Hyperfine Structure and Magnetic Fields (20 points)
Consider the ground state of hydrogen, which has energy Eo = -13.6 eV.
a. (5 pts) When we include the effects of proton spin, the single ground state splits into 4 different states labeled by (stot mstot), where stot = se + Sp. List these 4 states.
b. (5 pts) Relative to Eo, provide the energy of these 4 states when proton spin effects are included. What is the frequency splitting between these states (in units of GHz)?
c. (5 pts) Now, the atom is placed in a weak uniform magnetic field Boz, where Bo = 5 x 10^(-4) T. Without doing any calculations, describe whether each of the 4 states should go up in energy, down in energy, or have no change in their energy.
d. (5 pts) Experimental measurements show that when the Hydrogen atom is placed in the magnetic field of part (c), the frequency splitting between the |stot = 1, mstot = 1) and |stot = 1, mstot = 0) states is 7 MHz. Calculate the total Lande g-factor for Hydrogen, gstot: Note: The overall g-factor gstot will differ from the g-factor we derived in class (gJ) since in this case we are including the effects of the proton's spin angular momentum. The energy relationship we derived in class still holds when considering stot: eh Bextgstotmstot 2m (1)