130_notes.dvi

So adding this to the (easier) part above, we have

E(1)n =

eB

2 mc

(

mj ̄h±

mj ̄h

2 ℓ+ 1

)

=

e ̄hB

2 mc

mj

(

1 ±

1

2 ℓ+ 1

)

forj=ℓ±^12.

In summary then, we rewrite the fine structure shift.

∆E=−

1

2

mc^2 (Zα)^4

1

n^3

[

1

j+^12

−

3

4 n

]

To this we add the anomalous Zeeman effect

∆E=

e ̄hB

2 mc

mj

(

1 ±

1

2 ℓ+ 1

)

23.5 Homework Problems

1. Consider the fine structure of then= 2 states of the hydrogen atom. What is the spectrum
   in the absence of a magnetic field? How is the spectrum changed whenthe atom is placed in a
   magnetic field of 25,000 gauss? Give numerical values for the energyshifts in each of the above
   cases. Now, try to estimate the binding energy for the lowest energyn= 2 state including the
   relativistic, spin-orbit, and magnetic field.


2. Verify the relations used for^1 r,r^12 , andr^13 for hydrogen atom states up ton= 3 and for any
   nifl=n−1.


3. Calculate the fine structure of hydrogen atoms for spin 1 electrons forn= 1 andn= 2.
   Compute the energy shifts in eV.

23.6 Sample Test Problems

1. The relativistic correction to the Hydrogen Hamiltonian is H 1 = − p

4

8 m^3 c^2. Assume that

electrons have spin zero and that there is therefore no spin orbit correction. Calculate the

energy shifts and draw an energy diagram for the n=3 states of Hydrogen. You may use

〈ψnlm|^1 r|ψnlm〉=n (^21) a 0 and〈ψnlm|r^12 |ψnlm〉=n (^3) a 21
0 (l+^12 )

.

2. Calculate the fine structure energy shifts (in eV!) for then= 1,n= 2, andn= 3 states
   of Hydrogen. Include the effects of relativistic corrections, the spin-orbit interaction, and the
   so-called Darwin term (due to Dirac equation). Do not include hyperfine splitting or the effects
   of an external magnetic field. (Note: I am not asking you to derive the equations.) Clearly list
   the states in spectroscopic notation and make a diagram showing the allowed electric dipole
   decays of these states.


3. Calculate and show the splitting of then= 3 states (as in the previous problem) in a weak
   magnetic field B. Draw a diagram showing the states before and after the field is applied


4. If the general form of the spin-orbit coupling for a particle of massmand spinS~moving in a
   potentialV(r) isHSO= 2 m^12 c 2 L~·S~^1 rdVdr, what is the effect of that coupling on the spectrum of