The solution of the timehyphen;dependent Schrouml;dinger equation by the (t,trsquo;) method: Theory, computational algorithm and applications

摘要

A new powerful computational method is introduced for the solution of the time dependent Schrouml;dinger equation with timehyphen;dependent Hamiltonians (not necessarily timehyphen;periodic). The method is based on the use of the Floquethyphen;type operator in an extended Hilbert space, which was introduced by H. Sambe lsqb;Phys. Rev. A7, 2203 (1973)rsqb; for time periodic Hamiltonians, and was extended by J. Howland lsqb;Math Ann.207, 315 (1974)rsqb; for general time dependent Hamiltonians. The new proposed computational algorithm avoids the need to introduce the time ordering operator when the timehyphen;dependent Schrouml;dinger equation is integrated. Therefore it enables one to obtain the solution of the timehyphen;dependent Schrouml;dinger equation by using computational techniques that were originally developed for cases where the Hamiltonian is time independent. A timehyphen;independent expression for statehyphen;tohyphen;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.