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arXiv:quant-ph/9911064

Time-dependent Perturbation Theory in Quantum Mechanics

C. Y. Chen

Published Nov 14, 1999• 0 citations
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Abstract

After revealing difficulties of the standard time-dependent perturbation theory in quantum mechanics mainly from the viewpoint of practical calculation, we propose a new quasi-canonical perturbation theory. In the new theory, the dynamics of physical observables, instead of that of coefficients of wave-function expansion, is formulated so that the gauge-invariance and correspondence principles are observed naturally.

DOI
10.48550/arxiv.quant-ph/9911064

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After revealing difficulties of the standard time-dependent perturbation theory in quantum mechanics mainly from the viewpoint of practical calculation, we propose a new quasi-canonical perturbation theory. In the new theory, the dynamics of physical observables, instead of that of coefficients of wave-function expansion, is formulated so that the gauge-invariance and correspondence principles are observed naturally.
The previous time-dependent variation-perturbation theory, which is derived from the conventional time-dependent perturbation equations, contains two inconvenient features. When the perturbation becomes static, the wave function so obtained does not satisfy the time-dependent Schr\"odinger equation, and the perturbation equations cannot be reduced to the ordinary time-independent perturbation equations. To remove these difficulties, we introduce a perturbed time-dependent phase factor in deriving a new set of perturbation equations. We then construct the variation-perturbation functionals as in the previous theory; the perturbed wave functions are solved by minimizing these functionals. The same method is also applied to many-electron systems, using the coupled and uncoupled Hartree-Fock scheme.
Abstract The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory.
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.
We consider the classical limit of quantum mechanics from the viewpoint of perturbation theory. The main focus is time dependent perturbation theory, in particular, the time evolution of a harmonic oscillator coherent state in an anharmonic potential. We explore in detail a perturbation method introduced by Bhaumik and Dutta-Roy [J. Math. Phys. 16, 1131 (1975)] and resolve several complications that arise when this method is extended to second order. A classical limit for coherent states used by the above authors is then applied to the quantum perturbation expansions and, to second order, the classical Poincaré–Lindstedt series is retrieved. We conclude with an investigation of the connection between the classical limits of time dependent and time independent perturbation theories, respectively.