# Is temperature quantized?

I'm learning quantum mechanics on my own. I've known that energy is quantized and I've started wondering about temperature. From thermodynamics we have: $$U=\frac{3}{2}NkT$$ (for ideal gas, of course)
Both U and N aren't continous, so i think T shouldn't be, too. Is that formula correct also for quantum mechanics?
I'm really sorry because of my language, I'm still working on it.

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In the canonical case it's clear that $T$, if coming from outside, can be continous (the probability $0<p<1$ for some random state is continous too, after all). However, in the microcanonical case of a closed system, where $1/T:=\partial S/\partial E$, where $S$ depends on a counting quantity (some trace over an operator which counts different energies where $[E,E+\Delta]$ gives some energy interval, say $\Gamma (E)$), my question would be what guaranties you that there is a smooth counting quantity, such that the derivative makes sense. – NikolajK Jul 22 '12 at 23:05
Excuse me, what canonical and microcanonical cases are? And what does $\Gamma(E)$ mean? – user10768 Jul 24 '12 at 12:45