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Now, we know, from Sect. 12.8, that when we take electron spin
and spinorbit coupling into account the degeneracy of the six
states of the hydrogen atom is broken. In fact, these states are divided
into two groups with slightly different energies. There are four states
characterized by the overall angular momentum quantum number
these are called the states. The remaining two
states are characterized by , and are thus called the states.
The energy of the states is slightly higher than that of
the states. In fact, the energy difference
is

(1162) 
Thus, the wavelength of the spectral line associated with the
transition
in hydrogen is split by a relative amount

(1163) 
Note that this splitting is much greater than the natural linewidth estimated in Eq. (1161), so there really are two spectral lines.
How does all of this affect the rate of the
transition?
Well, we have seen that the transition rate is independent of spin, and
hence of the spin quantum number , and is
also independent of the quantum number . It follows that the
transition rate is independent of the component
of total angular momentum quantum number . However,
if this is the case, then the transition rate is plainly also independent of
the total angular momentum quantum number . Hence,
we expect the
and
transition rates to be the same. However, there are four states
and only two states. If these states are equally
populatedwhich we would certainly expect to be the case in thermal
equilibrium, since they have almost the same energiesand since they decay
to the state at the same rate, it stands to reason that the
spectral line associated with the
transition
is twice as bright as that associated with the
transition.
Next: Forbidden Transitions
Up: TimeDependent Perturbation Theory
Previous: Transitions in Hydrogen
Richard Fitzpatrick
20100720