24.3.3 Hydrogen in a Strong B Field
We need to compute the matrix elements of the hyperfine perturbation using|msmi〉as a basis with
energiesE=En 00 + 2μBBms. The perturbation is
Hhf=AS~·~I
̄h^2whereA=^43 (Zα)^4
(
me
MN)
mec^2 gNn^13.Recalling that we can write
S~·I~=IzSz+^1
2I+S−+
1
2
I−S+,
the matrix elements can be easily computed. Note that the terms likeI−S+which change the state
will give zero.
A
̄h^2〈
+−
∣
∣
∣~I·S~
∣
∣
∣+−
〉
=
A
̄h^2〈
+−
∣
∣
∣
∣IzSz+1
2
I+S−+
1
2
I−S+
∣
∣
∣
∣+−
〉
=
A
̄h^2〈+−|IzSz|+−〉=−A
4
〈−+|Hhf|−+〉=−A
4
〈+ +|Hhf|+ +〉=A
4
〈−−|Hhf|−−〉=A
4
We can write all of these in one simple formula that only depends on relative sign ofmsandmi.
E=En 00 + 2μBBms±A
4
=En 00 + 2μBBms+A(msmI)24.3.4 Intermediate Field
Now we will work the full problem with no assumptions about which perturbation is stronger. This
is really not that hard so if we were just doing this problem on the homework, this assumption free
method would be the one to use. The reason we work the problem all three ways is as an example
of how to apply degenerate state perturbation theory to other problems.
We continue on as in the last section but work in the states of|fmf〉. The matrix for〈fmf|Hhf+
HB|f′m′f〉is
1 1
1 − 1
1 0
0 0
A
4 +μBB^000
0 A 4 −μBB 0 0
0 0 A 4 μBB
0 0 μBB −^34 A