Tutorial: Perturbation Approximations to Asymptotic Quantum Transition Rates

Steven Kenneth Kauffmann

Published Aug 1, 2019• 0 citations

Abstract

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.

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