Was the Ultraviolet Catastrophe a Real Problem or Just a Fake?

[fox26]

Max Planck formulated the quantum hypothesis, that electromagnetic radiation was
emitted from heated bodies only in quanta of energy E = hf, where f was the frequency
of the radiation and h was a constant now called “Planck's Constant”, in order to solve
the Ultraviolet Catastrophe problem--that classical Electromagnetic and Statistical Mechanical
theory predicted that infinite power would be radiated from a finite material body at any
finite non-zero temperature in the form of electromagnetic radiation with frequencies
above any given value, which of course was observed not to occur.

For a long time I didn't know the reason that this was supposed to occur, but it seemed
to me that classical E.M., S.M., and Atomic theory wouldn't predict this. My reasoning was
that in any finite body at a finite temperature there were only a finite number of charged
particles each moving at a finite speed relative to the others, and while the collisions
between them would cause infinite accelerations which, according maybe to some
extension of Maxwell's E.M. theory, would result in the power of the E.M. radiation
emitted by such a body being infinite, for at least an instant, if the particles were
completely rigid, such a model was unrealistic. Instead of the collisions being ones with
infinite accelerations, they would be interactions between particles caused by finite E.M.
or other forces, so the accelerations would be bounded above over time for each
particle, so the power radiated by each particle would also be bounded above, so there
would be only a finite total amount of power radiated from acceleration of charged particles
in the body. Radiation from electrons in atoms falling to lower energy orbits would also be of
finite total power at all times.

When recently I looked up the reasoning which had led to the infinite power prediction, I
found that the finite body which was supposed to radiate infinite power was modeled as
a finite amount of matter inside a cavity, supposedly to simulate a black body, and the
matter and cavity were supposed to be at thermal equilibrium! In such a cavity there are
an infinite number of E.M. modes, all but a finite number being above any finite
frequency, each of which counts as a degree of freedom of the body, and so according
to the Equipartition Theorem of Statistical Mechanics, at equilibrium at any non-zero
temperature T, each degree of freedom accounts for an equal, non-zero amount kT/2 of
energy, so there would be an infinite total amount of E.M. energy in the body, and an
infinite amount of power radiated from it, most at frequencies above any given value.

While this infinite radiated power prediction was true according to classical theory under
the assumed conditions, why the body was modeled as one inside a cavity was unclear,
and much more importantly, that the material and the cavity were at thermal equilibrium
was as unrealistic an assumption as the infinite acceleration assumption, since it
requires an infinite amount of energy to be in the matter and cavity, in the E.M. radiation
in the infinite number of E.M. modes, which energy could be supplied to the
matter-cavity, which initially upon construction would contain only a finite amount of
energy, only from the environment at a finite rate, bounded above, so it would take an
infinite amount of time for the matter-cavity to reach the assumed equilibrium.

The Quantum Mechanics which was started by Planck's quantum hypothesis has been
very successful in making true predictions about the world, and can hardly be doubted to
be in many ways correct, but the supposed Ultraviolet Catastrophe which led to Planck's
hypothesis seems to be a total fake. Does anyone have an answer opposing, or at least
explaining, this?

 

[PeterDonis]

When recently I looked up the reasoning

Where? What source are you getting it from?

Also, do you really think this is a "B" level (high school level) question?

 

[PeterDonis]

It might also be worth noting that the question of whether pre-quantum electrodynamics can actually be used to derive the ultraviolet catastrophe is separate from the question of whether quantum mechanics is valid. As the OP notes, we have many other reasons to believe QM than just the fact that it correctly predicts the spectrum of black-body radiation. So the ultraviolet catastrophe as a motivation for Planck's hypothesis that led to QM is really a question of the history of physics, not physics. Historically, it was commonly believed in the late 19th century that the ultraviolet catastrophe was a prediction of classical electrodynamics.

 

[fox26]

Where? What source are you getting it from?

Also, do you really think this is a "B" level (high school level) question?

I Googled "Ultraviolet Catastrophe" months ago and got a variety of references, including often mostly correct Wikipedia, and just did the same, getting Wikipedia, Hyperphysics, and quite a few others, all of which (several) that I read which discussed in some detail the reason for the classical result of infinite radiated energy at high frequencies said it was due to the infinite number of E-M modes in the blackbody cavity which they were considering and the fact that each mode, after the cavity was in thermodynamic equilibrium, had the same energy--kT/2. One source,
https://www.cv.nrao.edu/course/astr534/PDFnewfiles/BlackBodyRad.pdf
even gave the reason that a cavity was considered--it was to prevent the modes from decaying before thermal equilibrium was reached. None of the references mentioned the obvious fact that it would take an infinite time for thermal equilibrium to be reached, so actual measurements of radiation from such cavities were never made with them being in thermal equilibrium, or even nearly so, and thus the classical prediction of infinite radiated power in that condition was irrelevant.

I didn't give my question a "B" level , I gave it an "A". Someone, maybe a Mentor, has since then changed it to "B".

 

[PeterDonis]

I Googled "Ultraviolet Catastrophe" months ago and got a variety of references, including often mostly correct Wikipedia

Wikipedia itself is not a valid source; it is "often mostly correct" only because when it is, it is referencing valid sources (textbooks or peer-reviewed papers).

I'll take a look at the link you gave, which does look like it's at least from a college level course.

I didn't give my question a "B" level , I gave it an "A". Someone, maybe a Mentor, has since then changed it to "B".

Yes, I see that. I'll look into it.

 

[vanhees71]

The link is pretty good, except that you shouldn't call "photons" ("light particles") if you don't want to confuse students.

 

[fox26]

Wikipedia itself is not a valid source; it is "often mostly correct" only because when it is, it is referencing valid sources (textbooks or peer-reviewed papers).

I'll take a look at the link you gave, which does look like it's at least from a college level course.
Yes, I see that. I'll look into it.

One correction to my original post--for the classical E-M oscillation degrees of freedom modes, the average energy per mode is kT, not kT/2, since there are both electric and magnetic fields oscillating.

Why consider matter in a (conducting-wall) cavity, which is never in thermodynamic equilibrium? As mentioned, before I found a description of such a situation that had been considered (badly, I am claiming) for finding the spectrum, classically, of blackbody radiation, I had thought about just a mass of material in equilibrium at a temperature T, maybe coupled to a large heat bath at T so it could maintain equilibrium, but with no cavity to make consideration of the E-M oscillation degrees of freedom necessary. Then the only causes of E-M radiation would be the accelerations of charged particles, mostly in atoms, due to their or their atoms mutual interactions, and the decay of electrons in atoms to lower energy levels, from which they had been raised by these same interactions. Correct? The only problem, as far as I can see, would be that such matter might not very closely approximate a black body, so calculating what should be its radiation classically might be somewhat more complicated in that respect than for a black body, but there would be no infinite wait for it to reach thermal equilibrium. The calculated spectrum presumably would differ from the measured one due to quantum effects, but the difference would be not the same as in the matter-in-a-cavity case, if the measurement in that case could be made, which it couldn't.

 

[fox26]

One correction to my original post--for the classical E-M oscillation degrees of freedom modes, the average energy per mode is kT, not kT/2, since there are both electric and magnetic fields oscillating.

Why consider matter in a (conducting-wall) cavity, which is never in thermodynamic equilibrium? As mentioned, before I found a description of such a situation that had been considered (badly, I am claiming) for finding the spectrum, classically, of blackbody radiation, I had thought about just a mass of material in equilibrium at a temperature T, maybe coupled to a large heat bath at T so it could maintain equilibrium, but with no cavity to make consideration of the E-M oscillation degrees of freedom necessary. Then the only causes of E-M radiation would be the accelerations of charged particles, mostly in atoms, due to their or their atoms mutual interactions, and the decay of electrons in atoms to lower energy levels, from which they had been raised by these same interactions. Correct? The only problem, as far as I can see, would be that such matter might not very closely approximate a black body, so calculating what should be its radiation classically might be somewhat more complicated in that respect than for a black body, but there would be no infinite wait for it to reach thermal equilibrium. The calculated spectrum presumably would differ from the measured one due to quantum effects, but the difference would be not the same as in the matter-in-a-cavity case, if the measurement in that case could be made, which it couldn't.

Clarification: What I meant by the matter-in-a cavity never reaching equilibrium was that according to classical physics, it would never do so, and the laws of classical physics would have to be used to properly test the laws of classical physics (in contradistinction to what a certain popular Japanese-heritage physicist once said on YouTube about a test of Special Relativity), so an infinite wait for equilibrium to occur would be required to test classical theory in the cavity case. Actually, the matter-in-a-cavity could get very close to equilibrium in a finite time, due to quantum effects.

 

[jerromyjon]

Why consider matter in a (conducting-wall) cavity, which is never in thermodynamic equilibrium?

The cavity is simply to exclude outside interference and regulate the resonance. Equilibrium is only as important as the precision of the observation demands.

 

[slow]

https://arxiv.org/pdf/1703.05635.pdf

Hi. Maybe we are omitting a methodological detail. Theoretical physicists, on different occasions, face challenges of different kinds.

Sometimes the challenge is to apply the principles and laws available to describe a phenomenon that has not been described to date.

Other times the challenge is to test the coherence, consistency and completeness of the principles and laws available to date. When this is the case, the discussion of practical details is irrelevant and inconclusive, it is a direct conduit towards error and the disarticulation of the whole theory. The coherence, consistency and completeness of the principles and laws are put to the test with abstract models that ideally represent the essential features of the phenomenon. That is to say, models where the postulates (principles), the logical methods and the mathematical methods are sufficient to reach in an unquestionable form an inevitable conclusion.

Hot matter is treated by Thermodynamics. Hot matter emits light. Light is treated by electrodynamics. Then the phenomenon called light is methodologically located at the intersection of two theories. At the end of the 19th century that intersection was being intensively studied. The experimenters accumulated data that the theoreticians could not explain. Then everything was in doubt. The macroscopic thermodynamics a little less than mechanics and electrodynamics, because the macroscopic variables are observable and measurable independently of arguments such as hypotheses, be conceptual hypotheses or equations that have been proposed in a hypothetical way, with much observational base in their support, but finally proposed by the intellectual method of inductive reasoning, which is a reliable way of elaborating and proposing hypotheses. Reliable hypotheses are equally hypothetical components of the theory and, if necessary, allow review.

The ideal model of the black body is a cavity with electromagnetic radiation inside. The first challenge is to transfer the thermodynamic definition of temperature to the terms of electromagnetic radiation. Energy is a very general term, which appears naturally in Thermodynamics and in Electrodynamics. That helps a lot. With these theoretical resources, Rayleigh-Jeans and Wien took the first steps, which were insufficient and defective, like walking through a thick forest at night. Planck noted that it was not enough to find a way to define the temperature in terms of electromagnetic radiation. It was also essential to define entropy in those terms. This second task is more difficult than the first. And Planck resorted to all available physics, including the statistical thermodynamics of Boltzmann and Gibbs, novel at that time and based on reasonable hypotheses but derived from inductive reasoning. In this task, Planck understood that if both versions of Thermodynamics, macroscopic and statistical, must be mutually coherent, entropy takes a form that corresponds to distributing the energy in discrete portions, which have a finite value directly proportional to the frequency of the correspondent oscillation mode.

All this was discussed, thought and resolved by examining an ideal model, which represents the essential features of the phenomenon independently of the data accumulated by the experimenters. Only the principles of physics and the laws established within the framework of these principles were of interest. That is, it only interested the set of statements and equations that make up theoretical physics. Were the experiments interesting? Yes, they were interested every time that the abstract work came to some conclusion, partial or general, that needed to be contrasted. But the way to reach that conclusion is independent of all the experiments. Knowing that I will never have before me an infinite amount of matter, neither of energy, nor of radiation, is not something that has a category of physical principle or of law framed in the principles. For principles and laws, knowing that is anecdotal, not a foundation.

 

[vanhees71]

Well, the black-body spectrum is an almost paradigmatic example for the case that experimental results (as often precision experiments in fact) lead the theorists to the correct result and in this case to a paradigm shift, leading to the discovery of quantum theory (from 1925 with Heisenberg's Helgoland paper and Born's and Jordan's mathematical clarification (including field quantization in the "Dreimännerarbeit", the third paper, this time together with Heisenberg) of Heisenberg's ingenious but incomprehensible ideas; followed by Schrödinger's wave mechanics and finally the representation independent "transformation theory" by Dirac, both in 1926).

Planck was well prepared for his discovery, being an expert in both thermodynamics and (very reluctandly in fact) statistical mechanics as well as the black-body-spectrum problem. He got the issue almost right by counting "field modes" in a way we call today Bose-Einstein statistics of the corresponding quanta of the field, and that's in my opinion the only justified way to teach this subject in the 21st century. A didactics, using a naive photon picture in the sense of localized "bullet-like particles" is highly misleading. One should also not teach the idea the photoeffect or the Compton effect "prove" the necessity for field quantization. All these effects can be understood by semiclassical theory, i.e., by treating only the matter (in this case electrons) quantum-theoretically, while the electromagnetic field is treated as a classical field.

The black-body spectrum in fact reveals the necessity for field quantization, as becomes clear by using Einstein's 1917 way to derive it using kinetic-theory arguments. To get the spectrum correct, in addition to stimulated emission and absorption (both proportional to the intensity of the already present em. field) one also needs the assumption of spontaneous emission, which is the most simple observable fact about electromagnetism which cannot understood with the semiclassical approach.

 

[Jano L.]

Max Planck formulated the quantum hypothesis, that electromagnetic radiation was
emitted from heated bodies only in quanta of energy E = hf, ... in order to solve
the Ultraviolet Catastrophe problem

Historians of science have repeatedly pointed out that the problem with the Rayleigh-Jeans spectrum (nicknamed 'UV catastrophe'), although it was known at the time Planck did his work, was not the motivation that lead Planck to his work and results on blackbody radiation. Reportedly what he was concerned about was the experimental curves of Lummer and Pringsheim. Planck most probably did not see the UV problem to be a serious concern for the physical theories involved - if your read his book Heat radiation, there is not much discussion of the UV problem.

the supposed Ultraviolet Catastrophe which led to Planck's
hypothesis seems to be a total fake. Does anyone have an answer opposing, or at least
explaining, this?

It is fake in the sense that it was not a necessary point in Planck's path towards his theories on heat radiation. It is not a fake as a problem of consistency of Boltzmann's statistical approach and macroscopic EM theory of radiation where energy is given by the Poynting expressions. Those two schemes of thought do not work together very well which posed a legitimate theoretical question, how to resolve that and a practical one, how to describe thermal radiation.

My guess is, Planck saw the R-J result as based on too simplistic and spurious assumptions (extrapolation of equipartition theorem to the realm of EM radiation field energy) and did not believe those were valid for high frequencies, but I only guess. I think that is how the result should have been regarded and how I regard it now.

 

[Jano L.]

One should also not teach the idea the photoeffect or the Compton effect "prove" the necessity for field quantization. All these effects can be understood by semiclassical theory, i.e., by treating only the matter (in this case electrons) quantum-theoretically, while the electromagnetic field is treated as a classical field.

The black-body spectrum in fact reveals the necessity for field quantization, as becomes clear by using Einstein's 1917 way to derive it using kinetic-theory arguments. To get the spectrum correct, in addition to stimulated emission and absorption (both proportional to the intensity of the already present em. field) one also needs the assumption of spontaneous emission, which is the most simple observable fact about electromagnetism which cannot understood with the semiclassical approach.

In fact, Planck invented and published a model that explains the blackbody radiation as a result of "quantized" oscillators interacting with "classical" radiation. "quantized" in the sense when the oscillators emit radiation, they do so only by losing integer multiples of energy $h\nu$, "classical" in the sense this radiation interacts with the oscillators in a continuous way described by differential equations. You can find this model in his book Heat radiation. So not even blackbody radiation is evidence for necessity of quantization of the EM field; in Planck's model, the only quantization that is assumed is quantization of energy release in the emission process.

 

[vanhees71]

Planck's original derivation is obscure in many ways. The point is that you cannot derive the Planck spectrum semiclassically but you need spontaneous emission which cannot be derived semiclassically. Planck's way to count the microstates in the canonical macrostate is of course, nowadays, well understood as using the correct Bose-Einstein statistics for the field modes. The most clear derivation of the black-body spectrum is by calculating the (generalized) partition sum of the free electromagnetic field.

 

[Dale]

that the material and the cavity were at thermal equilibrium was as unrealistic an assumption

Of course that is not an unrealistic assumption. Anybody who had used an oven was familiar with material at thermal equilibrium with a cavity. At most it was an idealization, but hardly unrealistic.