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Physical Review A

Semiclassical perturbation approach to quantum reflection

Neepa T. Maitra, Eric J. Heller

Published Dec 1, 1996• 45 citations
Publisher

Abstract

We consider the problem of quantum ``reflection above the barrier,'' introducing a distorted-wave Born approximation based on the primitive semiclassical Wentzel-Kramers-Brillouin wave function. The idea generalizes to the concept of an unperturbed Hamiltonian defined as the exact Hamiltonian plus small parts which ``turn off'' nonclassical processes such as barrier reflection and barrier tunneling. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.4763

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We consider the problem of quantum reflection above the barrier,'' introducing a distorted-wave Born approximation based on the primitive semiclassical Wentzel-Kramers-Brillouin wave function. The idea generalizes to the concept of an unperturbed Hamiltonian defined as the exact Hamiltonian plus small parts which turn off'' nonclassical processes such as barrier reflection and barrier tunneling. \textcopyright{} 1996 The American Physical Society.
We investigate tunneling through a barrier and reflection over a step potential in the limit of long waves, i.e., near the base of the barrier, or for barriers with a small action integral over the forbidden region, and at the top of the ramp where the usual semiclassical approximation breaks down systematically. General approximations of the probability of transmission and reflection are given and compared to competing ones showing that the present approach based on the comparison method can give far better results.
A curious effect is uncovered by calculating the time evolving probability of reflection of a Gaussian wave packet from a rectangular potential barrier while it is perturbed by reducing its height. A time interval is found during which this probability of reflection is larger (superarrivals) than in the unperturbed case. This nonclassical effect can be explained by requiring a wave function to act as a ``field'' through which an action, induced by the perturbation of the boundary condition, propagates at a speed depending on the rate of reducing the barrier height.
Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.
In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneling. We develop detailed semiclassical analyses in the complex domain and demonstrate that the key trajectories contributing to the fringed tunneling are quite different from the instanton-like trajectories, which works only in unperturbed or weakly-perturbed regimes. Indeed, such trajectories are traveling in the complex phase space being guided by the complexified stable and unstable manifolds which exhibit heteroclinic-like entanglement with the incoming wave manifolds.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
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.