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International Journal of Modern Physics A

AN INTERESTING MODEL OF QUANTAL TRANSMISSION

Zafar Ahmed

Published Sep 10, 2006• 2 citations
Publisher

Abstract

We present the case of quantal transmission through a smooth, single-piece exponential potential, V(x) = -V0ex/a, in contrast to the piecewise continuous potentials, as an intersecting model to demonstrate the analytic extraction of transmission (reflection) coefficient. Several other aspects of quantal transmission are also discussed in this simple and exactly solvable model.

DOI
10.1142/s0217751x0603299x

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We present the case of quantal transmission through a smooth, single-piece exponential potential, V(x) = -V0ex/a, in contrast to the piecewise continuous potentials, as an intersecting model to demonstrate the analytic extraction of transmission (reflection) coefficient. Several other aspects of quantal transmission are also discussed in this simple and exactly solvable model.
We present the case of quantal transmission through a smooth, single-piece exponential potential, V(x)=−V0ex/aV(x)=-V_0 e^{x/a}V(x)=−V0​ex/a, in contrast to the piece-wise continuous potentials, as a pedagogic model to demonstrate the analytic extraction of transmission (reflection) co-efficient.
The transmission coefficient for quantal tunneling through a rectangular, one-dimensional potential barrier is analyzed. By employing continuity of ‖ ψ ‖2 and flux the results, even with an asymmetrical barrier, are obtained easily. The asymmetrical barrier has the same transmission in either direction. Considerable insight into the form of the most physical quantity, ‖ ψ(x) ‖2, is afforded with little algebra. Graphs of transmission are given for general barriers, and properties of the solutions are discussed.
Disparate behaviour of above-the-step quantal reflection probability for smooth and composite semi-infinite potentials is shown. For a smooth semi-infinite potential (step) the reflectivity is usually a monotonically decreasing function of energy. But the composite two-piece (non-differentiable, e.g., at x = 0) potential step gives rise to a parameter-dependent pronounced single minimum in the reflectivity. Three analytically solvable and several other models of composite semi-infinite (step) potentials are shown supporting such a behaviour of the quantal reflection.
Abstract We present one-dimensional potentials V ( x ) = V 0 ( e 2 ∣ x ∣ / a − 1 ) as solvable models of a well ( V 0 > 0 ) and a barrier ( V 0 < 0 ). Apart from being a new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of the Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission ( R / T ) coefficients for the barrier have been derived. Interestingly, the crossover energy E c where R ( E c ) = 1 / 2 = T ( E c ) may occur above, below or at the barrier-top. A connection between the poles of these coefficients and the bound-state eigenvalues of the well is also demonstrated.