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Hyperfine Correction in Hydrogen

We start from the magnetic moment of the nucleus

Now we use the classical vector potential from a point dipole
(see (green) Jackson page 147)

We compute the field from this.

Then we compute the energy shift in first order perturbation theory for s states.

The second term can be simplified because of the spherical symmetry of s states.
(Basically the derivative with respect to x is odd in x so when the integral is done,
only the terms where
are nonzero).

So we have

Now working out the
term in spherical coordinates,

we find that it is zero everywhere
but we must be careful at
.
To find the effect at
we will integrate.

So the integral is nonzero for any region including the origin, which implies

We can now evaluate the expectation value.

Simply writing the
in terms of
and regrouping, we get

We will sometimes group the constants such that

(The textbook has numerous mistakes in this section.)

Jim Branson
2013-04-22