On the Reflection Coefficient in a Superbarrier Passage of a Quantum Particle

Abstract

The problem on the reflection coefficient is considered for a quantum particle passing over a potential barrier. A rigorous treatment of this problem is not available in the literature. We have developed a consecutive method of finding the pre-exponential multiplier in solving the problem on the probability of the passage in a quasiclassical case, including a correct choice of the singular point. Its novelty in comparison to the earlier used methods is that it involves some rules for the most expedient analytic continuation of the wave function to the complex region. Our method does not use the conventional subdivision of the incident wave function into two ones: penetrating and reflected. When considering the action integral L = ∫pdx = L ₁ + iL ₂, we obtain a bundle of trajectories with L ₂ = const: one extreme member of this bundle is the real axis and the other extreme member is a curve which is indefinitely close to one of the singular points. This singular point plays the leaging role in finding the asymptote of the reflection coefficient R having a physical meaning. Five examples that explain the theory are considered.