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THE LIGHT-QUANTUM 369

Even if Planck had stopped after October 19, he would forever be remembered

as the discoverer of the radiation law. It is a true measure of his greatness that he

went further. He wanted to interpret Eq. 19.6. That made him the discoverer of

the quantum theory. I shall briefly outline the three steps he took [P4].

The Electromagnetic Step. This concerns a result Planck had obtained some

time earlier [P5]. Consider a linear oscillator with mass m and charge e in inter-

action with a monochromatic, periodic electric field (with frequency oj) in the

direction of its motion. The equation of motion is

Let v denote the frequency of the free oscillator, f/m = (2irv)^2. Consider in par-
ticular the case in which the radiation damping due to the 'x term is very small,
that is, 7 <sC v, where 7 = Si^eV/Smc^3. Then one may approximate 'x by —
(2irv)^2 x. The solution of Eq. 19.8 can be written (see [P6]) x = C cos (2-irait —
a). One can readily solve for C and a. The energy E of the oscillator equals
m(2irv)^2 C^2 /2, and one finds that

(19.8)

(19.9)

(19.10)

;i9.11)

Next, let the electric field consist of an incoherent isotropic superposition of fre-

quencies in thermal equilibrium at temperature T. In that case, the equilibrium

energy U of the oscillator is obtained by replacing the electric field energy density

F^2 /2 in Eq. 19.9 by 4irp(o), T)du>/?> and by integrating over co:

Since 7 is very small, the response of the oscillator is maximal if w = v. Thus we
may replace p(w, T) by p(v, T) and extend the integration from — oo to + oo. This
yields

This equation for the joint equilibrium of matter and radiation, one of Planck's
important contributions to classical physics, was the starting point for his discovery
of the quantum theory. As we soon shall see, this same equation was also the point
of departure for Einstein's critique in 1905 of Planck's reasoning and for his quan-
tum theory of specific heats.
The Thermodynamic Step. Planck concluded from Eq. 19.11 that it suffices
to determine U in order to find p. (There is a lot more to be said about this
seemingly innocent statement; see Section 19b.) Working backward from Eqs.
19.6 and 19.11, he found U. Next he determined the entropy S of the linear