Let’s define our perturbationW as
W≡
A
̄h^2S~ 1 ·S~ 2 +w 1 S 1 z+w 2 S 2 zHere, we have three constants that are determined by the strength of the interactions. We include
the interaction of the “nuclear” magnetic moment with the field, which we have so far neglected.
This is required because the positron, for example, has a magnetic moment equal to the electron so
that it could not be neglected.
1 1
1 − 1
1 0
0 0
A
4 +̄h
2 (w^1 +w^2 )^000
0 A 4 − ̄h 2 (w 1 +w 2 ) 0 0
0 0 A 4 h ̄ 2 (w 1 −w 2 )
0 0 ̄h 2 (w 1 −w 2 ) −^34 A
E 3 =−
A
4
+
√
(
A
2
) 2
+
(
̄h^2
2
(w 1 −w 2 )) 2
E 4 =−
A
4
−
√
(
A
2
) 2
+
(
̄h^2
2(w 1 −w 2 )) 2
Like previous hf except now we take (proton) otherB~·S~term into account.
24.4 Derivations and Computations
24.4.1 Hyperfine Correction in Hydrogen
We start from the magnetic moment of the nucleus
~μ=
ZegN
2 MNcI.~
Now we use the classical vector potential from a point dipole (see (green) Jackson page 147)
A~(~r) =−(~μ×∇~)^1
r.
We compute the field from this.
B~=∇×~ A~
Bk=∂
∂xiAjǫijk=−∂
∂xiμm∂
∂xnǫmnj1
rǫijk=−μm∂
∂xi∂
∂xn(−ǫmnjǫikj)1
r=−μm∂
∂xi∂
∂xn
(δkmδin−δknδim)1
r=−
(
μk∂
∂xn∂
∂xn
−μi∂
∂xi∂
∂xk)
1
rB~=−
(
~μ∇^21
r−∇~(~μ·∇~)1
r