130_notes.dvi

We will sometimes group the constants such that

∆E≡

A

̄h^2

S~·~I.

(The textbook has numerous mistakes in this section.)

24.5 Homework Problems

1. Calculate the shifts in the hydrogen ground states due to a 1 kilogauss magnetic field.


2. Consider positronium, a hydrogen-like atom consisting of an electron and a positron (anti-
   electron). Calculate the fine structure of positronium forn= 1 andn= 2. Determine the
   hyperfine structure for the ground state. Compute the energyshifts in eV.


3. List the spectroscopic states allowed that arise from combining (s=^12 withl= 3), (s= 2 with
   l= 1), and (s 1 =^12 , s 2 = 1 andl= 4).

24.6 Sample Test Problems

1. Calculate the energy shifts to the four hyperfine ground states of hydrogen in a weak magnetic
   field. (The field is weak enough so that the perturbation is smaller than the hyperfine splitting.)


2. Calculate the splitting for the ground state of positronium due tothe spin-spin interaction
   between the electron and the positron. Try to correctly use the reduced mass where required
   but don’t let this detail keep you from working the problem.


3. A muonic hydrogen atom (proton plus muon) is in a relative 1sstate in an external magnetic
   field. Assume that the perturbation due to the hyperfine interaction and the magnetic field is
   given byW=AS~ 1 ·S~ 2 +ω 1 S 1 z+ω 2 S 2 z. Calculate the energies of the four nearly degenerate
   ground states. Do not assume that any terms in the Hamiltonian aresmall.


4. A hydrogen atom in the ground state is put in a magnetic field. Assume that the energy shift
   due to the B field is of the same order as the hyperfine splitting of theground state. Find the
   eigenenergies of the (four) ground states as a function of the B field strength. Make sure you
   define any constants (likeA) you use in terms of fundamental constants.