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So I have this problem where a particle is on a ring of perimeter L, and the coordinate of the particle is denoted by $s$, $0<s<L$. 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?

Qmechanic
  • 201,751
Ludwig van Dirac
  • 153
  • 4

2 Answers2

6

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.

knzhou
  • 101,976
1

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.

Qmechanic
  • 201,751