= Solution
Let $\rho=\psi^*\psi$ be the <probability density>. For a real <potential energy> $V(x)$, the <Schrödinger equation> and its complex conjugate are
$$
i\hbar\partial_t\psi=-\frac{\hbar^2}{2m}\partial_x^2\psi+V\psi,\qquad -i\hbar\partial_t\psi^*=-\frac{\hbar^2}{2m}\partial_x^2\psi^*+V\psi^*.
$$
Differentiate $\rho$ and substitute both equations. The potential terms cancel, leaving
$$
\partial_t\rho=\frac{i\hbar}{2m}(\psi^*\partial_x^2\psi-\psi\partial_x^2\psi^*)=\frac{i\hbar}{2m}\partial_x(\psi^*\partial_x\psi-\psi\partial_x\psi^*).
$$
Comparison with the <probability continuity equation> $\partial_t\rho+\partial_xJ=0$ identifies
$$
\boxed{J=\frac{i\hbar}{2m}[(\partial_x\psi^*)\psi-\psi^*(\partial_x\psi)]=\frac\hbar m\operatorname{Im}(\psi^*\partial_x\psi).}
$$
For a <plane wave> of <amplitude> $A$ and positive <wavenumber> $k$, $J=\hbar k|A|^2/m$, the density times the <velocity>. Integrating the continuity equation over an interval makes its <probability> change equal to the incoming current minus the outgoing current.
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