Like all the fine structure corrections, this is down by a factor of orderα^2 from the Hydrogen binding
energy.
The second term, due toSpin-Orbit interactions, is harder to derive correctly. We understand
the basis for this term. The magnetic moment from the electron’s spin interacts with the B field
produced by the current seen in the electron’s rest frame from the circulating proton.
H 2 =−~μe·B~We canderive(see section 23.4.2)B from a Lorentz transformation of the E field of a static proton
(We must also add in the Thomas Precession which we will not try to understand here).
H 2 =
1
2
ge^2
2 m^2 c^2 r^3~L·S~
This will be of the same order as the relativistic correction.
Now wecompute(see section 23.4.3)the relativity correction in first order perturbation theory.
〈ψnlm|H 1 |ψnlm〉= +E
(0)
n22 mc^2[
3 −
4 n
ℓ+^12]
The result depends onℓandn, but not onmℓorj. This means that we could use either the
ψnjmjℓsor theψnℓmℓsmsto calculate the effect ofH 1. We will need to use theψnjmjℓsto add in the
spin-orbit.
The first order perturbationenergy shift from the spin orbit correctioniscalculated(see
section 23.4.4)for the states of definitej.
〈ψnlm|H 2 |ψnlm〉=ge^2 ̄h^2
4 m^2 c^21
2
[j(j+ 1)−ℓ(ℓ+ 1)−s(s+ 1)]〈
1
r^3〉
nlm=
(g
2)E(0)
n22 mc^22
[ n
(ℓ+^12 )(ℓ+1)
−n
ℓ(ℓ+^12 )]
j=ℓ+^12
j=ℓ−^12Actually, theL~·S~term should give 0 forℓ= 0! In the above calculation there is anℓℓfactor which
makes the result forℓ= 0 undefined. There is an additional Dirac Equation contribution called
the“Darwin term”(see section 23.4.5) which is important forℓ= 0 and surprisingly makes the
above calculation right, even forℓ= 0!
We will now add these three fine structure corrections together for states of definitej. We start
with a formula which has slightly different forms forj=ℓ±^12.
Enjmjℓs=En(0)+En(0)22 mc^2
3 −^4 n
ℓ+^12+
{ 2 n
(ℓ+^12 )(ℓ+1)
−ℓ(ℓ^2 +n 1
2 )}(+)
(−)
Enjmjℓs=En(0)+
En(0)22 mc^2[
3 −
n
(ℓ+^12 ){
4 −ℓ+1^2
4 +^2 ℓ
}(+)
(−)