= Rectangular barrier transmission probability
{title2=$T(E,a)$}
For equal exterior <potential energies> and constant <mass>, the exact <transmission probability> is
$$
T(E,a)=\begin{cases}
\displaystyle\left[1+\frac{V_0^2}{4E(V_0-E)}\sinh^2(\kappa a)\right]^{-1},&0<E<V_0,\\[6pt]
\displaystyle\left[1+\frac{mV_0a^2}{2\hbar^2}\right]^{-1},&E=V_0,\\[6pt]
\displaystyle\left[1+\frac{V_0^2}{4E(E-V_0)}\sin^2(qa)\right]^{-1},&E>V_0,
\end{cases}
\qquad q=\frac{\sqrt{2m(E-V_0)}}{\hbar}.
$$
Below the barrier, the <hyperbolic sine> is positive for $a>0$, so $0<T<1$. At $E=V_0$, the interior <wavefunction> is linear in $x$; either neighbouring formula has the displayed finite <limit>, since $\sinh z/z$ and $\sin z/z$ tend to one. Above the barrier, the interior <wavefunction> oscillates: the <quantum reflection probability> generally remains nonzero, but $qa=n\pi$ for a positive integer $n$ gives perfect <transmission probability> $T=1$ through constructive <wave interference>. These are <scattering resonances>, not below-barrier <quantum tunnelling>.
At half height, $E=V_0/2$, we have $k=\kappa$ and $T=\operatorname{sech}^2(\kappa a)$, consistent with <finite square barrier transmission at half barrier height> after replacing this width $a$ by that example's width $2a$. In the illustration, the width is four times $\ell=\hbar/\sqrt{2mV_0}$ and $E/V_0=1/2$, giving $T\simeq0.01388$. The plotted <wavefunction> comes from solving the four interface equations; the <transmission probability> curve uses the closed formula.
\Image[/square-barrier-wavefunction-transmission.png]
{title=Matched square-barrier wavefunction and transmission below and above the barrier}
{description=Original model calculations with equal exterior <potential energies>, constant <mass> and incident <probability amplitude> one. The <real part> and <imaginary part> of the <wavefunction>, its <modulus> and the exact <transmission probability> are shown; these curves are not measurements. The conserved net <probability current> is small but nonzero throughout the barrier.}
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