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Journal of Mathematical Physics

Transmission through a system of potential barriers. II. Necessary condition for complete transparency. A maximum transmission problem

Örjan Dammert

Published Feb 1, 1986• 5 citations
Publisher

Abstract

A smooth one-dimensional system of N potential barriers of arbitrary shapes (unequal or equal) is considered. A general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system. A maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition. The sub- and superbarrier cases are treated in a unified way. The exact final formulas can readily be converted into accurate approximate ones by insertion of available phase-integral expressions (of an arbitrary order) for certain characteristic quantities appearing in the formulas.

DOI
10.1063/1.527244

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A smooth one-dimensional system of N potential barriers of arbitrary shapes (unequal or equal) is considered. A general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system. A maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition. The sub- and superbarrier cases are treated in a unified way. The exact final formulas can readily be converted into accurate approximate ones by insertion of available phase-integral expressions (of an arbitrary order) for certain characteristic quantities appearing in the formulas.
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.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.
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.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.