ls3_unit_8A - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY VIIIA. A PPENDIX : R ADIATIVE T RANSITION R ATES R EVISITED This appendix builds on the formulation presented in Review of Basic Quantum Mechanics: Dynamic Behavior of Quantum Systems, Section II of the lecture notes entitled The 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 Ψ S t ( 29 . According to Equation [ II-23 ] of IRM:ST, we may write Ψ S t ( 29 = T 0 t ( 29 Ψ I t ( 29 = exp - iH 0 t h ( 29 Ψ I t ( 29 [ VIIIA-1 ] where H 0 is the Hamiltonian of the uncoupled radiation and matter systems. Further, Equations [ I-36a ] and [ I-37a ] of IRM:ST inform us that Ψ I t ( 29 = T I t , t 0 ( 29 Ψ I t 0 ( 29 [ VIIIA-2 ] where T I t , t 0 ( 29 = 1 + - i h d t H int I t ( 29 T I t , t 0 ( 29 t 0 t = 1 + - i h d t T 0 - 1 t ( 29 H int T 0 t ( 29 T I t , t 0 ( 29 t 0 t [ VIIIA-3 ] In IRM:ST we showed how this integral equation can be iterated to yield T I t , t 0 ( 29 as a power series in H int I t ( 29 . If denote the wave functions of the uncoupled system as ψ , then the probability that the coupled system is in a state ψ f at a time t is given by
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY ψ f Ψ S t ( 29 2 = ψ f T 0 t ( 29
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