How Planck Discovered His Formula for Blackbody Radiation

An excursion into the history of quantum physics

How did I get it? Left: Planck, middle: Boltzmann. Collage by author

Have you ever wondered how Max Planck (1858–1947), the “father” of Quantum Physics (henceforth abbreviated as QP), arrived at his complicated formula for the radiation of a black body? He himself said: It was a “happily guessed interpolation”.

Now here are the facts you believe you know:

(1) Planck is the “father” of quantum physics.

(2) He discovered quanta, that is, energy packets.

(3) He arrived at his radiation formula by guessing (his own words).

Right?

Wrong!

History isn’t simple and never like it’s told in books. Let’s dive into reality.

(1) Although Planck may rightly be called QP’s father, there is at least one important grandfather or, perhaps better: godfather, Planck’s teacher Ludwig Boltzmann (1844–1906). He invented and applied the methods Planck so ingeniously adopted to get at the laws of QP. That’s what Wikipedia says:

“Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.”

And again:

“His 1877 paper on the kinetic theory of heat used discrete energy levels of physical systems as a mathematical device, and went on to show that the same approach could be applied to continuous systems. This might be seen as a forerunner to the development of quantum mechanics.”

That was Boltzmann’s idea and device that made possible Statistical Thermodynamics and, through his pupil, QP: to regard energy (Boltzmann) and radiation (Planck) not as continuous entities (although they look like it), but as packets, discrete units of energy, “cells”, as Boltzmann called them — not real spatial cells, but conceptual ones. To be precise: Both scientists tried to cast the transmission of energy (Boltzmann: kinetic energy through collisions, Planck: radiation energy through heat, and later through absorption and emission) into mathematical formulas.

Based on his considerations and calculations of microphysical states, Boltzmann was now able to derive macrophysical (= measurable) quantities and relationships such as the ideal gas law or the specific heat of a substance — a triumph of classical theoretical physics. His energy cells are the same as the quanta of quantum physics, but Boltzmann didn’t call them that; the term was coined by his student Planck, who applied Boltzmann’s ideas to the absorption of radiation.

(2) As we explicated: Boltzmann and Planck didn’t discover quanta, they assumed them, they invented them as a mathematical device, with great success. By pursuing this idea, Planck found the fundamental formula of quantum physics: E=nhν, today attributed to Planck, Einstein, Bohr, in different forms, all expressing the same notion: Energy depends on a certain constant (h), on frequency (ν), and it is not continuous but discrete (n). Back to blackbody-radiation.

Since the mathematical relations were correct and very useful, you may ask: Are quanta of energy real or a mathematical device only? That’s a question for philosophers we won’t tackle here. It belongs to nearly every aspect of physics, from the Higgs boson (does it really exist?) to curved spacetime (is it purely mathematical or something tangible?).

Now to Planck’s real enigma that he so graciously solved. He was tasked by his teacher with finding a formula for the distribution of energies or wavelengths of a black body. A black body or blackbody, as the name suggests, is a body that absorbs all radiation and thus appears black from the outside. Examples include the pupil of the human eye, or a well shaft. However, the reverse is also true: When a black body emits energy, in the form of radiation, for example, because it is heated, this energy emission is greatest, compared to other (non-black) bodies.

But the really surprising fact is this: The energy-density or the frequency-spectrum of emitted radiation is independent of the material composition of the body. Everyone is familiar with the phenomenon of heated iron (iron must be used because other materials would burn up): First, it glows dark red, then bright red, orange, white, blue, and violet. The same applies to stars: The higher the temperature, the more the mean color shifts towards violet. Beyond, but we can no longer see it.

A black body doesn’t just emit a single color; it sends an entire spectrum of colors into its surroundings. Scientists have been trying for some time to express the distribution of these frequencies, solely as a function of temperature, in a single formula. Two equations already existed, both with limited validity: In 1894, Willy Wien discovered the temperature dependence of the radiation frequency through thermodynamic considerations and analogies to the Maxwell-Boltzmann distribution. However, his law only applies to high frequencies (light, X-rays), where light exhibits more particle-like characteristics. Furthermore, he didn’t yet use the constant “h,” as this was later discovered and named by Planck. In 1900, John William Strutt, also known as Baron Rayleigh, and James Jeans found a different equation for the energy distribution as a function of frequency and temperature through purely classical considerations (the wave nature of light, not quanta). However, this law only applies to low frequencies (radio waves), where light exhibits more wave-like characteristics. Paul Ehrenfest recognized that this law leads to the “ultraviolet catastrophe” at higher frequencies, since the energy is not limited at the top.

(3) How did Planck find the correct mathematical relation? It could never have been a “guess”, a mere intuition. Look at the spectra, according to the graph of these three equations:

Three different curves for the radiation spectrum of a black body. But only one is correct — the green one.

Even a genius like Planck couldn’t see any interpolation. The mystery still gets deeper when looking at the mathematical expressions:

left: Wien, right: Rayleigh-Jeans, bottom: Planck. How do you derive the upper two formulas from the lower one? Or, more difficult: How do you combine the formulas above into the one below?

So how did he do it?

Here’s Planck secret.

Although the problem seemed hopeless, Planck turned to a physical quantity that his teacher had grappled with his entire life, about which Planck also had an excellent understanding: entropy. If you want to know more about it, especially about entropy’s connection to information, see here:

[embed]What is Information? It has to do with entropy — but it is something differentpeterripota.medium.com

Planck not only succeeded, but — above all — also gained an immense personal advantage, which he himself described as follows:

In my in-depth study of this problem, fate intervened, and an external condition I had previously found distasteful — the lack of interest among my colleagues in my chosen field of research — now proved to be quite the opposite, actually benefiting my work. At that time, a considerable number of outstanding physicists, both experimental and theoretical, had turned their attention to the problem of energy distribution in the normal spectrum. However, they all sought only to express the radiation intensity K as a function of temperature T, while I suspected a deeper connection lay in the dependence of entropy S on energy U. Since the concept of entropy had not yet received its due recognition, no one concerned themselves with the method I employed, and I was able to carry out my calculations at my leisure and with complete thoroughness, without fear of interference or obsolescence from any quarter.

Thus, Planck translated the radiation formulas into dependencies of entropy (S) on energy (E). S and E are connected via temperature T:

Because equilibrium states were important to him, he formulated an equation for the second derivative of entropy with respect to energy. If this derivative is negative, the equilibrium state will be reached gradually and then maintained. He arrived at the following results:

(“~” means: is proportional). The mathematical manipulations, in short:

• Wien’s formula must be logarithmized, resulting in 1/T, which can then be replaced by dS/dE and differentiated again.

• For the Rayleigh-Jeans formula, one only needs to take the reciprocal, which yields 1/T, which, as described, can again be replaced by dS/dE.

Now it was easy to find an interpolation formula that reduces to Wien’s equation for low energies (large values ​​of 1/E) and to Rayleigh-Jeans’ for high energies (small values ​​of 1/E). Here’s the result:

The denominator is E² + aE. For large E, E² predominates, and E can be neglected. This yields the Rayleigh-Jeans formula. For small E, E² becomes even smaller and can be neglected. Thus, aE remains, and we arrive at Wien’s formula. By back-translating his entropy equation, i.e., by integrating twice, Planck arrived at the radiation formula named after him, valid for all frequencies. As you can see: no guess, just painstaking logical reasoning.

Of course, the constant a and an integration constant still need to be determined through further considerations.

Planck’s genius, therefore, lay in translating from one physical domain to another, which made connections and similarities visible. So why did he say, it was a guess? He didn’t. If you cite somebody, changing one word (or deleting it) can invert the while meaning. Looking at Planck’s original Nobel Prize address (1920), he said:

Even if this radiation formula should prove to be absolutely accurate it would after all be only an interpolation formula found by happy guesswork. … I was, therefore, from the day of its origination, occupied with the task of giving it a real physical meaning.

Which he accomplished, rendering his “happy guesswork” invalid.

Here are the graphics for entropies. The similarity of the entropy curves is clearly visible, which made the calculation of intermediate values ​​possible in the first place. The curves for the dependence of frequency on energy — and especially the corresponding formulas — are, on the other hand, so different from each other that any interpolation becomes practically impossible.

Dependence of entropy (second derivative) on energy. The Planck curve merges into the Wien curve at low energies and into the Rayleigh-Jeans curve at high energies.

Literature:

Max Planck: The Origin of the Quantum Theory. Being the Nobel Prize Address. Oxford at the Clarendon Press 1922

Max Jammer: The Conceptual Development of Quantum Mechanics. McGraw-Hill Book Company 1966; American Institute of Physics 1989

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