= Solution
For a particle of mass $m$, the <probability density> and one-dimensional <probability current> are
$$
\boxed{\rho=\psi^*\psi,\qquad
j=\frac{\hbar}{2mi}(\psi^*\psi_x-\psi\psi_x^*)
=\frac{\hbar}{m}\operatorname{Im}(\psi^*\psi_x).}
$$
The <Schrödinger equation> and its <complex conjugate>, for real $V$, give
$$
\rho_t=\psi_t\psi^*+\psi\psi_t^*
=\frac{i\hbar}{2m}(\psi^*\psi_{xx}-\psi\psi_{xx}^*)=-j_x.
$$
The potential terms cancel precisely because $V$ is real. Thus \b[$\rho_t+j_x=0$], the <continuity equation> for probability. Integrating over an interval says that its probability changes by the incoming current minus the outgoing current.
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