Before claiming that this
formula explains the fine structure of the hydrogen atom, however, one
needs to be careful. The correction is of the order 
,
which means it of the order 
, where v is the electron speed. The
kinetic energy used in the Hamiltonian when solving the
Schrödinger equation was just 
, which contributed to order
. However, the next term in the expansion of the true
relativistic kinetic energy is of order 
and hence will contribute
to order . So if one wishes to quote the energy splittings
of the hydrogen atom accurate to order , one had better include
the contribution from this further correction.
The relativistic kinetic energy of the electron can be expanded in terms of momentum as
Therefore, the correction to the Hamiltonian is
At first sight, this looks quite complicated, since it involves the
operator 
. However, one can make use of the fact
that
to get
With 
, applying first-order perturbation theory to this
Hamiltonian reduces to the problem of finding the expectation values
of 
and 
. This can be done with some effort, and the
result is
Combining equations 57 and 52 and using the fact that
j=l-1/2,l+1/2, the complete energy correction to order 
may be
written
This energy correction depends only on j and is called the
fine structure of the hydrogen atom, since it is of order
times smaller than the principle energy
splittings. This is why 
is known as the fine-structure
constant. The fine structure of the hydrogen atom is illustrated
in figure 4. Note that all levels are shifted down from the
Bohr energies, and that for every n and l there are two states
corresponding to j=l-1/2 and j=l+1/2, except for s states. Also
note that states with the same n and j but different l have the same
energies, though this will be shown later not to be true due an effect
know as the Lamb shift. As an aside, these fine structure splittings
were derived by Sommerfeld by modifying the Bohr theory to allow
elliptical orbits and then calculating the energy differences between
the different states due to differences in the average velocity of the
electron. By using the wrong method he got exactly the right answer, a
coincidence which caused much confusion at the time.
Figure 4: The fine structure of the hydrogen atom. The diagram is not to
scale.
Strictly speaking, equation 58 has only been shown to be correct
for 
states, although it turns out to be correct for all l. To
do the calculation correctly for l=0, one needs to include the effect of
an additional term in the Hamiltonian known as the Darwin term, which
is purely an effect of relativistic quantum mechanics and can only be
understood in the context of the Dirac theory. It is therefore appropriate
to discuss the Dirac theory to achieve a more complete understanding of the
fine structure of the hydrogen atom.
