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arXiv:quant-ph/9709039 · Physics of Atomic Nuclei

Supersymmetry of the nonstationary schrödinger equation and time-dependent exactly solvable quantum models

B. F. Samsonov, L. A. Shekoyan

Published Apr 1, 2000• 6 citations
arXivHTMLPDF

Abstract

New, exactly solvable time-dependent quantum models are obtained with the help of the supersymmetric extension of the nonstationary Schrödinger equation.
DOI
10.1134/1.855682

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A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are reviewed.
We formulate a set of conditions under which the nonstationary Schrödinger equation with a time-dependent Hamiltonian is exactly solvable analytically. The main requirement is the existence of a non-Abelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.