Show that the spontaneous emission rate for a transition from n l to n’, l’ in hydrogen is...

Show that the spontaneous emission rate for a transition from n l to n’, l’ in hydrogen is

where

(The atom starts out with a specific value of m, and it goes to any of the states m' consistent with the selection rules: m' = m + 1, m, or m — 1. Notice that the answer is independent of m.) Hint: First calculate all the nonzero matrix elements of x, y, and z between |nlm) and |n'l'm') for the case l' = l + 1. From these, determine the quantity

Then do the same for l' = l – 1.