Problem 8.1: Strong and weak field Zeeman. (40 points) In this problem, consider only the n = 2 level of a hydrogen atom placed in a uniform static magnetic field B = B3, Moreover, ignore the proton spin and the hyperfine coupling. There are thus 8 states, all of which are degenerate if we ignore both fine structure and the Zeeman effect. You do not have to rederive formulas we have already obtained. a) Consider the strong-field (large Zeeman) limit. In the limit that we ignore the fine struc- ture completely, what are the first-order energy corrections for each of the states from the Zeeman effect? Use the basis appropriate to the strong-field limit. Indicate any degeneracies that remain. b) Now include the fine structure as a small perturbation to the strong-field limit. We found the formula for the first-order energy corrections 1 ( + 1) mm, By = ja'me g l - f'+1f?}{'+1)] #0. (1) The expression is not well-defined for = 0. A more careful treatment (which vou don't have to do) shows that for this case, EY = c:_ [i 1] F=0. (2) 2 4n Using the above, what are the energy splittings for each state, in terms of a'm.c?? Which degeneracies are split, and which remain? } Now go to the opposite limit, the weak-field limit. If we ignore the Zeeman term entirely, what are the first order corrections for the various states from the fine structure? Use the basis appropriate to the weak-field limit. Indicate any degeneracies that remain. (We've already done this, but it's setting up the next part.) d) Now include the Zeeman effect as a small (weak-field) perturbation to the system of hydrogen with fine structure. What is the first-order energy correction to each state? Which degeneracies are split, and which remain