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:

• Wmv, a relativistic correction term:

where

Given the stationary state wavefunctions of the hydrogen atom we find

,

Therefore

with

In general,

• WD, the Darwin term:

(integration by parts)

Therefore

For l¹0, |Rnl(0)|2=0, so WD is non zero only for l=0 states.

• Wso, the spin-orbit term:

For the spin-orbit interaction we have

Wso is a scalar operator.  It does not operate on the proton spin variables and commutes with J.  Its eigenvalues are therefore independent of mjWso also commutes with L2 and matrix elements between states with different values of l are zero.

The eigenfunctions of H0 may also be chosen as {|n l s j mj i mi>}.  These basis functions are eigenfunctions of Wso.

If l=0 then j=s and Wso=0. If l¹0, then

With j=l ±1/2 we have

Wso partially remove the six fold degeneracy of the 2p level.  The 2p level splits into a 2 fold degenerate 2p1/2 level and a 4 fold degenerate 2p3/2 level.

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.

#### Problem:

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).

• Solution:

(a)  If B=0, we have just the spin orbit interaction, WsoµL×S.

The eigenstates of H0 are the eigenstates of J2, L2, and S2 and they may be denotes by {|nls;jmj>}.

We have l=1, j=1/2 or j=3/2.

For the p1/2 level we have <Wso>=-A.  For the p3/2 level we have <Wso>=(1/2)A.  The p3/2 level lies (3/2)A above the p1/2 level.

(b)  Consider the case of a weak field. The splitting introduced by B is much smaller than the splitting between the p1/2 level and the p3/2 level.  This is called the Zeeman effect. We consider

a perturbation to

The eigenstates of H0 are the |E0ls;jmj>.  The corresponding eigenvalues are 2j+1 fold degenerate.  L+2S=V is a vector operator. In the subspace E(k, j) we have

using the projection theorem. Choose your coordinate system such that B points in the z-direction.  Then the matrix elements of Lz +2Sz are proportional to the matrix elements of Jz.  To evaluate the proportional constant we use

For the p1/2 level we have g=2/3.  For the p3/2 level we have g=4/3.

The B field completely removes the degeneracy of each level.  The energy differences are proportional to B and a constant gJ, the Landé g - factor.

(c)  Consider the case of a strong field.  The splitting introduced by B is much larger than the splitting introduced by the spin orbit interaction.  This is called the Paschen-Back effect.  Assume the spin orbit interaction can be ignored.  We consider V’ a perturbation to H0.  The eigenstates of H0 are |nlm;sms>.  They are (2l+1)(2s+1) fold degenerate. V’ is diagonal in this basis.

The possible values for m are -1, 0, 1, and the possible value for 2ms are -1, 1.  Therefore the possible values for (m+2ms) are -2, -1, 0, 1, 2.  The value 0 is two fold degenerate.