U. Peskin et N. Moiseyev, THE SOLUTION OF THE TIME-DEPENDENT SCHRODINGER-EQUATION BY THE (T,T')METHOD - THEORY, COMPUTATIONAL ALGORITHM AND APPLICATIONS, The Journal of chemical physics, 99(6), 1993, pp. 4590-4596
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.