The solution of the time-dependent Schrödinger equation by the (t,t′) method: Theory, computational algorithm and applications

A new powerful computational method is introduced for the solution of the time dependent Schrödinger equation with time-dependent Hamiltonians (not necessarily time-periodic). The method is based on the use of the Floquet-type operator in an extended Hilbert space, which was introduced by H. Sambe Phys. Rev. A 7, 2203 (1973) for time periodic Hamiltonians, and was extended by J. Howland Math Ann. 207, 315 (1974) for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the time-dependent Schrödinger equation is integrated. Therefore it enables one to obtain the solution of the time-dependent Schrödinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A time-independent expression for state-to-state transition probabilities is derived by using the analytical time dependence of the time evolution operator in the generalized Hilbert space. Illustrative numerical examples for complex scaled time periodic model Hamiltonians are given.