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Time fractional Schrödinger equation. (English) Zbl 1071.81035
Summary: The Schrödinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrödinger equation. The resulting Hamiltonian is found to be non-Hermitian and nonlocal in time. The resulting wave functions are thus not invariant under time reversal. The time fractional Schrödinger equation is solved for a free particle and for a potential well. Probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative. New identities for the Mittag–Leffler function are also found and presented in an Appendix.
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
26A33Fractional derivatives and integrals (real functions)
35K57Reaction-diffusion equations