Some thoughts around Radiation

Apr 21

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This was originally published as an article on our SeanForScience Notebook. It was a fun, historical account, so we’re reposting it here.

Introduction

A few months ago, I started writing some course notes on Quantum Mechanics. I wanted them to be complimentary to the current offerings. The best approach, I figured, was too really fill in the gaps that I had after taking my first series on the subject.

There's a lot of "lore" in Physics that gets communicated to students poorly in classes with hand waving explanations because the details aren't pertinent to direct calculations that can be both easily measured on an exam and be framed as a simple case of an actual research problem.

Ironically, the student of physics will eventually have to go back and master this material anyway in order to grasp the material when confronted with actual, research problems.

What I've learned studying these sort of "lost" details is that the main sources of this disconnect are twofold.

First. Material is siloed into courses, and those courses have be codified into something that borders on dogma. technocratic in the extreme.

Second, although related to the first, nobody in an academic setting has time to study the context around these details, and so they don't get passed down directly.

Whence, there is an opportunity to compliment the existing material. I'm hopeful that these discussions will.

Today I'm going to talk about the first bit of theoretical physics that landed in what we might call "Quantum Theory". For historical reasons, it's called Blackbody Radiation, but really, it's the story of Max Planck and his description of the colors of light set off by the glow by a hot object. If you've ever used a coiled stovetop, or stuck a metal poker into a fire, you know what I'm talking about.

Disclaimer

Despite aiming at the holes in an undergraduate physics education, for this article I tried as much as possible to make it a general audience piece.

Of course, you'll find references to basic physical ideas like the wavelength and frequency of light. I'll also mention entropy a few times, so I might as well discuss quickly before we get started.

For the non physicists, people often say "entropy is a measure of disorder in a system". Which is a silly definition to give the uninitiated. It's silly because by order physicists often mean something very specific. Something mathematically precise. Something that entire textbooks have been written about. Who is going to know that? I think a better way to frame entropy is in terms of options. How many options do you have available to you? How many different ways can we pack people into a bus, or a movie theatre?

Persistent Vagueness

One topic that seemed to be glossed over in every course was blackbody radiation. All we were ever told was "because Planck assumed that light comes in little packets", he was able to derive his famous distribution function.

That persistent vagueness really bugged me. Worse. It was confusing. In a standard quantum mechanics class, you learn about the oscillator algebra and expect discrete energy levels. Quanta, if you will. Because that formalism translated directly to quantum field theory, and therefore exciting and relevant things like particle physics, nobody has time to go back and explain the difference.

That said, It's a hard disconnect for me to explain in words - it was muddled in my brain some 18 years ago - but the thing that bugged me at the time was roughly this:

Planck's formula for the spectrum of ideal thermal radiation was continuous, not discrete. And energy was proportional to frequency. Fine, but that alone does not make a photon. It doesn't make it a quantum of anything. Where are the discrete things that I can index with a whole number? What the hell are people talking about?

So I set out to write a piece on how Planck derived his famous distribution function. And I was stoked with what I found! This story offers great connections to statistical mechanics and even shows how messy the process of actual physics research is.

A Confused Narrative

After poking around in textbooks and popular accounts, I stumbled into a conference proceedings written on the history of the $ T^{4} $ radiation law by some Mechanical Engineering professor in Idaho.

It's always fun to revisit history of your field and try to glean some perspective from the cast of characters. Maxwell, Rayleigh, Jeans and Boltzmann. The piece started out strong and clear and focused heavily on the thermodynamics. That wasn't too surprising. Mechanical engineers get far better training on thermo than physicists do.

As the piece passed from Willy Wien's work into Max Planck's it started to get harder to read. I figured I was just tired and so I put down the tablet and traded up for a glass of Syrah.

The next few days I would revisit the article. The same confusion would arise. I wasn't able to fully understand what the author was saying, and the logical progression seemed. Well. Distorted? Maybe, random at best.

At this point I had start work on some other projects, so I lost track of the paper and the ideas. But after a few weeks, I carved some time out to revisit the idea. This time I dug a lot deeper and pulled out the primary sources. I found myself sounding out the old german aloud, using google translate to help as needed.

Reading Wien and Planck's original work, the trouble with that review article became clear. The guy's math was wrong! Worse. He was clearly referencing the WRONG paper.

You see. In the year 1900. Planck first published the functional form of his distribution, but he hadn't yet justified it - other than comparing with experiment. He was doing basic phenomenology. Planck's seminal paper wasn't written until 1901.

Let me share with you the error in the review article.

Thermodynamics is all about relating different physical observables to each other. In this case entropy and energy. You can approximate the entropy as a Taylor polynomial of the internal energy, and the various terms in that expansion can be assigned physical meaning. Importantly, the linear coefficient is the inverse of the temperature of the system. And so on.

Planck had originally tried to justify Wien's old work by guessing a complete, functional form of Entropy in terms of energy, so he could perform this expansion and get the required results.

Okay so here's the thing: On dimensional grounds, he argued that the second derivative of the entropy should go like one divided by the internal energy, squared. Of course. Duh. Tautology.

But that wasn't the formula of interest. It was just the motivation for it. You see, that's the hard part of translating these old papers from the German. They did these kind of wonky approximations all the time, and without the context latent in their Scientific culture, you could easily get confused by current diction and conventions. There's a LOT of that kind of stuff flying around in those works.

Unfortunately for the review article, the simply copied the 1/energy^2 term. Without reading the rest of the work.

You see, after Planck discusses the dimensional analysis, he goes on to say that he's currently interested in explicit functions like

$$ \frac{\partial^2 S}{\partial U^2} \propto \frac{1}{U(U+\beta)}.$$

It's the simple dimensional term with a slight deformation of the denominator. When you write entropy as a function of internal energy in THAT way, you can integrate to find the temperature using the method of partial fractions. And low and behold, you find a schematic representation of Planck's distribution.

But again. THATS NOT THE PHYSICS. That's just another step along the way. This isn't the main paper. It looks like some kind of workshop proceedings. We probably wouldn't publish something like that today. Let alone present it at a conference.

Again. The author of the review article simply wrote down the $S = 1/U^2$ squared relation, and suggested that it was Planck's big insight. Which is of course totally wrong because integrating that gets you just a linear relationship between $U$ and $T$. That little deformation was what was needed.

So at least my confusion in reading that history of the $T^4$ radiation law was resolved. Dude probably didn't translate the correct paper, and probably didn't do the math to confirm.

To be fair, it was almost certainly the case that the piece wrapped up too quickly because of a deadline. Having taught math and physics for over a decade, I can confirm that it's not uncommon.

But what's great about that error is it afforded me the chance to explore the old literature and reveal all that old culture and context. And of course, I finally got to read Planck's seminal paper.

Planck's idea

Planck was originally trying to put Willy Wien’s spectral distribution function on a microscopic footing. Willy himself had the insight that the energy of an electromagnetic wave should be proportional to its frequency.

You see. Before Wien, folks tried to argue that the Equipartition Theorem of statistical mechanics ascribes a uniform distribution to all possible modes of the electromagnetic spectrum. Because you can always pack more smaller wavelengths into a box of fixed size, you were doomed from the beginning. So folks like Rayleigh and Jeans tried some hacks like mode counts per unit wavelength and stuff like that.

Wien fixed that by arguing that the energy of an electromagnetic wave is related to its frequency. He simply copied the structure of the Maxwell Boltzmann distribution for the distribution of molecular velocities in a gas, and made it relevant to the spectrum of radiation by analogy.

He was guessing.

It was a pretty good guess because, well one it gave a finite answer in the UV and two it had basically the correct form. The peak of the distribution - the wavelength with the MOST intensity - was predicted to be inversely related to the temperature. In other words he got the peak of the distribution mostly right. This observation is now his famous “displacement law”.

Just try teaching anyone about this so-called "displacement law". The thing being displaced is the peak of the curve. Like on a plot. Nothing physical. What a terribly confusing lexical artifact to bring with us into the 21st century.

What Planck took from Willy’s work was that observation. That the entropy - or alternatively the internal energy - should be some function of wavelength times temperature. Or, as Planck would put it, temperature divide by frequency.

That observation - which afforded Willy his displacement law - effectively amounted to reweighing the different modes of the electromagnetic spectrum by a Boltzmann factor, with energy now directly proportional to frequency. In other words, the thermal spectrum turns over because despite their being MORE high frequency modes available in a given box or cavity, they are LESS energetically available because they cost more energy to produce. Just like in a gas, you’re not going to find all the internal energy of a system devoted to the kinetic energy of a single particle, you’re not going to find all the internal energy devoted to the high frequency spectrum. The peak moves with the temperature.

So. Given the displacement law and his phenomenological solution, Planck set out to find a microscopic explanation for his formula for entropy. And the interpretation of what he found kick started quantum mechanics.

Planck's Seminal Paper

In his seminal paper, On an Improvement of Wien’s Equation for the Spectrum, Planck posits the following description of the the microscopic states of the theory.

Suppose you had a bunch of electromagnetic resonators in a cavity. Some folks would probably just call these radiation modes.

Suppose further that you had a budget of P bits of energy to distribute amongst those resonators. Assuming $P$ is greater than $N$, what do you find?

Each WAY you can distribute the $P$ quanta of energy amongst the N resonators corresponds to a different micro state of the system. That’s the celebrated insight, however derived.

What’s funny is, this is a basic problem from combinatorics. It’s actually the reciprocal of the famous beta function, restricted to natural numbers.

In the limit where $N$ - and there $P$ - is extremely large, Planck did the usual thing and applied Stirling’s approximation where \[N! \approx N^N. \] Entropy goes like the log of the count of these micro states. Do the thermodynamics and require that energy should go like the frequency of light. That, plus a little algebra gives you his famous distribution.

Now. Planck almost certainly worked backwards to get his answer. There’s a clear path that way, and that’s how almost all of us would do it. If he didn’t, he probably would not have gone through the exercise of explicitly writing out solutions for small N and would have referenced the relationship to the Beta function. But still. The interpretation was clear. The physical states of the system are all the different ways you can distribute $P$ quanta of energy amongst $N$ electromagnetic resonators.

Postscript on Oscillators

It’s kind of fun to point out that Planck’s insight also demonstrates the differences between actual quantum mechanics and thermal, quantum statistics.

When I was an undergraduate I had trouble with Planck’s distribution at first glance because it seemed to have NOTHING to do with individual quanta. Those things that we studied as excitations of the simple harmonic oscillator. Or even modes of the electron orbiting hydrogen.

But actually. What Planck did was describe a THERMAL state. A state where the entropy is huge. The micro states associated to, say the quantum harmonic oscillator, aren’t thermal. They’re fundamental. At least when we try to take the number of quanta to be large in some way that looks like those simple, microscopic counts.

To put in plain English: quantum states are lasers, thermal states are charcoal. A quantum state that you would study and derive in quantum mechanics, at large N, would look like a laser. A so-called coherent state. A thermal state, even an idealized one like Planck’s distribution for Blackbody Radiation, would be considerably more smeared out.

It's an entropy of zero vs entropy of. well. A whole lot.

So if you’re like me and you didn’t quite understand what the hell folks were talking about when they talked about Planck’s result. It’s probably because you didn’t know statistical mechanics yet.

Yet another reason to rethink the way we train our students.