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Journal of Physics A Mathematical and General

Lower and upper bounds for time-smoothed total transition probabilities and their rates

Sidney Golden

Published Jun 1, 1976• 3 citations
Publisher

Abstract

A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.

DOI
10.1088/0305-4470/9/6/012

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A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.
Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.
The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the SSS matrix and an appropriate definition by the time-dependent method.
Summary form only given. We have developed a general theory of single-photon bound-free transitions, which takes into account all the states of the quantum system of interest and treats the rotating- and counterrotating-wave terms on equal footing. With the help of the smoothed-dynamics approximation (SDA), which holds until the laser field is much weaker than the intramolecular field, a closed analytical result for the Laplace transform of the initial-state amplitude has been obtained. To perform the inverse Laplace transform, the pole approximation, extended to obtain higher-order corrections, has been employed.
Abstract Although the discussion in the preceding two chapters concentrated on the stationary states of an electronic system, the need to consider perturbations which mix stationary states and thereby excite transitions between these states was noted. In this chapter we apply time-dependent perturbation theory to estimate the rates of such transitions, first by developing a general formula for the transition probability between initial |i〉 and final |f〇 states; if the initial state |i〉 is lower in energy than |f〇 an absorption transition is involved, whereas if |f〇 is the lower state emission takes place. We then consider in some detail how such transitions are caused by interactions between the electronic system and the radiation field.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.