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852HW6 - PHYS852 Quantum Mechanics II Spring 2009 HOMEWORK...

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PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 6: 1. The fine structure consists of the relativistic mass correction, the spin-orbit interaction, and the Darwin term. Based on the fact that j = + 1 / 2 or j = 1 / 2, show that for all n , , and j , the combined fine-structure effect is given by E (1) nj = m e c 2 α 4 4 n 4 parenleftbigg 2 n j + 1 / 2 3 / 2 parenrightbigg . You do not need to derive this formula, just show that it works for all cases. 2. In lecture, we derived expressions for degenerate perturbation theory by including V D into H 0 , and then applying non-degenerate perturbation theory. Now you are going to derive the same result by the more direct method. Begin from the expression ( H 0 + λV E nm ) | nm ) = 0 and as usual expand E nm and | nm ) in powers of λ . At each order you will get an equation, which you can then hit with one of three bra’s: ( nm | , ( n m | , where n negationslash = n , or ( nm | , where m negationslash = m . Each of these cases will give a different piece of the final answer. Be sure to also enforce normalization at each order to obtain additional information.
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