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While deriving the Planck's radiation formula, why do we use MB statistics when we calculate the average energy of oscillators? Shouldn't we use BE?

Is this because temperatures concerned are very high and quantum stat goes to classical at high temperatures?

But again, in a similar calculation, while calculating the lattice specific heat of the solid using Einstein's model or the Debye model, I have seen books calculate the average energy of oscillators using MB distribution only, but we use this formula to predict specific heats at very low temperatures also. So I don't understand, what is the reason?

Qmechanic
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Shikhar Chamoli
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1 Answers1

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Planck's Law is the first place BE statistics shows up, and so it is necessary to have some way to derive BE statistics.

It is not just that we are treating each QHO separately, but that we are also treating each energy level of any one QHO separately, and those follow MB statistics. This is equivalent to treating each QHO with BE statistics; it is how we derive BE statistics in the first place.

Another way to put it is this: I know you want to get $$f_i=\frac1{e^{E_i/k_BT}-1}$$ But with some manipulations, you can see that $$f_i=\frac{e^{-\beta E_i}}{1-e^{-\beta E_i}}=\frac1{1-e^{-\beta E_i}}-1\\ 1-e^{-\beta E_i}=\frac1{1+f_i}\\ e^{-\beta E_i}=1-\frac1{1+f_i}=\frac{f_i}{1+f_i}$$ i.e. Essentially we are temporarily working with this last expression, so that we can apply MB statistics, and thereby derive BE statistics. With a little changes here and there, we can get the same with FD statistics.

This is a very standard derivation, just that it is often not stated this explicitly.

naturallyInconsistent
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