= Solution
\b[The <WKB approximation> describes a wave whose local <wavelength> changes slowly compared with one <wavelength>.] For the one-dimensional stationary <Schrödinger equation>, put $p(x)=\sqrt{2m[E-V(x)]}$. In a <classically allowed region>,
$$
\psi(x)\simeq\frac{C_+}{\sqrt{p(x)}}\exp\!\left[\frac{i}{\hbar}\int^x p(s)\,\mathrm ds\right]
+\frac{C_-}{\sqrt{p(x)}}\exp\!\left[-\frac{i}{\hbar}\int^x p(s)\,\mathrm ds\right].
$$
The <phase> is the classical action divided by $\hbar$. The amplitude varies as $p^{-1/2}$ so that a single travelling component has constant <probability current>. A useful local validity condition is $|\hbar p'/p^2|\ll1$.
For $V>E$, write $\kappa=\sqrt{2m(V-E)}/\hbar$. The wave instead has growing and decaying components proportional to $\kappa^{-1/2}\exp(\pm\int\kappa\,\mathrm dx)$. At a simple <turning point>, $p$ vanishes and neither local WKB expression is valid. Linearizing the potential gives an Airy-function solution that connects the two sides and supplies the turning-point <phase> shift. It is not legitimate to carry the divergent WKB amplitude straight through the <turning point>.<Topics in Quantum Mechanics by David Tong>[§2.2, semiclassical expansion and linear-potential matching]{ref}
\b[A useful application is tunnelling through a smooth opaque barrier.] With <turning points> $x_1,x_2$, the leading <transmission probability> is
$$
\boxed{T\simeq\exp\!\left[-2\int_{x_1}^{x_2}\kappa(x)\,\mathrm dx\right].}
$$
The exponent is twice the forbidden-region action in units of $\hbar$, because transmission is a <probability> rather than a wave amplitude. For an ideal rectangular barrier of height $V_0>E$ and width $d$, the same leading exponential is $\exp[-2d\sqrt{2m(V_0-E)}/\hbar]$; matching at the abrupt edges supplies an additional prefactor.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2016/paper-1-wkb-tunnelling.png]
{title=Original smooth-barrier construction and leading WKB transmission exponent; the shaded interval is classically forbidden}
For a bound state between two smooth <turning points>, the related WKB quantization rule is $\int_{x_1}^{x_2}p\,\mathrm dx=(n+1/2)\pi\hbar$. In a <harmonic potential> the integral equals $\pi E/\omega$, recovering $E_n=(n+1/2)\hbar\omega$. This exact agreement of energies for the harmonic oscillator does not make WKB <wavefunctions> uniformly exact near its <turning points>.
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