For the hydrogen ground state we are just adding two spin^12 particles so the possible values are
f= 0,1. The transition between the two states gives rise to EM waves withλ= 21 cm.
We will work out theeffect of an external B field on the Hydrogen hyperfine statesboth
in the strong field and in the weak field approximation. We also work theproblem without a field
strength approximation. Thealways applicable intermediate field strength resultis that the
four states have energies which depend on the strength of the B field. Two of the energy eigenstates
mix in a way that also depends on B. The four energies are
E = En 00 +A ̄h^2
4±μBBE = En 00 −A ̄h^2
4±
√(
A ̄h^2
2) 2
+ (μBB)^2.1.32 The Helium Atom
The Hamiltonian for Helium (See section 25) has the same terms as Hydrogen but has a large
perturbation due to the repulsion between the two electrons.
H=
p^21
2 m+
p^22
2 m−
Ze^2
r 1−
Ze^2
r 2+
e^2
|~r 1 −~r 2 |Note that theperturbation due to the repulsion between the two electronsis about the
same size as the the rest of the Hamiltonian so first order perturbation theory is unlikely to be
accurate.
TheHelium ground state has two electrons in the 1s level. Since the spatial state is
symmetric, the spin part of the state must be antisymmetric sos= 0 (as it always is for closed shells).
For our zeroth order energy eigenstates, we will useproduct states of Hydrogen wavefunctions
u(~r 1 ,~r 2 ) =φn 1 ℓ 1 m 1 (~r 1 )φn 2 ℓ 2 m 2 (~r 2 )and ignore the perturbation. The energy for two electrons in the (1s) state forZ = 2 is then
4 α^2 mc^2 = 108.8 eV.
We can estimate the ground state energy infirst order perturbation theory, using the electron
repulsion term as a (very large) perturbation. This is not very accurate.
We can improve the estimate of the ground state energy using thevariational principle. The main
problem with our estimate from perturbation theory is that we are not accounting forchanges in
the wave function of the electrons due to screening. We can do this in some reasonable
approximation by reducing the charge of the nucleus in the wavefunction (not in the Hamiltonian).
With the parameterZ∗, we get a better estimate of the energy.
Calculation Energy Zwfn
0 thOrder -108.8 2
1 stOrder perturbation theory -74.8 2
1 stOrder Variational -77.38^2716
Actual -78.975