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Journal of Mathematical Physics

Energy eigenvalues for an arbitrary potential well with N minima

Örjan Dammert

Published Jul 1, 1991• 3 citations
Publisher

Abstract

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.

DOI
10.1063/1.529247

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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.
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.
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.
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.
The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.
The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.
The determination of the Eigenwerte, and the normalization of the Eigenfunktionen of the one-dimensional wave equation, with a potential energy that has a single minimum, is customarily based on the use of the so-called phase integral and W.K.B. formulas. These formulas are asymptotic in character. As they have usually been applied, they accordingly lead to conclusions which can be regarded as established only when the respective quantum numbers are sufficiently large. This restriction has been both unfortunate and puzzling. For on the one hand, the cases of small or moderate quantum numbers are often peculiarly interesting while, on the other hand, the phase integral formulas have been found by trial, at least in a variety of simple cases, to give surprisingly good results even in the lower quantum number range.It is shown in this paper how the asymptotic method should be applied when small or moderate quantum numbers are in question. In the case of the Eigenwerte, the characteristic equation appropriate to the lower range is derived, and a comparison of this new equation with the familiar one based on the phase integral is made. It is thus theoretically established that under certain very liberal conditions upon the potential energy the older method does give good approximations. In the special case of the harmonic oscillator the two equations are, in fact, identical. For the normalization of the Eigenfunktionen a method which has been used in principle by W. H. Furry is applied. It is, however, framed in a manner appropriate to the smaller quantum number range. Formulas are given for the normalized Eigenfunktionen over the several intervals of the variable, over the critical interval which includes the minimum of the potential energy and the turning points, as well as over the intervals remote from the turning points.In the special case of the harmonic oscillator, the method here used gives certain exact---not merely asymptotic---results. These, therefore, supply a basis for the appraisal of the corresponding results that are obtainable by other means.