The One-Electron Atom
113
Finally, the contribution of the Darwin term depends on the wave function
at the origin. Detailed analysis shows that ψ(0) =
3/ 2
0
1/ 2
1
1
l
Z
a n

 δ


π 

which then
leads to
〈 V 3 〉 =
2 2 | |
n
Z
E
n
α
for l = 0
= 0
for l ≠ 0
(4.61)
These relations allow us to obtain ∆E n ,
∆E n =
2 2
2
| |
4
3
1/ 2
4
n
Z
E
n
j
n
α


−


+


(4.62)
for the fine structure of the energy levels of the hydrogen atom.
The important properties of the fine structure given in Eq. (4.62) and
demonstrated in Fig. 4.2 are:
1. As expected, the fine structure corrections are smaller than E n by a factor
of about α
2
/4 ~ 10
–5
. The hydrogen atom energies E n themselves may be
written as E n = –α
2
mc
2
/2n
2
, which means that they are smaller than the
rest energy by a factor of about α
2
. All the shifts in the energy levels due
to fine structure corrections are negative and the shift decreases as
j increases. Furthermore, the corrections decrease rapidly as n increases,
so that its effect is more easily noticeable for small-n states.
2. The fine structure corrections remove some of the degeneracy of the
energy levels E n . The states with different j values now have different
energies. For a given n, the allowed j values range from 1/2 to (n –1/2) so
that each n level is now split into n levels.
3. Some degeneracy still survives. For a given n, the level with j = n – 1/2 is
nondegenerate but all the other levels have a degeneracy of order two
corresponding to l = j ± 1/2. For example, for n = 2, 2
2
P 3/2 is nondegenerate
but 2
2
P 1/2 and 2
2
S 1/2 are degenerate. Actually there is a small separation
between the 2
2
P 1/2 and 2
2
S 1/2 states also, known as the Lamb shift, which
can be satisfactorily explained in terms of quantum electrodynamics.