= Finite rectangular barrier matching
Take a <nonrelativistic particle> of constant <mass> $m$ with <energy> $E>0$, and choose the same zero of <potential energy> on both sides of a <rectangular potential barrier>:
$$
V(x)=\begin{cases}V_0,&0<x<a,\\0,&x<0\text{ or }x>a,\end{cases}
\qquad V_0>0,\quad a>0.
$$
The equality of the exterior <potential energies> matters: it gives the same exterior <wavenumber> $k=\sqrt{2mE}/\hbar$ and makes $T=|t|^2$. Here $\hbar=h/(2\pi)$ is the reduced <Planck constant>. This model has a finite jump in <potential energy>, no <delta potential>, and no absorbing term.
For $0<E<V_0$, the <Time-independent Schrödinger equation> gives the spatial <wavefunction>
$$
\psi(x)=\begin{cases}
e^{ikx}+re^{-ikx},&x<0,\\
Ae^{\kappa x}+Be^{-\kappa x},&0<x<a,\\
te^{ikx},&x>a,
\end{cases}
\qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.
$$
The full <stationary state> is $\Psi(x,t)=\psi(x)e^{-iEt/\hbar}$. We set the incident <probability amplitude> to one and impose no incoming wave from the right; this <scattering wavefunction> is not a <normalizable wavefunction>. The <tunnelling decay constant> $\kappa$ describes <exponential decay>, rather than an oscillatory <wavenumber> in the barrier. Both <exponential functions> are needed on the finite interval: the growing term is not excluded by a condition at infinity.
The <wavefunction> and its <derivative> are <continuous> at each face, giving four linear matching conditions:
$$
\begin{aligned}
1+r&=A+B,&ik(1-r)&=\kappa(A-B),\\
Ae^{\kappa a}+Be^{-\kappa a}&=te^{ika},&
\kappa(Ae^{\kappa a}-Be^{-\kappa a})&=ikte^{ika}.
\end{aligned}
$$
Their solution supplies the <transmission amplitude>
$$
t=\frac{e^{-ika}}{\cosh(\kappa a)+i\frac{\kappa^2-k^2}{2k\kappa}\sinh(\kappa a)}.
$$
A short derivation is to use the two right-face equations to express $Ae^{\kappa a}$ and $Be^{-\kappa a}$ in terms of $t$, then substitute into the left-face equations. Taking the squared <modulus> gives the <rectangular barrier transmission probability>.
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