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Journal of Mathematical Physics

Time-Dependent Perturbation Theory

C. L. Hammer, T. A. Weber

Published Oct 1, 1965• 30 citations
Publisher

Abstract

A time-dependent perturbation theory which is based upon the U-matrix approach is presented using the interaction e-αt(1 + e-βt)−1V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time-independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|-δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete-state time-dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy-level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time-dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.

DOI
10.1063/1.1704699

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A time-dependent perturbation theory which is based upon the U-matrix approach is presented using the interaction e-αt(1 + e-βt)−1V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time-independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|-δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete-state time-dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy-level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time-dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.
This chapter presents a modernized version of a method originally due to Dirac. This approach depends on a straightforward expansion in powers of the perturbing interaction. The chapter examines a problem where the unperturbed Hamiltonian had a discrete spectrum, and the perturbation was explicitly time dependent. The situation where the spectrum is continuous and the perturbation is independent of time is encountered much more frequently in physics. The chapter explores the derivations of some rather general formulas for collision cross sections. The photo-effect in hydrogen constitutes a very interesting example of a perturbation problem. The dramatic behavior of the matrix element demonstrates that at low energy the Coulomb attraction is very effective in keeping the slow photo-electron near the origin. The chapter examines the result of a collision experiment. It aims to illustrate the power and limitations of the technique, and to establish its relationship to the treatment of elastic scattering.
Employing an infinite hierarchy of equations for the Green functions of a many particle system in its ground state mod psi N ), the well known non-relativistic, time-dependent, ground-state perturbation theory of a normal system of interacting fermions is derived, without introducing the hypothesis that mod psi N ) is the adiabatic transform of a non-interacting state. This new formulation of the problem allows us to renormalise conventional perturbation theory by introducing an effective interaction Gamma and the final result is a highly summed and manifestly self-consistent perturbation expansion in powers of Gamma . The authors calculate Gamma to second order and demonstrate that many of the well known results of time-dependent perturbation theory may be obtained by choosing the simplest approximations for Gamma . The results are readily generalised to finite temperatures and the formalism provides a generalised form of Hartree-Fock theory which may have important applications in many areas of physics.
A novel and readily understandable derivation of the Golden Rule of time dependent perturbation theory is presented. The derivation is based on adiabatic turning on of the perturbation as used, for instance, in some formal developments of scattering theory. Energy conservation is expressed in terms of an intuitively and physically appealing Lorentzian line shape rather than the artificial, oscillatory sin(x)=x type line shape that appears in conventional derivations. The conditions for the derivation's validity are compactly and conveniently expressed in the frequency/energy domain rather than in the usual time domain. The derivation also highlights how, along with energy conservation, the transition rate faithfully and instantaneously follows the variations in the square of the perturbing potential as one may expect in the adiabatic limit. In the first instance, the adiabatic turning on is achieved, as usual, by a single exponential time variation. But we demonstrate that the instantaneous following of the square of the perturbing potential by the transition rate is more general and that one can derive the Golden Rule for a general slowly varying time dependent perturbation. This allows one to derive generalisations of the simple decay law, originally derived in the classic paper by Weisskopf and Wigner; a tutorial exposition of the essence of this classic work is provided. The Oppenheimer method for applying the Golden rule to problems, such as the electric field ionisation of atoms, in which the perturbing potential can also create the final states, is reviewed. No use of an energy gap condition is needed to derive our results on adiabatic behaviour in contrast to the origianl derivations of the adiabatic theorem in quantum mechanics.
We develop a perturbative framework for time-dependent point interactions in quantum systems with purely discrete unperturbed spectrum. In two and three dimensions the point interaction is described through the renormalized resolvent obtained from heat-kernel regularization, while in one dimension the diagonal Green function is finite and no renormalization is required. Time dependence is introduced either through the interaction parameter or through the motion of the support point. In both cases we expand the non-autonomous Hamiltonian in the spectral basis of the corresponding static point-interaction problem and derive the first- and second-order pole shifts, projection corrections, and transition amplitudes. We also describe how finite degeneracies are treated by separating the coupled point-interaction direction from the unaffected orthogonal subspace in each degenerate eigenspace. The method is illustrated by explicit examples: point interactions with time-dependent coupling, harmonic oscillators perturbed by moving point interactions, and a particle on a sphere with a moving interaction center. We also discuss the one-dimensional harmonic oscillator with a moving delta potential, showing that in the non-renormalized case the present formulation reduces to the standard time-dependent perturbation theory. Finally, an exactly solvable delta center moving with constant velocity on a circle is used as a discrete-spectrum example for the local moving-center expansion and its velocity-dependent dynamical corrections.
This chapter presents approximation methods in quantum mechanics, focusing on both time-independent and time-dependent perturbation theory. The time-independent formulation is developed up to second-order corrections for both nondegenerate and degenerate systems. Applications include the analysis of the Zeeman and Stark effects, where energy level shifts due to magnetic and electric fields are calculated. The time-dependent perturbation theory is introduced, with harmonic perturbations used to derive the first-order transition probabilities and to establish Fermi's golden rule. As an application, the ionization probability of a hydrogen atom under a linearly polarized electromagnetic wave is computed in detail. The treatment is extended to include the case of an unpolarized electromagnetic field. Metadata/Keywords: perturbation theory, Zeeman effect, Stark effect, time-dependent perturbation, harmonic perturbation, Fermi's golden rule, hydrogen ionization
Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.