I am looking at this: http://physics.ucsc.edu/~drip/5D/photons/photons.pdf Counting the Modes in the Box and I understand everything up to the point when it talks about spherical shell in m-space. I know what it means by m-space, but I don't understand why the number of points lying in the shell is equal to its volume.

Also, in general, what dm refers to. Isn't dm the difference of m that goes infinitely close to zero? Or am I misunderstanding it since m-space is discrete? I assume that the equation with dm and dw does not have physical meaning unless they are integrated, is it correct?


OK, if I take into account that the "lattice" of modes contains one point per unit cube, the number of points is actually the volume. But why can I do this on a discrete space?


1 Answer 1


OK, I asked my Physics TA today. The best explanation from him, up to now, is that this method is nowhere an exact solution, but an approximation. Generally, when the black body is heated up enough, there are many, many different oscillator modes happening at the same time, so the m in the m-space is very large, and thus approximates a continuous space. Therefore, computing the volume of the octant spherical shell is approximately computing the number of m points inside. Therefore, again, when the temperature is high, and thus m is high, the Planck's law is a good approximation.


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