= Solution
Inside a rectangular barrier with $E<V$, the stationary <Schrödinger equation> gives
$$
\Psi''=\frac{2m(V-E)}{\hbar^2}\Psi.
$$
Its independent solutions are $e^{\pm\alpha x}$ with
$$
\boxed{\alpha=\frac{\sqrt{2m(V-E)}}\hbar.}
$$
For an opaque barrier the amplitude is suppressed by approximately $e^{-\alpha L}$ and the probability by $e^{-2\alpha L}$. Matching contributes a prefactor, so this exponential approximation is not an exact <transmission coefficient> for a transparent or very thin barrier. For two moving particles the relative-coordinate <mass> is their <reduced mass>.
Back to article page
