What is the paradox in Planck's Law?

[Almanzo]

The problem with a histogram is the arbitraryness of the division of parameter space into different slots. This arbitraryness can be removed -- for the classical case, at any rate -- by using a accumulation curve.

In the ball-bearing experiment this curve would have height zero for diameter or volume zero, and increase by one step at every diameter or volume which is found. (If two or more balls have exactly the same diameter or volume, the curve would increase by two or more steps.) At infinite diameter or volume the height of the curve would be equal to the number of balls tested.

Now, instead of finding a peak in the distribution, one has to find the region of greatest steepness in the accumulation curve. Finding this region for the volume-curve and the diameter curve, we find a volume and a diameter. The question is now whether the relation between this diameter and volume are the same relation which we would expect in any individual ball.

In the quantum case, one would have to test two equivalent samples, measuring wavelength for one sample and frequency for the other one. In principle one of them should be measurable exactly if one forgoes measuring the other one. But if exact measurements are not possible, the accumulation curve will change from a sharp curve consisting of little steps into a more fuzzy, "S-shaped" band. (Not really S-shaped, of course; more like a hillside.) A region of greatest steepness should still be there to be found, though not to be pinpointed exactly.

 

[apolski]

\dot n(\nu) = \frac{2\pi\nu^2}{c^2} \frac{1}{e^{h\nu/kT}-1}

How you found that expression?

 

[apolski]

\dot n(\nu,T) = \frac{J(\nu,T)}{h\nu}

But why?