# Semiclassical state counting

• I
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)?