23.2 Hydrogen Atom in a Weak Magnetic Field
One way to break the spherical symmetry is to apply an external B field. Lets assume that the field
is weak enough that the energy shifts due to it are smaller than the fine structure corrections. Our
Hamiltonian can now be written asH=H 0 + (H 1 +H 2 ) +H 3 , whereH 0 =p
2
2 μ−Ze^2
r is the normal
Hydrogen problem,H 1 +H 2 is the fine structure correction, and
H 3 =
eB~
2 mc·(~L+ 2S~) =
eB
2 mc(Lz+ 2Sz)is the term due to the weak magnetic field.
We now run into a problem becauseH 1 +H 2 picks eigenstates ofJ^2 andJzwhileH 3 picks eigenstates
ofLzandSz. In the weak field limit, we can do perturbation theory using the states of definitej.
A directcalculation(see section 23.4.6)of the Anomalous Zeeman Effect gives the energy shifts
in a weak B field.
∆E=
〈
ψnℓjmj∣
∣ 2 eBmc(Lz+ 2Sz)∣
∣ψnℓjmj〉
=e 2 ̄hBmcmj(
1 ± 2 ℓ^1 +1
)
This is the correction, due to a weak magnetic field, which we should add to the fine structure
energies.
Enjmjℓs=−1
2
α^2 mc^2(
1
n^2+
α^2
n^3[
1
j+^12−
3
4 n])
Thus, in a weak field, thethe degeneracy is completely broken for the statesψnjmjℓs. All
the states can be detected spectroscopically.