THE SOLUTION OF THE TIME-DEPENDENT SCHRODINGER-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 Schrodinger equation with time-dependent Hamiltonia ns (not necessarily time-periodic). The method is based on the use of the Floquet-type operator in an extended Hilbert space, which was intr oduced by H. Sambe [Phys. Rev. A 7, 2203 (1973)] for time periodic Ham iltonians, and was extended by J. Howland [Math Ann. 207, 315 (1974)] for general time dependent Hamiltonians. The new proposed computationa l algorithm avoids the need to introduce the time ordering operator wh en the time-dependent Schrodinger equation is integrated. Therefore it enables one to obtain the solution of the time-dependent Schrodinger equation by using computational techniques that were originally develo ped for cases where the Hamiltonian is time independent. A time-indepe ndent expression for state-to-state transition probabilities is derive d by using the analytical time dependence of the time evolution operat or in the generalized Hilbert space. Illustrative numerical examples f or complex scaled time periodic model Hamiltonians are given.