Time fractional Schrödinger equation

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.