4

I was studying the derivation of Planck's Law. But I found confusing texts in different books.

In the book "Quantum Physics of Atoms" by Eisberg and Resnick, it states that Planck considered the cavity radiation to be composed of electromagnetic standing waves and then calculated the number density of those standing waves. He then found the average total energy of all those standing waves by multiplying the possible discrete energies by the weights, using Boltzmann probabilty distribution $P(\epsilon)=\frac{\exp(-\frac{\epsilon}{kT})}{kT}$ and computing the summation to get the Planck's Blackbody distribution. It is written below.
enter image description here

In the book "A Treatise on Heat" by Saha and Srivastava and also in my professor's lecture notes, it states that Planck considered the cavity radiation to be composed of simple harmonic hertzian oscillators and then calculated the number density of those oscillators. He then found the average total energy of all those standing waves by multiplying the possible discrete energies by the weights, using Maxwell probabilty distribution $P(\epsilon)=\exp(-\frac{\epsilon}{kT})$ and computing the same summation to get the Planck's Blackbody distribution. $$N=\sum_{i=1}^\infty N_i$$ $$\bar E=\frac{1}{N}(N_1\cdot\epsilon+N_2\cdot2\epsilon+\ldots)=\frac{\epsilon}{N}\sum_{r=1}^\infty rN_r$$ $$N_r=N_1\exp(-\frac{\epsilon}{kT})$$ In Kenneth Krane's book on Modern Physics, it states the radiation is composed of resonators and that Planck used the Maxwell-Boltzmann probabilty distribution $P(\epsilon)=\frac{N\exp(-\frac{\epsilon}{kT})}{kT}$ for his corresponding calculations.

QUESTION:

All these however correctly yield the same result. So which is the correct assumption regarding the form of radiation in the cavity? Which one did Planck use? And which probability distribution did he use? Are any of the books wrong?

I am eagerly waiting for an answer. Thanks!

SchrodingersCat
  • 4,923
  • 3
  • 23
  • 49
  • 1
    These all look essentially the same to me, with minor convention differences. Can you pinpoint exactly where you think the contradiction is? – knzhou Aug 21 '16 at 18:41
  • @knzhou I mean, they are using different probabilities as weights... one uses maxwell...one uses Boltzmann... different pre-exponential factors... and one considers the radiation as standing waves... the other considers it as a series of harmonic hertzian oscillators....Can you help now? – SchrodingersCat Aug 21 '16 at 18:44
  • Interesting: possible ultraviolet catastrophe from including "vacuum" energy, https://physics.stackexchange.com/q/704105/226902 – Quillo Jul 27 '23 at 10:47

1 Answers1

3

All of your sources are correct.

  • The Boltzmann distribution states that the probability of a single state being occupied is proportional to $e^{-E/kT}$.
  • To get the probability of a range of energies to be occupied, we have to integrate this, giving the Maxwell-Boltzmann distribution $e^{-E/kT}/kT$.
  • Some sources give different names to these two distributions, but it don't make a difference.
  • There are various prefactors you can add, but they don't make a difference.

As for the language about what they're calculating, they're all the same thing. Individual modes of the electromagnetic field are standing waves, and they are harmonic oscillators too.

knzhou
  • 101,976