Posted by Steve Simon on May 24, 2014, 7:59 am, in reply to "Debye mode"
I'm not sure why you would want 3/2 there. If you compare to eq. 2.1 (of free lecture notes) you see that for each oscillator you should have energy hbar omega ( n + 1/2) and when you consider the thermal occupancy of the mode you end up with an expectation of the energy hbar omega (n_Bose(beta hbar omega) + 1/2) What you are probably thinking of is that sometimes people will write, for a three dimensioanl harmonic oscillator E = hbar omega (nx + 1/2 + ny + 1/2 + nz + 1/2) adding the energy of the three modes together. Then defining ntotal = nx + ny + nz, you might write E = hbar omega (ntotal + 3/2) But then if you were to do this, the thermal expectation of ntotal is 3 n_Bose(beta hbar omega) so it is much better to keep all the modes separate and write it as E = 3 hbar omega(n_Bose(beta hbar omega) + 1/2) 
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