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Physical Review A

Tunneling near the base of a barrier

Christopher Eltschka, Hanno Friedrich, M. J. Moritz, Johannes Trost

Published Aug 1, 1998• 21 citations
Publisher

Abstract

Generalized WKB connection formulas are used to derive the transmission amplitude describing tunneling through a potential barrier via two isolated classical turning points. The resulting formulas correctly reproduce the behavior in the vicinity of the base of the barrier where the tunneling probability vanishes exactly, and where formulas available to date fail.

DOI
10.1103/physreva.58.856

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Generalized WKB connection formulas are used to derive the transmission amplitude describing tunneling through a potential barrier via two isolated classical turning points. The resulting formulas correctly reproduce the behavior in the vicinity of the base of the barrier where the tunneling probability vanishes exactly, and where formulas available to date fail.
In this paper, we calculate the transmission and reflection amplitudes of wave functions for different potentials such as the delta function, the rectangular barrier, the Eckart potential, and the Hulthen potential. We describe the relationship between these amplitudes and compute the reflection resonances between each potential. We describe the transmission and reflection probabilities using the WKB formula and compare the results with ones obtained from matching the boundary conditions. Furthermore, we use a two by two transfer matrix to calculate a rigorous bound on the transmission and reflection probabilities.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.