EXAMPLE PROBLEM

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Posted by student team #5 (Anthony Kuchera, Jeff Klatsky, Chelsey Morien)


QUESTION:

Consider a transition from between two states of a nucleus with spins and , respectively. The transition probability is proportional to the squared matrix element where is a hermitian tensor operator of rank responsible for the process. Define the reduced transition probability

as a sum of squared matrix elements over final projections and operator projections .


a) Express in terms of the reduced matrix element and show that it does not depend on the initial projection .

b) Establish the detailed balance between the reduced transition probabilities of the direct , and inverse processes. Hint: This is just the ratio between and


SOLUTION:

a) According to the Wigner-Eckert theorem, the entire dependence of the matrix element of a tensor operator on the magnetic quantum numbers is concentrated in the vector coupling coefficients,


We obtain the rate by squaring this and summing over and


Using the orthogonality condition:


Which leads us to our final result:

It is obvious that this result does not depend on


b) All that is missing to find the detailed balance relation is . This is done in the same way as part a).

Note, the only difference is the sum over

Thus, we have

And the detailed balance relation is: