The Interaction of Radiation and Matter: Quantum Theory (cont.)

Back to topThis appendix builds on the formulation presented in Review of Basic Quantum Mechanics: Dynamic Behavior of Quantum Systems, Section II of the lecture notes entitledThe Interaction of Radiation and Matter: Semiclassical Theory(hereafter referred to as IRM:ST) to obtain explicit and reasonably general expressions for radiative transition rates. Let us suppose that a coupled system of radiation and matter is described in the Schrödinger picture by a wave function . According to Equation [ II-23 ] of IRM:ST, we may write

[ VIIIA-1 ] where is the Hamiltonian of the uncoupled radiation and matter systems. Further, Equations [ I-36a ] and [ I-37a ] of IRM:ST inform us that

where

[ VIIIA-2 ]

[ VIIIA-3 ] In IRM:ST we showed how this integral equation can be iterated to yield as a power series in . If denote the wave functions of the uncoupled system as , then the probability that the coupled system is in a state at a time is given by

[ VIIIA-4 ] To be more specific, if the system happened to be in state at a time , the probability that the coupled system is in a state at a time is given by

[ VIIIA-5 ] In the most experimental circumstances, we are interested in the rate at which transitions take place from some particular initial state to a set of final states --

i.e.

[ VIIIA-6 ]

( for ):The first order approximationUsing the second term in the iteration set forth in Equation [ II-27c ] of IRM:ST, we can write

[ VIIIA-7 ] where the factor was introduced to avoid the transient effects which might otherwise result from an apparent sudden application of the interaction between systems. Therefore, the first order approximation for is given by

[ VIIIA-8 ] which is, of course, a generalized form of the

Fermi golden rule.

( for ):The second order approximationUsing the third term in the iteration set forth in Equation [ I-27c ] of IRM:ST, we can write

[ VIIIA-9a ] or in the Schrödinger picture

[ VIIIA-9b ] Using the closure theorem we see that

[ VIIIA-10 ] Integrating and writing the matrix elements in more concise notation, we see that

[ VIIIA-11 ] Therefore, following the arguments presented above, we see that the second order approximation for is given by

[ VIIIA-12 ]

( for ):The nth order approximationWe then may make the obvious extrapolation of these results and write the nth order approximation for as

[ VIIIA-13 ] where ,,

etc.could be real imtermediate states or so calledfor the transitions to the state . The conservation of energy holds only between and .virtual intermediate states

This page was prepared and is maintained by R. Victor Jones,

Last updated May 8, 2000