yields the energy eigenvalues exactly. That the Bohr–Sommerfeld half-integer quantization condition is applicable to potentials with very steep walls may at first sight seem surprising in view of the well-known fact that the energy eigenvalues of a square-well potential with infinitely high walls are obtained exactly from the Bohr–Sommerfeld integer quantization condition, i.e., the quantization condition obtained by replacing (s+1/2) π by (s+1) π in the right-hand member of the Bohr–Sommerfeld half-integer quantization condition. From the study of a potential with horizontal bottom and linearly rising walls, which goes over into a square well when the inclination of the walls tends to zero, it can easily be understood, why the half-integer quantization condition is appropriate when the steep walls have a finite slope.
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 double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.