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

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 . 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
 

[ VIIIA-2 ]

where
 

[ 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 ]

The first order approximation ( for  ):

Using 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.

The second order approximation ( for  ):

Using 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 ]

The nth order approximation ( for  ):

We 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 called  virtual intermediate statesfor the transitions to the state . The conservation of energy holds only between  and .