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

The eigenfunctions of H₀ may be chosen as {|n,l,m,s,m_s,i,m_i>}, and the corresponding eigenvalues are E_n=-E_I/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, m_s, i, m_i is therefore essential, the degeneracy with respect to l is accidental.  We assume that <H₀> >> <W_f> and that <W_f> >> <W_hf> and that we can use stationary perturbation theory to find the eigenvalues and eigenfunctions of H.

= W_mv + W_D +W_so .

.

W_mv and W_D 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:

• W_mv, a relativistic correction term:

  

   

  where

  

  Given the stationary state wavefunctions of the hydrogen atom we find

  ,

  

  Therefore

  

  with

  

  In general,

  

• W_D, the Darwin term:

  

  

  

  (integration by parts)

  Therefore

  

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

  

• W_so, the spin-orbit term:

  For the spin-orbit interaction we have

  

  

  W_so is a scalar operator.  It does not operate on the proton spin variables and commutes with J.  Its eigenvalues are therefore independent of m_j.  W_so 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 m_j i m_i>}.  These basis functions are eigenfunctions of W_so.

  

  

   

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

  

  

  With j=l ±1/2 we have

  

  W_so partially remove the six fold degeneracy of the 2p level.  The 2p level splits into a 2 fold degenerate 2p_1/2 level and a 4 fold degenerate 2p_3/2 level.

The fine structure of the hydrogen atom:

W_f = W_mv + W_D + W_so.

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 2s_1/2 level with respect to the 2p_1/2 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 W_hf remove this 4 fold degeneracy?

With  and and we have

We choose {|n l s j i f m_f>} 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 W_hf partially removes the degeneracy, splitting the f=0 and f=1 levels.  The result is the 21cm line of atomic hydrogen.

Problem:

(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, W_soµL×S.

  

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

  

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

  For the p_1/2 level we have <W_so>=-A.  For the p_3/2 level we have <W_so>=(1/2)A.  The p_3/2 level lies (3/2)A above the p_1/2 level.

  (b)  Consider the case of a weak field. The splitting introduced by B is much smaller than the splitting between the p_1/2 level and the p_3/2 level.  This is called the Zeeman effect. We consider

  

  a perturbation to

  

  The eigenstates of H₀’ are the |E₀ls;jm_j>.  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 L_z +2S_z are proportional to the matrix elements of J_z.  To evaluate the proportional constant we use

  

  

  

  

  For the p_1/2 level we have g=2/3.  For the p_3/2 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 g_J, 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;sm_s>.  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 2m_s are -1, 1.  Therefore the possible values for (m+2m_s) are -2, -1, 0, 1, 2.  The value 0 is two fold degenerate.

Link:

• Everything you always wanted to know about the hydrogen atom