= Solution
In a <classically forbidden region> the local <wavevector> becomes imaginary. For a slowly varying barrier, the WKB <transmission probability> has leading exponential
$$
\mathcal T\sim e^{-2G},\qquad
G=\frac1\hbar\int_{r_1}^{r_2}\sqrt{2\mu[V(r)-Q]}\,dr,
$$
where $r_1,r_2$ are turning points and $\mu$ is the <reduced mass>. This is a <probability> exponent; the wave <amplitude> has only half that exponent. Near a turning point one uses matching formulas rather than the unmodified local WKB expression.
For <alpha decay>, an attractive nuclear interior supports a quasibound state behind the repulsive <Coulomb barrier> of the daughter nucleus. For zero <orbital angular momentum>, write the exterior potential as $C/r$, $C=2Z_de^2/(4\pi\epsilon_0)$, with nuclear matching radius $R$ and outer turning point $b=C/Q>R$. The exterior <action> can be integrated:
$$
G=\frac{\sqrt{2\mu Q}\,b}{\hbar}
\left[\arccos\sqrt{R/b}-\sqrt{(R/b)(1-R/b)}\right].
$$
The decay rate is approximately $\lambda=\nu P_{\rm pre}e^{-2G}$ and the <half-life> is $\ln2/\lambda$. Here $\nu$ is the interior assault-frequency scale and $P_{\rm pre}$ is the alpha preformation <probability>. They are model-dependent; barrier penetration alone does not determine them. Nonzero <orbital angular momentum> adds a <centrifugal barrier>.
For $R/b\ll1$, the leading dependence is $G\propto Z_d/\sqrt Q$. Thus a modest increase in decay <energy> can cause an enormous increase in decay rate, giving the <Geiger-Nuttall law> relating logarithmic <half-life> to inverse square-root <energy> along a suitable isotope sequence. Tunnelling conserves the total <energy>: the escaping <alpha particle> does not acquire <energy> from the barrier.
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