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

Abstract

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.

Adiabatic theorem for non-Hermitian time-dependent open systems

“…It has been shown by Peskin and Moiseyev that by regarding time as an extra coordinate t ′ , one can obtain another Schrödinger equation with a time independent Hamiltonian in the extended Hilbert space (r, t ′ ), whose solution Ψ(r, t ′ , t) has an analytical time dependence (given by the analytical time evolution operator associated with time-independent Hamiltonians) [7]. Our desired wavfunction Ψ(r, t) can be deduced from this wavfunction by a simple operation (that will be shown later).…”

“…In order to answer this question we use the (t,t') formalism [7] together with the non-hermitian quantum mechanics (NHQM) formalism. The use of NHQM formalism to describe the dynamics of atoms/molecules subjected to CW laser fields is essential, since only then can the dynamics be described in terms physical, square integrable, resonance Floquet states.…”

Numerical method for integrodifferential generalized Langevin and master equations

“…Here we show that these integro-differential equations can be converted to ordinary-differential equations at the expense of introducing a new time variable which is treated as if it is of spatial type. [Similar schemes are employed to numerically solve the Schrödinger equation for time-dependent Hamiltonians [6] and as analytical tools [7]. There is also some resemblence to schemes for solving intego-differential equations of viscoelasticity [8].…”

Preparation and control of aligned cyclic rotational states

“…Eigenstates of F belong to an extended Hilbert space formed by the composition of the Hilbert space corresponding to the spatial part of the Hamiltonian operator and the space formed by all possible periodic Fourier basis functions of the time coordinate, with finite norm [26]. Yajima [27] and Howland [28] showed that for periodic systems, the so-called Floquet operator, that is, the time-evolution operator during one period T of the external perturbation U (T + t 0 ,t 0 ), and the Floquet Hamiltonian are spectrally equivalent.…”