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arXiv:2601.06369 · International Journal of Theoretical Physics

Exact Solutions for Compact Support Parabolic and Landau Barriers

Peter Collas, David Klein

quant-phmath-phSubmitted Jan 10, 2026• Announced Jan 13, 2026• 0 citations
v1 · Jan 10, 2026v2 · Jan 28, 2026v3 · Aug 20, 2026
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Abstract

We derive exact solutions to the one-dimensional Schrödinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest
Journal reference
International Journal of Theoretical Physics, (2026) 65:242
DOI
10.1007/s10773-026-06448-5
Comments
27 pages, 9 figures

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Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.
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