"Packing of energy" question related to the ultraviolet catastrophe

[brajesh]

TL;DR

Packing the energy into frequencies - how is that done?

Hi,

I'm curious about the ultraviolet catastrophe, and how Planck decided to hypothesize the quantization of energy integrally proportional to frequency to explain the drop off of energy in the higher frequencies.

So what I don't understand is, when the energy is transmitted as a photon, does it show up as a fixed size, and then the best frequency says "this one's for me?".

Or to put it another way, let's say X amount of energy needs to be transmitted.

Do the frequencies divvy up the energy X until they have all used it up?

Or they say, freq Y can carry X size energy, so this X energy is going to be loaded up exactly onto frequency Y.
And there is never any "dividing up" function amongst the frequencies prior to the transmission.

Hope this question makes sense, my first post :)

 

[mpresic3]

The best historical account for Planck's solution, that I have seen is in a book by Oliver Darrigol, From C-numbers to Q-numbers. The book illustrates correspondence between Planck and Einstein. Apparently, Planck was one of the first to improve and advance thermodynamic arguments set forth by Boltzmann, and others. The book makes it clearer, (as ,many QM books do not do), that Planck did not merely fit the experimental results to a curve to obtain the quantization of energy, but had a strong theoretical foundation for his thermodynamic arguments. Another good book that does not go in as much depth is by Malcolm Longair, A course in Theoretical physics (or some such title).

Planck's solution essential regards the cavity as a collection of harmonic oscillators, (like masses on springs). The electromagnetic energy, (light), supplied into the cavity, excites these oscillators, which reradiate the light, and achieve an equilibrium, for a prescribed temperature. I will have to think about this in greater depth as it is not completely clear to me all at once at this time. It is probably best to consult Longair, or Darrigol.

 

[vanhees71]

In short: Planck's great discovery (which he never fully accepted himself, always trying until the end of his long life to find a derivation of the conservation law from classical physics), went in two steps.

First he got high-precision data from the Technische Reichsanstalt, where Rubens, Kurlbaum et al measured the black-body spectrum over a wide range of frequencies to establish a standard for the lightning industry. Planck had tought about the problem before that for some years before. So he was very well prepared, and using an ingenious argument, based on phenomenological thermodynamics, to guess an expression for the entropy of the radiation in the cavity which extrapolated between the classical result by Jeans, which was a good approximation in the long-wavelength limit but failed for short wavelength, leading to the UV catastrophe, and the semi-empirical formula by Wien, which was valid in the short-wavelength region but failed in the long-wavelength region. In this way Planck came to the radiation formula in a semi-empirical way based on very precise data.

In a second step he (reluctantly!) turned to the statistical approach. To count the macrostates of the em. field as a system of oscillators he introduced energy quanta and then counted in a way we'd call today "Bose-Einstein statistics" and then soon realized that choosing the energy quanta to be proportional to the frequency of the field modes lead to the correct radiation formula. He never liked this conclusion, but there was no way out, and indeed this triggered the discovery of quantum theory. In this sense the birthday of QT is Dec/14/1900, when Planck gave his famous talk at the German Physical Society about the derivation of the radiation formula.

 

[brajesh]

My understanding is that a high frequency will only take a large energy quanta.
So if a small quanta of energy comes in, won't it also oscillate the high frequency just a little bit?
How does the matching of the quanta of energy with the "correct" frequency that will carry it work?

@vanhees71 and @mpresic3 if you answered this in your replies above, then I didn't follow it, could you expound to me a little more?

thank you !

 

[mpresic3]

The best historical account for Planck's solution, that I have seen is in a book by Oliver Darrigol, From C-numbers to Q-numbers. The book illustrates correspondence between Planck and Einstein. Apparently, Planck was one of the first to improve and advance thermodynamic arguments set forth by Boltzmann, and others. The book makes it clearer, (as ,many QM books do not do), that Planck did not merely fit the experimental results to a curve to obtain the quantization of energy, but had a strong theoretical foundation for his thermodynamic arguments. Another good book that does not go in as much depth is my Malcolm Longair.

Planck's solution essential regards the cavity as a collection of harmonic oscillators, (like masses on springs). The electromagnetic energy, (light), supplied into the cavity, excites these oscillators, which reradiate the light and

 

[mpresic3]

It seems from your question, that you are taking a given photon, and measuring the energy (mentally), and assigning the photon to an "energy bin", and then doing that to all the photons in the black body cavity. The bins do not select the energy of the photon. I can give a explanation but do not expect this to be a full answer either. My explanation leaves many questions, at least in my mind, and I will have to think about it further.

My explanation begins with, take light which is made up of quadrillions (?), of photons. Suppose the light was all one energy. When it hits the cavity surface, it excites an oscillator which starts vibrating. Assume there is some damping in the oscillator, so that the oscillator reradiates the photon with less energy, and in a random direction, and still has some added energy. This happens quadrillions of times and over and over again, but eventually (at equilibrium, which is reached in perhaps microseconds, or sooner?), you get light with a random direction, and with random energy. By random, I invoke the central limit theorem to motivate the following assumptions:

The light energy is the same in all directions (this direction-free dependence is called isotropic, but the magnitude of the energy is normally distributed, because not every interaction with every oscillator was identical. Some oscillators kept more of the incident energy than others. The bottom line is you have radiation (light) that has no dependence in direction, and it has a normal distribution in the dependence in all three equivalent directions with x, y, z, equivalent components. The equivalence in all three components indicate there is not preferred direction dependence, which we have already assumed.

Now (these are no longer assumptions but consequents). When you have a normal distribution in energy (components) in the three x, y, z directions, and you are looking for the energy distribution in the magnitude of the energy which is Maxwell Boltzmann. Already you have lower probability of higher (magnitude) of energy for the light in the cavity.

I have to think a little further to get the Rayleigh-Jeans distribution, but these are the broad strokes. I think I am not too far off, but Vanhees, may want to add his comments. The full story is probably better explained in the references.

 

[mpresic3]

Postscript. I now see from Vanhees71 post I glossed a lot under the rug. First of all the photons are indistinguishable leading to Bose-Einstein, not Maxwell Boltzmann statistics. I know I left out many other considerations, with entropy and others. To my mind, Planck should occupy a place just below Newton and Einstein, and the is one of the most underrated physicist. The chemistry and some ohysics lectures I have attended regarding quantum mechanics seem to suggest all he did was "fit" curves. He deserves better than that.