Apply the WKB approximation to the energy levels below the top of the barrier in a symmetric double well, and show that the energy eigenvalues are determined by a condition of the form
$$
\tan \left(\int k d x+\alpha\right)= \pm 2 \theta
$$
where $\theta$ is the quantity defined in (7.48) for the barrier, $\alpha$ is a constant dependent on the boundary conditions, and the integral $\int k d x$ is to be extended between the classical turning points in one of the separate wells. Show that at low transmission the energy levels appear in close pairs with a level splitting approximately equal to $\hbar \omega / \pi \theta$ where $\omega$ is the classical frequency of oscillation in one of the single wells.