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Journal of Physics A Mathematical and General

Semiclassically weak reflections above analytic and non-analytic potential barriers

Michael Berry

Published Dec 1, 1982• 97 citations
Publisher

Abstract

The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross) -1 )); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross) N . If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n ) gives r approximately exp(-h(cross) -n (n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10 -1000 .

DOI
10.1088/0305-4470/15/12/021

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The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross) -1 )); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross) N . If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod -n ) gives r approximately exp(-h(cross) -n (n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod 2 approximately 10 -1000 .
We show how quantum mechanical barrier reflection and transmission coefficients ℛ and 𝒯 can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V ( x ) = ∑ j V ( j ) ( x ), where each V ( j ) goes to zero for x →±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual ℛ ( j ) and 𝒯 ( j ) . The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.
A potential barrier of the kind studied by Fowler and others may be represented by the analytic function VVV (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy WWW is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W<VmaxW<{V}_{max}W<Vmax​.
Quantum tunneling of a nonrelativistic particle through a singular potential barrier V is studied on the line. The Hamiltonian is a self-adjoint extension of the operator H1=−d2/dx2+V(x). If H1 is essentially self-adjoint on its natural domain, the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers. If H1 is not essentially self-adjoint, the Friedrichs extension of H1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones. In general, the occurrence of tunneling is not excluded and depends on the self-adjoint extension we choose as the Hamiltonian of our problem. As an example, we evaluate the transmission coefficient for all self-adjoint extensions of the operator H1 referring to V(x)=gx−2 with 0<g< 3/4 .
Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ 1/3 , including those that are O (ℏ 0 ) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.
The reflection and transmission of a wave on a two-dimensional barrier is studied quantum mechanically and semiclassically. The simple model consists of two degrees of freedom. One of them contains a barrier (of the Eckart type); the second allows only a bound motion in discrete quantum states. These two degrees of freedom are coupled. The transition probabilities for transitions between the different quantum states can, in the model considered here, be evaluated analytically, both quantum mechanically and semiclassically. Several of the so called uniform semiclassical approximations and also the initial value integral representation for the semiclassical transition amplitudes are considered and compared to the exact quantum mechanical results. Some new insight on the semiclassical treatment of the tunneling process is gained, specially concerning the proper choice of the integration path for the initial value integral representation of the transition amplitudes.
Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter.