Semiclassical quantization of bouncing ball

Question

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?

Answer 1

Unfortunately I cannot tell you what went wrong on your first try, since I don't exactly know what you did. However, I sat down and tried to solve the system you describe:

We are looking for Eigenstates of the Hamiltonian $$ H = \frac{p^2}{2} + V(x),\qquad \text{where }V(x)=\left\{\begin{array}{ll}\infty & \text{if } x<0 \\ x & \text{otherwise}\end{array}\right. $$

For $x<0$ we have $\psi(x)=0$, because the potential is infinite there.

For $x\geq 0$, we have a little more work to do. With the definition of $p = -i\frac{d}{dx}$, we can write $H\psi(x) = E_n\psi(x)$ as:

$$ -\frac{1}{2}\psi'' + x\psi = E_n\psi $$

I don't think this equation has an analytic solution. Wolphram Alpha says the solution is called the Airy function $\text{Ai}(x)$, which looks like this:

The solution is\

$$ \psi(x) = c \cdot\text{Ai}\left(2^\frac{1}{3}(x-E_n)\right) $$

To be able to stitch together both parts of the solution, they must coincide at $x=0$, which means we need $\psi(0) = 0$. So we have the condition

$$ E_n = -2^{-\frac{1}{3}}\cdot (n\text{th zero of Ai}) $$

The first few values are 0.826645, 1.44531, 1.95181, 2.39946, 2.80867 etc. (consult Wolphram Alpha for more values). These values closely resemble $E_n = \left(n+\frac{1}{\sqrt{2}}\right)^{\frac{2}{3}}$, but they (probably) diverge ultimately. For the last statement I have no prove, however, only a hunch. So, with your second try, you got the exponent somewhat right, but there is no direct proportionality.

Answer 2

I) Ignoring the metaplectic correction/Maslov index, the Bohr-Sommerfeld quantization rule reads

$$\begin{align} N ~\approx~&\int_a^b \!\frac{\mathrm{d}x}{\pi\hbar} p(x)\cr ~=~& \int_a^b \! \frac{\mathrm{d}x}{\pi\hbar} \sqrt{2m(E-V(x))},\end{align} \tag{1} $$

so that

$$\begin{align} \frac{dN}{dE} ~\stackrel{(1)}{\approx}~&\int_a^b \! \frac{\mathrm{d}x}{\pi\hbar} {\sqrt{\frac{m}{2(E-V(x))}}}\cr ~=~&\int_a^b \! \frac{\mathrm{d}x}{\pi\hbar v(x)}\cr ~=~&\frac{T}{2\pi\hbar }\cr ~=~&\frac{1}{\hbar\omega }, \end{align} \tag{2}$$

or

$$ \frac{dE}{dN} ~\stackrel{(2)}{\approx}~\hbar\omega, \tag{3} $$

in agreement with OP's recollection.

Example: If

$$ E~\propto~N^{\frac{2}{3}}\tag{4}$$

then

$$\frac{dE}{dN}~\stackrel{(4)}{\propto}~N^{-\frac{1}{3}}~\stackrel{(4)}{\propto}~E^{-2},\tag{5} $$

and vice-versa.

II) Now consider the potential

$$ V(x)~=~\left\{\begin{array}{ll} mgx & \text{if } x>0, \\ \infty & \text{if } x<0, \end{array}\right. \tag{6}$$

with $m=g=\hbar=1$. Next let us include the metaplectic correction/Maslov index. The turning point at an infinitely hard wall and an inclined potential wall have Maslov index $2$ and $1$, respectively, cf. e.g. this Phys.SE post. In total $3$. We should then adjust the Bohr-Sommerfeld quantization rule with a fraction $\frac{3}{4}$.

$$\begin{align} n+\frac{3}{4}~\approx~&\int_a^b \!\frac{\mathrm{d}x}{\pi} k(x)\cr ~=~& \int_0^{E_n} \!\frac{\mathrm{d}x}{\pi} \sqrt{2(E_n-x)}\cr ~=~&\frac{(2E_n)^{\frac{3}{2}}}{3\pi}, \end{align} \tag{7}$$

where $n\in\mathbb{N}_0$. Inverting eq. (7) yields

$$ E_n~\stackrel{(7)}{\approx}~ \frac{1}{2}\left(3\pi( n+\frac{3}{4})\right)^{\frac{2}{3}}. \tag{8} $$

Answer 3

I think your error is in assuming that $E_{n+1} - E_{n}$ is proportional to $n$. At least, I assume you assumed it; it's the only way I can see that you could go from the statement $$ E_{n+1} - E_n \propto E_n^{-1/2} $$ to the statement $$ E_n \propto n^{-2}. $$ Really, what the first proportionality above implies is that $$ \frac{dE}{dn} \propto \frac{1}{\sqrt{E}} $$ in the limit of large $n$; and if you solve this equation, you get $$ \sqrt{E} \, dE \propto dn \quad \Rightarrow \quad E^{3/2} \propto n \quad \Rightarrow \quad E \propto n^{2/3} $$ as expected.