EXAMPLE PROBLEM
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: