0
0 500
E
B
full calc.
strong field
weak field
We can make a more general calculation, in which the interaction of the nuclear magnetic moment
is of the same order as the electron. This occurs in muonic hydrogenor positronium. See Example
24.3.6:The Hyperfine Splitting in an Intermediate B Field.
24.3 Examples
24.3.1 Splitting of the Hydrogen Ground State
The ground state of Hydrogen has a spin^12 electron coupled to a spin^12 proton, giving total angular
momentum state off= 0,1. We have computed in first order perturbation theory that
∆E=
2
3
(Zα)^4
(
m
MN
)
(mc^2 )gN
1
n^3
(
f(f+ 1)−
3
2
)
.
The energy difference between the two hyperfine levels determinesthe wave length of the radiation
emitted in hyperfine transitions.
∆Ef=1−∆Ef=0=
4
3
(Zα)^4
(
m
MN
)
(mc^2 )gN
1
n^3
Forn= 1 Hydrogen, this gives