PHY 491 HW Assignment #2; September 12-19, 2011 (1) 4. Using the 1st order perturbation results for πΈππ£ , where mv denotes mass velocity and (1) πΈπ π , where so denotes spin-orbit show that (1) (1) (1) (πΈ )2 4π π πΈππ£ + πΈπ π = πΈππ = 2ππ (3 − π+1/2), 2 where fs denontes fine structure and j is the total angular momentum, orbital plus spin. (1) πΈππ£ = − (1) πΈπ π = (πΈπ0 )2 4π [ − 3] 2ππ 2 π + 1/2 (πΈπ0 )2 π(π + 1) − π(π + 1) − 3/4 π[ ] 1 ππ 2 π (π + 2) (π + 1) (1) If we add the above two we get πΈππ . However for s=1/2, j=l+1/2 or j=l-1/2. This means l=j-1/2 and l=j+1/2. Eliminate l from the above equation for each value of l. Do the algebra and you will get the answer in terms of j.