7.2: The Darwin term. (30 points) The formula (1) we found for the first-order correction to the hydrogen energies from fine structure is true for any l. However, for l=0 the way we derived it does not work. In this problem we explore how the derivation does go for this case. a) Argue that the matrix element (:nlm|vec(L)*vec(S)|nlm:) is zero for the case l=0. What does this tell you about the spin-orbit contribution to the fine structure in the l=0 case? There is more than one way to do this; one way involves remembering that for the angular momentum ladder operators, L_( )^()=L_(-)