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Let us calculate the rate of spontaneous emission between the
first excited state (i.e., ) and the groundstate (i.e., ) of a hydrogen
atom. Now the groundstate is characterized by . Hence, in
order to satisfy the selection rules (1149) and (1150),
the excited state must have the quantum numbers and .
Thus, we are dealing with a spontaneous transition from a to a
state. Note, incidentally, that a spontaneous transition from a to a state
is forbidden by our selection rules.
According to Sect. 9.4, the wavefunction of a hydrogen atom takes the form

(1151) 
where the radial functions are given in Sect. 9.4,
and the spherical harmonics are given in Sect. 8.7.
Some straightforward, but tedious, integration reveals that
where is the Bohr radius specified in Eq. (679).
All of the other possible
matrix elements are zero because of
the selection rules. If follows from Eq. (1144) that the modulus
squared of the dipole moment for the
transition takes the same value

(1155) 
for , , or . Clearly, the transition rate is independent
of the quantum number . It turns out that this is a general result.
Now, the energy of the eigenstate of the hydrogen atom characterized
by the quantum numbers , , is , where
the groundstate energy is specified in Eq. (678).
Hence, the energy of the photon emitted during a
transition is

(1156) 
This corresponds to a wavelength of
m.
Finally, according to Eq. (1131), the
transition rate
is written

(1157) 
which reduces to

(1158) 
with the aid of Eqs. (1155) and (1156). Here, is the finestructure constant.
Hence, the mean
lifetime of a hydrogen state is

(1159) 
Incidentally, since the state only has a finite lifetime, it follows from the
energytime uncertainty relation that the energy of this
state is uncertain by an amount

(1160) 
This uncertainty gives rise to a finite width of the spectral
line associated with the
transition. This natural
linewidth is of order

(1161) 
Next: Intensity Rules
Up: TimeDependent Perturbation Theory
Previous: Selection Rules
Richard Fitzpatrick
20100720