10.6 The Zeeman effect
A magnetic field on a (Hydrogen like) atom interacts with both the electron spin as well
as its orbital angular momentum generated magnetic moment. To first order in a constant
magnetic fieldBalong thezaxis, the perturbing Hamiltonian is given by
H 1 =−
eB
2 mec
(Lz+ 2Sz) (10.40)
The factor of 2 in front of spin is due to the fact that the gyro-magnetic ratio for the electron
is 2 (up to orderαcorrections). This time, spin needs to be taken into account, so weshall
label the states by|n,ℓ,mℓ,ms〉wheremℓis the orbital magnetic quantum number, satisfying
|mℓ|≤ℓ, whilemsis the spin magnetic quantum number given byms=± 1 /2.
Conveniently,H 1 is diagonal in this basis. So, although in excited states this is degenerate
perturbation theory, it is straightforward to evaluate thecorrections. The diagonal matrix elements
ofH 1 are given by
E|^1 n,ℓ,mℓ,ms〉 = −
eB
2 mec〈n,ℓ,mℓ,ms|(Lz+ 2Sz)|n,ℓ,mℓ,ms〉= −
eB
2 mec(mℓ+ 2ms) (10.41)This same correction may of course also be evaluated in the basis|j,ℓ,m〉but the formula is slightly
more involved,
E|^1 n,j,ℓ,m〉 = −
eB
2 mec〈n,j,ℓ,m|(Jz+Sz)|n,j,ℓ,m〉= −
eB
2 mecm(
1 ±1
2 ℓ+ 1
)
(10.42)where the above±refers to the two casesj=ℓ± 1 /2, andm=mℓ+msis the total magnetic
quantum number.
10.7 Spin orbit coupling
Alkali atoms, such as Li, Na, K, etc, show many of the properties of single electron atoms and
ions, because only one electron occurs on top of a completely filled shell (characteristic of the
noble gases). But the cloud of inner electrons does interactwith the outer electron via spin orbit
coupling. The magnetic moment of the outer electron couplesto the orbital magnetic moment, via
a perturbing Hamiltonian of the type
H 1 =φ(r)L·S (10.43)In the basis of states given by the tensor product of the spin and orbital quantum numbers, this
Hamiltonian is not diagonal. One could evaluate the matrix elements ofH 1 in this basis and