Consider an elastically bouncing ball of mass $m$ and energy $E$. This has a triangular potential $$ V(x)~=~\left\{\begin{array}{ll} mgx & \text{if } x>0, \\ \infty & \text{if } x<0, \end{array}\right. $$ where the $x$-axis points upwards. Let $\hbar = m = g = 1$, so that the maximum height reached is $E$.
The classical frequency of motion is $$\omega = \frac{\pi}{\sqrt{2 E}}.$$ I can recall there is a quantization rule that says that the spacing between energy levels equals the classical frequency (in the semiclassical regime). This would imply that $$ E_{n+1} - E_n \propto E_n^ {-1/2} $$ so $$ E_n \propto n^{-2}. $$ However, the area in phase space enclosed by the orbit is proportional to $E^{3/2}$. This area must be an integer times $2\pi$, which gives a different quantization condition $$ E_n \propto n^{2/3}. $$ What went wrong? I am pretty sure the second result is correct, but why is the first result wrong?
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