# ZPE and Planck distribution

1. Nov 5, 2008

### FranzDiCoccio

Hi all,

in a couple of books I read that Planck's original derivation of the blackbody radiation
had some problem related to the zero point energy, which was solved after quite
a few years (when Schroedinger formulated his equation).

I sort of see this problem, but I'm not sure the above story has some real basis.
The point seems to be the ability of Planck's distribution to result in equipartition
at large temperatures.
It seems to me that formally equipartition can be equivalently obtained independent
of the presence of the ZPE, by expanding the exponential in the average energy of a single
oscillator. The somewhat tricky point in this respect is that the expansion is truncated at
different powers of $$\beta \hbar\omega$$ in the two cases.

In particular, if one does not consider the ZPE and does not stop at the first perturbative order the energy is

$$\langle\epsilon\rangle = k T - \frac{\hbar \omega}{2}+ ...$$

where the dots are a power series of $$T^{-1}$$. If the ZPE is considered the second term is canceled out.

Now my question is: before Schroedinger was people really disturbed by that second term?
Because if one naively thinks to the limit of "infinite temperatures" it surely does not matter, and equipartition is safe.

I see that $$k T$$ is comparable with $$\hbar \omega$$ e.g. considering room temperature (or even ten times larger) and visible light.
But in that case, does it really make sense to consider so few perturbative terms?
I mean, if they are comparable $$\hbar \omega/k T$$ is not small, and the expansions
found in books are not really correct. On the other hand, as I say, if the temperature is really large, the ZPE does not really matter.

2. Nov 6, 2008

### FranzDiCoccio

I did some research and read a paper by Einstein and Stern (1913) referred to by Wikipedia

They show that the experimental data for the specific heat of a gas of rotating molecules fit the formula including the ZPE,
under the assumption that $$\nu$$ depends on the temperature , where $$\nu$$ is the frequency of the oscillators involved in Planck's calculation.
If the same calculation is performed without ZPE the result has nothing to do with the experimental data.
On the other hand, if one ignores the temperature dependence of the frequency the two formulas give the same result, which is not so bad...

Uhm... I do not know... Einstein and Stern's assumptions look a bit simplistic and weird to me...
And anyway they do not sound particularly sure about the conclusiveness of their results, hence I wonder whether this is an appropriate reference. By the way, after reading the article I am under the impression that the introduction of the ZPE has to be credited to Planck himself, and not to Stern and Einstein as claimed by Wikipedia. I have some recollection about this from courses I took ages ago. I'll look it up.

It seems to me that the hints to this problem that I found so far in undergraduate books are useless or even a bit misleading (Alonso & Finn "Fundamental University Physics", Greiner & Stocker "Thermodynamics and Statistical Mechanics").
Does anybody know a book where this issue is discussed satisfactorily?