= Solution
The global time-dependent phase cancels from the <probability current>. Put $\chi=e^{ikx}+Re^{-ikx}$. Direct multiplication gives
$$
\chi^*\chi_x=ik(1-|R|^2)+ik(R^*e^{2ikx}-Re^{-2ikx}).
$$
The second term is real, because the expression in parentheses is purely imaginary. Hence
$$
\boxed{j=\frac{\hbar k}{m}(1-|R|^2).}
$$
The first <plane wave> travels to the right with incident current $\hbar k/m$; the second travels to the left with reflected current $-(\hbar k/m)|R|^2$. Their <wave interference> changes the probability density but cancels from the net current. The current vanishes for $|R|=1$, is positive for $|R|<1$, and is negative for $|R|>1$. These plane waves are scattering states, not individually normalizable bound states.
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