= Solution
For a real potential, the <Time-dependent Schrödinger equation> and its <complex conjugate> give
$$
\partial_t|\psi|^2=\frac{i\hbar}{2m}(\psi^*\psi_{xx}-\psi\psi^*_{xx}).
$$
The potential terms cancel. The expression on the right is a total derivative, giving
$$
\boxed{\rho=|\psi|^2,\qquad j=\frac{\hbar}{2mi}(\psi^*\psi_x-\psi\psi_x^*)=\frac\hbar m\operatorname{Im}(\psi^*\psi_x),\qquad\rho_t+j_x=0.}
$$
For the <plane wave>, $\psi_t=-i\omega\psi$ and $\psi_{xx}=-k^2\psi$. With zero potential, substitution gives
$$
\boxed{\omega(k)=\frac{\hbar k^2}{2m},\qquad\rho=1,\qquad j=\frac{\hbar k}{m}.}
$$
It is a free-particle <momentum eigenstate> with momentum $\hbar k$, <energy> $\hbar\omega$ and uniform <probability> flux. On the whole real line it is a generalized state, not a normalized square-integrable <wavefunction>; wave packets or box normalization give the corresponding physical interpretation.
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