= Solution
For the oppositely travelling components, write $u=c_1e^{ikx}$ and $v=c_2e^{-ikx}$ after removing the common time phase. Then $\partial_x\psi=ik(u-v)e^{-i\omega t}$. In $\operatorname{Im}(\psi^*\partial_x\psi)$ the <interference> terms contribute no imaginary part, so
$$
\boxed{J=\frac{\hbar k}{m}(|c_1|^2-|c_2|^2).}
$$
A standing-wave <interference> pattern can therefore change density without adding a cross-term to the net current.
For \b[<barrier-top scattering>], the stationary equation inside the barrier is $-\hbar^2\psi_2''/(2m)=0$. Thus its spatial part is linear: $\psi_2=(bx+c)e^{-i\omega t}$. Take incident amplitude one, $k=\sqrt{2mE}/\hbar$ and $\omega=E/\hbar$. Continuity of the spatial <wave function> and its derivative at the two finite steps gives
$$
\begin{aligned}
e^{-ika}+re^{ika}&=-ab+c,& ik(e^{-ika}-re^{ika})&=b,\\
te^{ika}&=ab+c,& ikte^{ika}&=b.
\end{aligned}
$$
Solving these four equations yields
$$
\boxed{r=-\frac{ika\,e^{-2ika}}{1-ika},\qquad t=\frac{e^{-2ika}}{1-ika}.}
$$
The equal exterior velocities mean flux ratios are squared amplitude ratios:
$$
\boxed{T=|t|^2=\frac1{1+k^2a^2}=\left(1+\frac{2mEa^2}{\hbar^2}\right)^{-1},\qquad R=\frac{k^2a^2}{1+k^2a^2},\qquad R+T=1.}
$$
This is the threshold limit of finite-barrier scattering, not exponential decay with a nonzero interior decay constant.
The interior constants are
$$
\boxed{b=\frac{ik e^{-ika}}{1-ika},\qquad c=e^{-ika}.}
$$
Since $\operatorname{Im}[(bx+c)^*b]=\operatorname{Im}(c^*b)=k/(1+k^2a^2)$,
$$
\boxed{J_2=\frac{\hbar k/m}{1+k^2a^2}=J_{\rm incident}T.}
$$
It is positive and independent of <position>, matching the transmitted flux. Although the classical kinetic energy inside is zero, the complex linear quantum <wave function> carries conserved <probability current>; its phase varies with <position>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2016/paper-1-barrier-top-current.png]
{title=At the barrier-top energy the complex amplitude is linear while its probability current remains constant}
{source=paper-1-barrier-top-current.py}
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