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arXiv:2601.06369 · International Journal of Theoretical Physics

Exact Solutions for Compactly Supported Parabolic and Landau Quantum Barriers

Peter Collas, David Klein

quant-phmath-phSubmitted Jan 10, 2026• Announced Jan 13, 2026• 0 citations• Revised Aug 20, 2026 (v3)
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: For published full text go to https://rdcu.be/fAnz7

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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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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.
Abstract The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.