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Spontaneous emission
So far, we have calculated the rates of radiation induced transitions
between two atomic states. This process is known as absorption when the energy of the final state exceeds that
of the initial state, and stimulated emission when the energy of the final state is less than that
of the initial state. Now, in the absence of any external radiation, we would
not expect an atom in a given state to spontaneously jump into an
state with a higher energy. On the other hand, it should be possible for
such an atom to spontaneously jump into an state with a lower energy
via the emission of a photon whose energy is equal to the difference
between the energies of the initial and final states. This process is known as spontaneous emission.
It is possible to derive the rate of spontaneous emission between two atomic states
from a knowledge of the corresponding absorption and stimulated
emission rates using a famous thermodynamic argument due to Einstein.
Consider a very large ensemble of similar atoms placed inside a closed cavity whose walls (which are assumed to be perfect emitters and absorbers of radiation) are held at
the constant temperature . Let the system have attained thermal equilibrium.
According to statistical thermodynamics, the cavity is filled with socalled ``blackbody'' electromagnetic
radiation whose energy spectrum is

(1104) 
where is the Boltzmann constant. This wellknown result was first
obtained by Max Planck in 1900.
Consider two atomic states, labeled and , with . One
of the tenants of statistical thermodynamics is that in thermal equilibrium
we have socalled detailed balance. This means that, irrespective
of any other atomic states, the rate at which atoms in the ensemble leave
state due to transitions to state is exactly balanced by the
rate at which atoms enter state due to transitions from state .
The former rate (i.e., number of transitions per unit time in the ensemble) is written

(1105) 
where
is the rate of spontaneous emission
(for a single atom) between
states and , and is the number of atoms in the ensemble
in state . Likewise, the latter rate takes the form

(1106) 
where is the number of atoms in the ensemble in state .
The above expressions describe how atoms in the ensemble make transitions from
state to state due to a combination of spontaneous and stimulated emission, and make the opposite transition as a consequence of absorption.
In thermal equilibrium, we have
,
which gives

(1107) 
According to Eqs. (1102) and (1103), we can also write

(1108) 
Now, another famous result in statistical thermodynamics is that
in thermal equilibrium the number of atoms in an ensemble occupying
a state of energy is proportional to
. This implies
that

(1109) 
Thus, it follows from Eq. (1104), (1108), and (1109) that the rate of spontaneous emission between states
and takes the form

(1110) 
Note, that, although the above result has been derived for
an atom in a radiationfilled cavity, it remains correct even in the absence
of radiation.
Finally, the corresponding absorption and stimulated emission rates
for an atom in a radiationfilled cavity are
respectively.
Let us estimate the typical value of the spontaneous emission rate for a
hydrogen atom. We expect the dipole moment to be
of order , where is the Bohr radius [see Eq. (658)].
We also expect to be of order , where
is the energy of the groundstate [see Eq. (657)]. It thus
follows from Eq. (1110) that

(1113) 
where
is
the finestructure constant. This is an important result, since our perturbation
expansion is based on the assumption that the transition rate between different energy
eigenstates is much slower than the frequency of phase oscillation of these states: i.e., that
(see Sect. 13.2). This is indeed the
case.
Next: Radiation from a harmonic
Up: Timedependent perturbation theory
Previous: The electric dipole approximation
Contents
Richard Fitzpatrick
20061212