ls3_unit_8A

ls3_unit_8A - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY VIIIA. APPENDIX: RADIATIVE TRANSITION RATES REVISITED 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 ( ) . According to Equation [ II-23 ] of IRM:ST, we may write Ψ S t ( ) = T 0 t ( ) Ψ I t ( ) = exp i H 0 t h ( ) Ψ I t ( ) [ 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 ( ) = T I t , t 0 ( ) Ψ I t 0 ( ) [ VIIIA-2 ] where T I t , t 0 ( ) = 1 + i h Λ Ν Ξ Π d ʹ t H int I ʹ t ( ) T I ʹ t , t 0 ( ) t 0 t = 1 + i h Λ Ν Ξ Π d ʹ t T 0 1 ʹ t ( ) H int T 0 ʹ t ( ) T I ʹ t , t 0 ( ) t 0 t [ VIIIA-3 ] In IRM:ST we showed how this integral equation can be iterated to yield T I t , t 0 ( ) as a power series in H int I t ( ) . 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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ls3_unit_8A - THE INTERACTION OF RADIATION AND MATTER:...

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