= Solution
For a finite potential step and constant <mass>, $\psi$ and $\psi'$ are continuous. A finite jump of $\psi'$ would require a delta-function contribution in the integrated equation, while a jump of $\psi$ would be still more singular. Matching the preceding amplitudes gives
$$
\boxed{\begin{aligned}
A+B&=C+D,\\
ik(A-B)&=\kappa(C-D),\\
Ce^{\kappa a}+De^{-\kappa a}&=Fe^{ika},\\
\kappa(Ce^{\kappa a}-De^{-\kappa a})&=ikFe^{ika}.
\end{aligned}}
$$
The <transmission probability> is $|F/A|^2$ because the incident and transmitted regions have equal <wave vector> and <mass>; in unequal leads a velocity ratio is also required. These matching conditions do not apply unmodified to infinite walls, delta barriers or <mass> discontinuities.
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