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arXiv · 1706.05275

Solvable models of an open well and a bottomless barrier in 1-D

Zafar Ahmed·Dona Ghosh·Sachin Kumar·Nihar Turumella
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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 new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of Schr{\"o}dinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission $(R/T)$ co-efficients for the barrier have been derived. Interestingly, the crossover energy $E_c$ where $R(E_c)=1/2=T(E_c)$ may occur below/above or at the barrier-top. A connection between poles of these co-efficients and bound state eigenvalues of the well has also been demonstrated.

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Zafar Ahmed, Dona Ghosh, Sachin Kumar, Nihar Turumella. 2017-06-16. Solvable models of an open well and a bottomless barrier in 1-D. https://doi.org/10.1088/1361-6404%2Faa8c0c

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