Fine structure and hyper-fine structure

The hydrogen atom is a bound state of two spin ½ particles.  Its Hamiltonian may be written as H=H0+Wf+Whf, with

The eigenfunctions of H0 may be chosen as {|n,l,m,s,ms,i,mi>}, and the corresponding eigenvalues are En=-EI/n2.  Here i denotes the proton spin.  H0 does not act on the spin variables at all.  In addition, [H,L]=0.  The degeneracy with respect to m, s, ms, i, mi is therefore essential, the degeneracy with respect to l is accidental.  We assume that <H0> >> <Wf> and that <Wf> >> <Whf> and that we can use stationary perturbation theory to find the eigenvalues and eigenfunctions of H.

= Wmv + WD +Wso .


Wmv and WD do not operate on the spin variables and commute with L.  In a given subspace E(nl) they are therefore represented by multiples of the unit matrix, i.e. their eigenvalues do not depend on the spin variables or on m.

Fine structure:

The fine structure of the hydrogen atom:

Wf = Wmv + WD + Wso.

is the general expression.

 are degenerate.  This is an accidental degeneracy, and it remains in the exact solution of the Dirac equation neglecting the proton spin.  However, QED corrections raise the 2s1/2 level with respect to the 2p1/2 level by a quantity called the Lamb shift.

Hyperfine structure:

If we do neglect the proton spin, then the 1s level is 4-fold degenerate in the state space of two spin 1/2 particles.  Can Whf remove this 4 fold degeneracy?

With  and and we have

We choose {|n l s j i f mf>} as the eigenbasis of H0.  If n=1, l=0, s=j=i=½, f can take on the values 0 and 1.  We find that Whf partially removes the degeneracy, splitting the f=0 and f=1 levels.  The result is the 21cm line of atomic hydrogen.


A valence electron in an alkali atom is in a p-orbital (l=1).  Consider the simultaneous interactions of an external magnetic field B and the spin-orbit interaction.  The two interactions are described by the potential energy

(a)  Describe the energy levels for B=0.

(b)  Describe the energy levels for weak magnetic fields (Zeeman effect).  What are the Lande' g-factors?

(c)  Describe the energy levels for large magnetic fields (Paschen-Back effect).