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The Journal of Chemical Physics

A new approach to time-dependent perturbation theory

S. C. Mehrotra, James E. Boggs

Published Apr 1, 1976• 20 citations
Publisher

Abstract

A time-dependent perturbation theory for the solution of the time-dependent Schrödinger equation is developed and is applied to systems having two states with pulse and Lorentzian types of perturbations. It is shown in the case of the pulse perturbation that the expression corresponding to the Nth order of the perturbation theory converges to the exact solution when N tends to infinity. The results obtained from this perturbation theory, the Magnus approximation, and the time-dependent perturbation theory developed by Dirac are compared with the exact results. This theory is found to be good for larger values of the perturbations and for larger intervals of time.

DOI
10.1063/1.432578

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A time-dependent perturbation theory for the solution of the time-dependent Schrödinger equation is developed and is applied to systems having two states with pulse and Lorentzian types of perturbations. It is shown in the case of the pulse perturbation that the expression corresponding to the Nth order of the perturbation theory converges to the exact solution when N tends to infinity. The results obtained from this perturbation theory, the Magnus approximation, and the time-dependent perturbation theory developed by Dirac are compared with the exact results. This theory is found to be good for larger values of the perturbations and for larger intervals of time.
It is shown by calculations on several model systems of physical importance that exponential time-dependent perturbation theory (the Magnus expansion and similar expansions) does not converge in the Schr\"odinger representation for time intervals larger than the natural period of the system. This result has serious implications for the application of exponential perturbation theory in the Schr\"odinger representation to scattering and adiabatic turn-on problems. It is argued that this result also applies generally to more realistic systems.
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.
Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations respectively. In this chapter we consider the time-dependent perturbation theory applicable for small time-dependent perturbation. This is the case when we wish to investigate the response of systems to incident radiation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927 [1]. In this chapter we discuss the theory for constant and harmonic perturbations applied during a finite time interval, adiabatic and sudden perturbations.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
There are few potentials V (r) for which the Schrödinger equation is exactly solvable. We must therefore develop approximation techniques to obtain the eigenvalues and eigenfunctions for potentials that do not lead to exactly soluble equations. One of these methods is the perturbation theory, which was developed both for time independent and time dependent phenomena. We will discuss the time independent perturbation theory. The formulation of the time independent perturbation theory is different for nondegenerate and degenerate states.