# WKB approximation for a particle on a ring $(E>V)$

So I have this problem where a particle is on a ring of perimeter L, and the coordinate of the particle is denoted by $$s$$, $$0. There is a nonzero potential which varies with $$s, V(s)$$, and is always smaller than the energy of the particle. For $$E>V$$, the WKB wave function is $$\psi(x) = \frac{1}{\sqrt{p(x)}}(C_+e^{i\phi(x)} + C_-e^{-i\phi(x)}),$$ $$\phi(x) := \frac{1}{\hbar}\int_{x_0}^{x}\sqrt{2m(E - V(x'))}dx',$$ with $$p(x) := \sqrt{2m(E - V(x))}.$$

What should I do if I want to get the quantization condition for the energies $$E_n$$? I've tried doing $$\psi(s) = \psi(s + nL),$$ $$n$$ being an integer, like with regular "particle on a ring" problems, but since we don't have any other boundary conditions, and $$V(s + nL) = V(s)$$ (because all we did was go around the loop $$n$$ times), I just get $$0 = 0$$ and can't seem to get anywhere with this. How could I get an expression for the quantized energies?

You don't use $$\psi(s) = \psi(s + n L)$$ for $$n$$ integer, because the ring always has length $$L$$, its length has nothing to do with $$n$$. Instead you impose the condition $$\psi(s) = \psi(s + L)$$, which implies that $$\phi(L) - \phi(0) = 2 \pi n$$ which $$n$$ is the energy level. Then you get $$n h = \int_0^L \sqrt{2 m (E_n - V(x))} \, dx$$ which is a typical WKB quantization integral, from which you compute the $$E_n$$ in the usual way.
• @LudwigvanDirac Yes; I used $2 \pi \hbar = h$. – knzhou Apr 1 '20 at 2:31
Let us modify OP's notation to acknowledge the dependence of the lower integration bound $$\phi(x_2,x_1) ~:=~ \frac{1}{\hbar}\int_{x_1}^{x_2}\!dx\sqrt{2m(E - V(x))}.\tag{1}$$ Then $$\phi(x_3,x_1)~=~\phi(x_3,x_2)+\phi(x_2,x_1).\tag{2}$$ From the periodicity of the potential $$V$$ we have $$\phi(L+x,x)~=~\phi(L,0).\tag{3}$$ From the single-valueness of the wavefunction, we get $$\psi(x)=\psi(x+L),\tag{4}$$ or equivalently, \begin{align} \sum_{\pm} C_{\pm}e^{\pm i\phi(x,x_0)} ~\stackrel{(4)}{=}~& \sum_{\pm} C_{\pm}e^{\pm i\phi(x+L,x_0)} ~\stackrel{(2)}{=}~ \sum_{\pm} C_{\pm}e^{\pm i\phi(x+L,x)}e^{\pm i\phi(x,x_0)}\cr ~\stackrel{(3)}{=}~& \sum_{\pm} C_{\pm}e^{\pm i\phi(L,0)}e^{\pm i\phi(x,x_0)}.\end{align}\tag{5} Eq. (5) are infinitely many equations for 2 unknowns $$e^{\pm i\phi(L,0)}$$. By picking at least 2 values of $$x$$, it becomes clear that the only solution to (5) is $$e^{\pm i\phi(L,0)}~=~1, \tag{6}$$ or equivalently, $$\phi(L,0)~\in~2\pi\mathbb{Z},\tag{7}$$ which leads to the well-known WKB quantization rule.