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Physical Review

Decay Theory of Closely Coupled Unstable States

Lyman Mower

Published Feb 25, 1966• 227 citations
Publisher

Abstract

A treatment of decay theory is presented which generalizes existing treatments so as to apply to closely coupled unstable states. The treatment is based upon a Green's function formulation for the transition amplitude in which the states of interest are selected by suitable projection operators. As an indication of the practicality of the resulting formulation a number of examples are given involving the stimulated transition between closely coupled unstable states.

DOI
10.1103/physrev.142.799

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A treatment of decay theory is presented which generalizes existing treatments so as to apply to closely coupled unstable states. The treatment is based upon a Green's function formulation for the transition amplitude in which the states of interest are selected by suitable projection operators. As an indication of the practicality of the resulting formulation a number of examples are given involving the stimulated transition between closely coupled unstable states.
A theory of sequential decay of an unstable system is presented using the projection-operator algebra suggested by the recent projection-operator approach to reaction theory. The formulation is an attempt to provide an alternative to perturbation theory. While exact, the theory is cast in a phenomenological form that is suggestive in its interpretation and may be extended so as to apply to sequential transitions in general. Examples treated include the transition between unstable states of an atom induced by an intense electromagnetic field as well as a problem in resonance fluorescence. Higher-order corrections to the more familiar theory are a natural consequence of the description of sequential decay and sequential transitions presented here.
The theory of sequential decays of an unstable system is studied. Examples include the sequential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate levels, and the decay of an unstable particle into other unstable particles. To describe these phenomena, a factorization of the Green's function is introduced. This leads to a simple, and intuitively obvious, description of sequential decays. It also makes possible an assessment of the accuracy of this description.
A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.
The theory of se~uential decays of an unstable system is studied.Examples include the se~uential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate leveis} and the decay of an unstable particle into other unstable particles.To describe these phenomena} a factorization of the Green's function is introduced.This leads to a simple} and intuitively obvious} description of se~uential decays.It also makes possible an assessment of the accuracy of this description.
Abstract A quantum model of the scattering of a particle via absorption and re-emission at the origin is described. The model is capable of exhibiting an arbitrary number of resonances, and the associated unstable states are discussed. The emphasis is on obtaining exact results. It is found that the exponential decay law is of limited application, and that in all cases of non-zero coupling the decay goes ultimately as t−3. Finally, it is pointed out that the treatment of unstable states in this type of model depends only on a knowledge of the phase shift as a function of radial momentum, and a possible phenomenological method is suggested.