= Solution
The <WKB approximation>, also called JWKB, describes a short local de Broglie wavelength in a slowly varying <potential energy>. For $p(x)=\sqrt{2m[E-V(x)]}$ in a classically allowed region, its two local waves are
$$
\boxed{\psi(x)\simeq\frac1{\sqrt{p(x)}}\left[Ae^{(i/\hbar)\int^xp(s)ds}+Be^{-(i/\hbar)\int^xp(s)ds}\right].}
$$
The phase follows the classical <momentum>; its amplitude varies to conserve <probability current>. In a forbidden region, putting $\kappa(x)=\sqrt{2m[V(x)-E]}$ gives $\psi\simeq\kappa^{-1/2}[Ce^{\int^x\kappa/\hbar}+De^{-\int^x\kappa/\hbar}]$, with boundary conditions selecting the physical combination.
The <WKB validity condition> is $\hbar|p'|/p^2\ll1$, with the analogous condition on $\kappa$. It fails at a <turning point> where $E=V$. For a simple turning point, locally linearizing V yields an <Airy function>; matching fixes the relative amplitudes and a $\pi/4$ phase. Two such turning points give the <two-turning-point WKB quantization condition>
$$
\boxed{\int_{x_1}^{x_2}p(x)\,dx=\left(n+\tfrac12\right)\pi\hbar,\qquad n=0,1,\ldots.}
$$
For an opaque barrier, the <WKB tunnelling exponent> instead gives
$$
\boxed{T\sim\exp\left[-\frac2\hbar\int_{x_1}^{x_2}\kappa(x)\,dx\right].}
$$
The factor two distinguishes probability from amplitude. Matching determines the prefactor; the exponential alone is not accurate near the barrier top. These formulas connect <bound state> <energy eigenvalues> and <quantum tunnelling> to classical actions. <Topics in Quantum Mechanics by David Tong>[§2.2, printed pp.37–49, semiclassical expansion, Airy matching and tunnelling]{ref}
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2012/paper-1-aqp2012-wkb-barrier.png]
{title=An illustrative smooth barrier separates two oscillatory WKB regions by an evanescent interval}
{source=paper-1-aqp2012-wkb-barrier.py}
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