Planck challenge

I was giving an oral exam today in Modern Physics, and I was inspired to write this post. A student was up front doing this standard: “I can compare and contrast the Rayleigh Jeans and Planck approaches to Blackbody radiation.” She was doing a great job of explaining the two major pieces needed to do either calculation:

1. Count the number of ways light can exist in a cavity. This is done by counting standing waves.
2. Determine the energy in each way (or mode) of light. Rayleigh and Jeans say “Oh, every mode just gets kT on average, duh, just like 19th century physics tells us about systems that can share/swap energy.” Planck says “Well, let’s pretend that the oscillators in the walls can only have certain energies and see what happens. Huh, you don’t get kT anymore, you get a nasty formula that needs a constant. I think I’ll name it after me!”

The student did a great job talking about how the kT decision by R/J leads to the ultraviolet catastrophe (infinite amount of energy coming out of the blackbody, concentrated at the low end of wavelengths). Really, she did a great job with just about the whole conversation.

One thing I asked her about was whether R/J should have not bothered telling anyone their results, given that, while it fits long wavelengths well, it really messes up at small wavelengths. She thought it was still useful in certain circumstances, so, yeah, R/J should be proud. We then explored how similar that was to the notion that kinetic energy isn’t 1/2 m v^2 but that it’s useful in certain circumstances.

Then I asked if she would be brave enough to suggest Planck’s approach, given that he seemed to suggest that pendula can’t have any starting amplitude. She laughed and said she wasn’t sure, since she still has a hard time buying that.

So that leads to my challenge:

What if Planck had decided to make a crazy suggestion about how to count standing waves? In other words, leave the kT part alone, and come up with a crazy scheme for deciding which waves are allowed so that the data still fit.

I asked my student about this, and at first she said “but we know how many standing waves there are. If you said that some weren’t allowed for some crazy reason, it seems we could easily disprove that.” But I reminded her about the crazy notion that pendulum can’t have just any amplitude, and that launched a great discussion.

So, what do you think? Can you come up with a way to correct the standing wave count so that you can match the observed blackbody spectrum, assuming that each standing wave has an average of kT energy? How implausible would it be? Would it be “worse” than Planck’s approach?

Is “correcting” the standing wave count effectively alterring the density of states to include the factor $\frac{\beta\epsilon}{e^{\beta\epsilon}-1}$?