Consider any two internal states, s1 and s2, of an atom.

Chapter 7, Problem 41P

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Problem 41P

Consider any two internal states, s1 and s2, of an atom. Let s2 be the higher-energy state, so that E(s2) − E(s1) =ϵ for some positive constant ϵ. If the atom is currently in state s2, then there is a certain probability per unit time for it to spontaneously decay down to state s1, emitting a photon with energy ϵ. This probability per unit, time is called the Einstein A coefficient:

A = probability of spontaneous decay per unit time.

On the other hand, if the atom is currently in state s1 and we shine light on it with frequency f = ϵ/h, then there is a chance that it will absorb a photon, jumping into state s2. The probability for this to occur is proportional not only to the amount of time elapsed but also to the intensity of the light, or more precisely, the energy density of the light per unit frequency u(f) (This is the function which, when integrated over any frequency interval, gives the energy per unit volume within that frequency interval. For our atomic transition, all that matters is the value of u(f) at f = ϵ/h The probability of absorbing a photon, per unit time per unit intensity, is called the Einstein B coefficient:

Finally, it is also possible for the atom to make a stimulated transition from s2 down to s1, again with a probability that is proportional to the intensity of light at frequency f. (Stimulated emission is the fundamental mechanism of the laser: Light Amplification by Stimulated Emission of Radiation.) Thus we define a third coefficient, B′, that is analogous to B:

As Einstein showed in 1917, knowing any one of these three coefficients is as good as knowing them all.

(a) Imagine a collection of many of these atoms, such that N1 of them are in state s1 and N2 are in state s2. Write down a formula for dN1/dt. in terms of A, B, B′ N1, N2, and u(f).

(b) Einstein’s trick is to imagine that, these atoms are bathed in thermal radiation, so that u(f) is the Planck spectral function. At equilibrium, N1 and N2 should be constant in time, with their ratio given by a simple Boltzmann factor. Show, then, that the coefficients must be related by

B′ = Β and