Einstein’s Deduction of Planck’s Theorem

At the beginning of his 1905 paper, "Concerning an Heuristic Point of View Toward the Emission and Transformation of Light", in which he used Max Planck’s quantum hypothesis to explain the photoelectric effect, Albert Einstein wrote, "While we consider the state of a body to be completely determined by the positions and velocities of a very large, yet finite, number of atoms and electrons, we make use of continuous spatial functions to describe the electromagnetic state of a given volume, and a finite number of parameters cannot be regarded as sufficient for the complete determination of such a state." In that statement Einstein is not disputing the validity of the wave theory of light: he notes that it has worked perfectly in representing purely optical phenomena, such as diffraction, reflection, refraction, dispersion, and so on. But "optical observations refer to time averages rather than instantaneous values". If we want to look at the emission and the transformations of light (as through collisions with matter), we need instantaneous values of the energy distribution in the electromagnetic field. Einstein proposes that those instantaneous values must reflect a discontinuous distribution of energy in space; that is, that "the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units". He then offers a derivation of that proposition as preparation for explaining the photoelectric effect.

Einstein’s derivation proceeds in two easy steps. First he calculates the entropy possessed by a given amount of electromagnetic radiation trapped in a container. And second he shows that the description he obtains conforms exactly to Ludwig Boltzmann’s description of the entropy of a gas consisting of a finite number of discrete particles trapped in an identical container.

He begins by, in essence, reminding his readers of the relationship between entropy and heat through the absolute temperature,

(Eq’n 1)

Light must obey the laws of thermodynamics, so that Clausius relation must apply to electromagnetic radiation as well as it does to material systems. He then refers to Wilhelm Wien’s distribution law of 1896, the precursor to Planck’s blackbody radiation law,

(Eq’n 2)

He notes that this is not exactly valid, but that it closely mimics Planck’s law for large values of ν/T. Because we are free to make T as small as we like, the inference that we draw from the analysis, that light is quantized, is valid at all frequencies. Using simple algabra, Einstein solves Equation 2 for the reciprocal of the absolute temperature,

(Eq’n 3)

Next he wants to calculate the entropy of monochromatic radiation (with frequency lying between ν and ν+dν) carrying energy E and confined in a container of volume V. For that calculation he has Equation 1 in the form

(Eq’n 4)

That equation integrates readily to

(Eq’n 5)

in which E=Vρdν
(the energy contained in the narrow band of spectrum under consideration) and e
represents the base of the natural logarithms. If we let S_{0} denote
the entropy of the radiation when the radiation occupies a volume V_{0},
then we have

(Eq’n 6)

In that equation the ratio of the volumes reflects the ratio of the corresponding densities of the radiation.

In statistical thermodynamics we have Boltzmann’s law
(which Einstein called the Boltzmann Principle) describing the relationship
between entropy and probability. If W represents the relative probability
between some state of a system with entropy S and some initial state of that
system with entropy S_{0}, then

(Eq’n 7)

expresses Boltzmann’s law. In that equation R/N_{A}, the universal
gas constant divided by Avogadro’s number, equals Boltzmann’s constant.

Imagine, as Einstein does, a container enclosing a volume
V_{0} and filled with n particles that interact with each other so
rarely that they are effectively independent of each other in their motions.
Imagine some small part of the container’s interior with volume V<V_{0}.
We can conceive the idea of all the particles coming, purely by chance, to
occupy the volume V. What is the probability that they will actually do so at
some randomly chosen instant of time? The probability of finding any single
particle within that volume equals V/V_{0}, so for n mutually
independent particles to occupy the smaller volume together we have the
probability

(Eq’n 8)

Thus the difference between the entropy of all the particles in volume V and
the entropy of all the particles spread throughout volume V_{0} comes to
us, via Equation 7, as

(Eq’n 9)

In the last step in drawing that equation I replaced R/N_{A} by
Boltzmann’s constant.

Look again at Equation 6, noting that β=h/k, Planck’s constant divided by Boltzmann’s constant. Because of the nature of the natural logarithm we can rewrite it as

(Eq’n 10)

Comparing that equation with Equations 9 and 8 tells us that if we have
perfectly reflecting walls completely enclosing a volume V_{0} and
containing monochromatic radiation of frequency
ν
and total energy E, then the probability that we will find all of the radiation
and its energy within a volume V<V_{0} at any randomly chosen instant
conforms to

(Eq’n 11)

Thus we conclude, with Einstein, that "Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of independent energy quanta of magnitude Rβν/N"; that is,

(Eq’n 12)

which is Planck’s theorem.

Of course Einstein’s reasoning is more than a little circular. To the extent that Wien’s radiation law mimics Planck’s radiation law, to that same extent it tacitly incorporates Planck’s theorem. Thus Einstein proved and verified what he already knew. But in doing so he gave us some extra insights into the thermodynamic nature of light.

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