Since the field is weak we work in the states
in which the hyperfine
perturbation is diagonal and compute the matrix elements for
But to do the computation, we will have to write those states in terms of
which we will abbreviate like
, which means the electron's spin is up and the
proton's spin is down.
Hyperfine Splitting in a Weak B Field
Now since the three
states are degenerate, we have to make sure all the
matrix elements between those states are zero, otherwise we should bite the bullet
and do the full problem as in the intermediate field case.
matrix is diagonal, as we could have guessed.
The only nonzero connection between states is between
and we are
assuming the hyperfine splitting between these states is large compared to the
So the full answer is
which is correct for both