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Supersymmetry of the Nonstationary Schrödinger Equation and Time-Dependent Exactly Solvable Quantum Models 1

Boris F. Samsonov, L. A. Shekoyan

Published Jan 1, 1997• 6 citations
OpenAlex

Abstract

Abstract — New exactly solvable time-dependent quantum models are obtained with the help of the supersymmetric extension of the nonstationary Schrödinger equation. 1.

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Abstract Here we report a project in which time-dependent supersymmetry has been employed to derive a new potential and eigenfunctions that satisfy the Schrödinger equation. The supersymmetry method is outlined and we apply it to a wavefunction obeying the free-particle Schrödinger equation. This leads to an exactly soluble model in which a quantum particle is seen to ‘surf’ on a time-dependent potential. The model can be solved and understood within both classical and quantum mechanics and the relationship between the two approaches is discussed. The mathematics of this formalism is accessible to a final year British undergraduate making supersymmetry derived Hamiltonians suitable as a final year theoretical physics research project.
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.