How Does Semiclassical State Counting Relate to Quantum Energy Eigenstates?

[jostpuur]

Long time ago I encountered a claim that if you fix some energy interval $[E_A,E_B]$, the measure of the set

$$\{(x,p)\;|\;E_A\leq H(x,p)\leq E_B\}$$

where $H(x,p)$ is some classical Hamiltonian, is going to be approximately proportional to the number of energy eigenstates contained in the energy interval in the quantized model. It could be that you had to divide this measure by $\hbar$, and that's where the approximate number would come from. Or perhaps by some power of $\hbar$ depending on the dimension?

Do you know this result, and does it have a recognizable name? How is it justified (proven)?

 

[DrDu]

On the Quantum Correction For Thermodynamic Equilibrium
link.aps.org/doi/10.1103/PhysRev.40.749

 

[jostpuur]

I don't have ability to download that pdf file, but based on the title I would guess that that reply got accidentally sent to a wrong thread. I had recently opened another thread where I requested information about Boltzmann distribution in quantum mechanical setting:

https://www.physicsforums.com/threads/boltzmann-with-degenerate-levels.902321/

When a PF user has multiple tabs opened in his or her browser, I guess it can happen that a message gets accidentally sent to a wrong thread?