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Journal of Hubei Normal University

Calculation on correction of the wave function and energy level in the time-dependent perturbation theory

Zhang Shao

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Based on the theory of series expansion,the energy and wave function of the time dependent perturbed system have been discussed,and optics Stark shifts is examined.

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We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schrödinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime.
The stimulated radiative corrections (or ac-Stark shift, or light shift...) of the energy levels of a hydrogenic atom irradiated by an intense, nonresonant, monomode laser field, are evaluated within the framework of the time-dependent perturbation theory, up to and including fourth order. The calculation is performed by using a Sturmian representation of the Coulomb Green's function. Whenever it is possible, comparison is made with other results, and several limiting cases are considered. If the laser frequency $\ensuremath{\omega}\ensuremath{\rightarrow}0$, one recovers the results of the dc-Stark effect. If $2\ensuremath{\omega}$ becomes larger than the ionization energy of the atomic state considered, the fourth-order level shift acquires an imaginary part which may be connected, via an extension of the optical theorem, to the two-photon ionization cross section. We point out also the difficulties encountered, when trying to get sensible estimates from partial summation of the infinite sums entering the perturbative expression of the fourth-order amplitudes. Finally, the order of magnitude of these various corrections is discussed.