Einstein's Derivation of Planck's

Blackbody Radiation Law

In 1909 Albert Einstein presented an alternative way to derive the equation describing the spectral density of electromagnetic radiation emanating from a perfectly black body, a path of reasoning slightly different from the one that Max Planck had laid out in 1901 in Annalen der Physik in his paper "On the Law of Distribution of Energy in the Normal Spectrum". In that paper Planck showed his readers how to work out a description of the average entropy of a large number of monochromatic electromagnetic resonators immersed in a stationary field of heat radiation, the kind of field that one might create and maintain inside a large cavity with perfectly black walls that remain at some absolute temperature. When the resonators come to equilibrium with the radiation field they contain a certain amount of energy and in order to calculate the number of ways in which that energy could be distributed over the resonators, a sine qua non for calculating the entropy of the system, Planck had to assume into his premises a statement that the energy came into the resonators and came out of them in energy elements of magnitude

E=hν

(Eq'n 1)

At any given instant a resonator would possess an amount of energy

E=nhν,

(Eq'n 2)

in which n represents any positive integer, though Planck did not write that fact out so explicitely.

At this point Planck's and Einstein's derivations go their separate ways. Where Planck took the entropy of the system as his key concept, Einstein took the partition function.

Devised as part of
Ludwig Boltzmann's statistical thermodynamics, the partition function provides
the effective number of ways (microstates) in which a given system can manifest
a certain macrostate. It comprises a sum of terms, each of which gives an
account of the microstates with a certain energy. If a given state of the system
possesses energy E_{i}, then its contribution to the partition function
(Z=Σz_{i}) equals

(Eq'n 3)

in which β=1/kT (k=Boltzmann's constant and T=the absolute temperature of the system). The entropy S of the system then conforms to

(Eq'n 4)

Einstein looked at Planck's imaginary system of resonators in a cavity and described the energy of its n-th microstate as

E_{n}=nhν

(Eq'n 5)

Thus he gave the system a nicely simple partition function,

(Eq'n 6)

for

(Eq'n 7)

When the system achieves thermal equilibrium its entropy becomes a constant, impervious to change until something pushes the system out of equilibrium. In that case we can differentiate Equation 4 with respect to and thereby calculate the average energy of a resonator in the system,

(Eq'n 8)

which is just what Planck got.

When the resonators achieve thermal equilibrium with the radiation field in the cavity, then, and Planck and others deduced, the average energy of the resonators equals the average energy per oscillation mode of the radiation. Thus, we calculate the spectral density of the radiation by multiplying the number of modes per unit volume by Equation 8 to obtain

(Eq'n 9)

which is the equation of Planck's radiation law.

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