Posted by Steve Simon on January 20, 2013, 10:22 pm, in reply to "Apparent loss of quantisation of sound waves in Debye model"
Right. I don't think I answered this clearly in lecture.
There are two places where quantization enters. First, given a particular wave mode, that mode acts like an oscillator. So if its frequency is omega, the energy stored in that mode at temperature T is
hbar omega (nbose(beta hbar omega) + 1/2)
This is a quantum mechanical result (note the hbar).
The second place that "quantization" occurs is that the wavevector must be discrete --- i.e., only certain values of k are allowed. Now in fact, this is not really really quantum mechanics, it is just some boundary condition on the waves.
What we want to do is to sum over all possible wave modes (over all possible allowed k's). This sum should be discrete if we are being very exact. However, as long as the argument of the sum (the thing you are summing) is not changing rapidly with k you can convert a sum over k into an integral, which is what we did in the class and in the notes.
You might wonder then why it was important that we started with a sum in the first place if we were just going to convert into an integral anyway. The point is that we need to know the coefficient of the integral --- which is related to the density of the discrete points. So you have to start with the discrete points and then convert to the integral and in the conversion you get the proper prefactor of the integral.
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