130_notes.dvi

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