On The Law Of Distribution Of Energy In The Normal Spectrum

Planck 1901

Summary

In this paper Max Plank uses Wein’s Displacement Law and a quantized energy to derive a relation for the black body energy distribution which matches empirical data
This paper shows that assuming a quantized energy results in a relation between the frequency of light and the energy that light has \(E=hf\)

Entropy

Entropy is the number of possible arrangments of microscopic states which could result in the macroscopic state. It is directly related to the temperature and thermal energy of the system.

\(\dfrac{dS}{dU} = \dfrac{1}{T}\)

If you view the system as N microscopic oscilators, then the total energy and entropy of the system is a multiple of the energy / entropy of each oscilator (assuming they are in phase / same freq).

\(U_N = NU\)

\(S_N = NS\)

Quantized Energy

If we quantize the energy and define the smallest unit of energy as ε, then the total energy of the system must be equal to the number of packets of energy, P, times ε. This energy must also be divided across the N oscilators in the system.

\(U_N = NU = P\epsilon\)

From combinatorics we can determine the number of possible arrangments for this energy, which (by definition) is porpotional to the entropy of the system. Using some math, the following can be shown,

\(S = f(\frac{U}{\epsilon})\)

From known thermodynamics (using Wein’s Displacement Law), the temperature for a black body can be shown to be a function of the heat energy divided by the frequency,

\(T= f(\frac{U}{f})\)

Using the relation between entropy, temperature and thermal energy, you can show that entropy must then also be a function of \(\frac{U}{f}\).

Since ε has been assumed to be discrete and frequency is continuous, \(\epsilon = hf\), where h is some constant (now known as Planck’s Constant).

Following this line of logic you can then determine the proper relation for the black body energy distribution.

Thoughts

My whole life I have known energy to be a quantized value so I cannot truly appreciate how puzzling this ad hoc assumption must have been. To me, with my limited knowledge of physics, quantized energy seems to play so nicely with the ideas of statistical mechanics that Planck uses in this paper.
The thought of quantized energy also imposes such a natural restriction on the infinite energy problem introduced by the Rayleigh Jean law that it would be hard to argue against such an aproach. Of course, this assumption turned out to be so groundbreaking that it redefined physics, but that is not at all aparent in this paper alone.