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 so-called ``black-body'' electromagnetic radiation whose energy spectrum is

where is the Boltzmann constant. This well-known 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 so-called *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

(901) |

where and are the rates of spontaneous and stimulated emission, respectively, (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

(902) |

where is the rate of absorption (for a single atom) between states and , and 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

(903) |

Equations (891) and (892) imply that

where , and the large angle brackets denote an average over all possible directions of the incident radiation (because, in equilibrium, the radiation inside the cavity is isotropic). In fact, it is easily demonstrated that

where stands for

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

Thus, it follows from Equations (900), (904), (905), and (907) that the rate of spontaneous emission between states and takes the form

Note, that, although the above result has been derived for an atom in a radiation-filled cavity, it remains correct even in the absence of radiation.

Let us estimate the typical value of the spontaneous emission rate for a hydrogen atom. We expect the matrix element to be of order , where is the Bohr radius. We also expect to be of order , where is the ground-state energy. It thus follows from Equation (908) that

(909) |

where is the fine structure constant. This is an important result, because 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 . This is indeed the case.