= WKB quantization condition for a symmetric double well
{c}
{title2=$\tan\theta=\pm2e^\phi$}
For a smooth symmetric confining <potential energy> with four separated simple <classical turning points> $-x_2<-x_1<x_1<x_2$, define
$$
\theta(E)=\frac1\hbar\int_{x_1}^{x_2}\sqrt{2m(E-V)}\,dx,\qquad\phi(E)=\frac1\hbar\int_{-x_1}^{x_1}\sqrt{2m(V-E)}\,dx.
$$
The <Airy turning-point connection formula> applied to a decaying exterior <wavefunction> gives a central <WKB approximation> proportional to $\chi^{-1/2}[2\cos\theta\,e^K+\sin\theta\,e^{-K}]$, where $\chi=\sqrt{2m(V-E)}$ and $K(x)=\hbar^{-1}\int_x^{x_1}\chi\,dx$. Since $\chi'(0)=0$ and $2K(0)=\phi$, the even <Neumann boundary condition> gives $\tan\theta=2e^\phi$, while the odd <Dirichlet boundary condition> gives $\tan\theta=-2e^\phi$. For $\phi\gg1$, $\theta=(n+\tfrac12)\pi\mp\tfrac12e^{-\phi}+O(e^{-3\phi})$, with the minus correction for even <parity>. Since $\theta'=\mathcal T/(2\hbar)$ for the single-well classical <oscillation period> $\mathcal T$, the leading <double-well tunneling splitting> is $2\hbar e^{-\phi}/\mathcal T$.
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