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arXiv:1706.05275 · European Journal of Physics

Solvable models of an open well and a bottomless barrier: one-dimensional exponential potentials

Zafar Ahmed, D. C. Ghosh, Sachin Kumar, Nihar Turumella

Published Sep 13, 2017• 9 citations
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Abstract

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.

DOI
10.1088/1361-6404/aa8c0c

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We derive analytic expressions of the recursive solutions to the Schrödinger's equation by means of a cutoff potential technique for one-dimensional piecewise constant potentials. These solutions provide a method for accurately determining the transmission probabilities as well as the wave function in both classically accessible region and inaccessible region for any barrier potentials. It is also shown that the energy eigenvalues and the wave functions of bound states can be obtained for potential-well structures by exploiting this method. Calculational results of illustrative examples are shown in order to verify this method for treating barrier and potential-well problems.
Abstract A thorough analysis of one-dimensional rectangular potential systems, encompassing both barriers and wells, is provided within the framework of the phase-space propagation method for the Time-Independent Schrödinger Equation (TISE). An initial value representation approach is adopted, involving an ensemble of initial conditions invariant under phase-space flow on the left side of the barrier or well. The system’s periodicity in the free-motion regions, coupled with the simplicity of the potential, facilitates the derivation of analytical expressions for the evolution of the phase-space state across the barrier or well and onto the right side. The transfer matrix method is employed to obtain explicit analytical expressions for the phase-space state of the one-dimensional barrier and its transmission coefficient. A similar approach is extended to the potential energy well, where transcendental equations for both bound and virtual (anti-bound) states are derived through a geometrical analysis of the phase-space flow of the TISE and the stable and unstable manifolds. Furthermore, a detailed analysis of the transmission coefficient is conducted. It is shown that the energy curves for the nth bound state and the (n+2)(n+2)(n+2) th virtual state tend toward the same asymptotic limit. This limit corresponds to the energy curve (or line) of the (n+1)(n+1)(n+1) th resonance, offering deeper insights into the system’s energy spectrum and its behavior in the limit of infinite potential wells.