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If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy between $E_0$ and $E_0 + \delta E$. In the limit of large quantum numbers, the result is well known to coincide with the available volume of the phase space of the corresponding classical system of N newtonian free particles in a box, namely

$$

\Omega(E_0,V,N; \delta E)_{\textbf{quantum}} \to \frac{1}{h^N} \int_{E_0<E<E_0 +\delta E} d^{3N}x d^{3N}p

$$

in the limit of large quantum numbers.

My question is the following. Is there any proof, besides this specific example of the quantum gas in a box, that the quantum expression is always going to approach the classical one in phase space, for any given physical system (and thus for some generalized coordinates), provided some classical limit is used?

This does not seem a trivial statement to me, and I can't find the proof in textbooks.

Many thanks.

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# A Quantum versus classical computation of the density of state

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