 Solution

For a real potential, the Time-dependent Schrödinger equation and its complex conjugate giveThe potential terms cancel. The expression on the right is a total derivative, givingFor the plane wave, ${\psi }_t=-i\omega \psi$ and ${\psi }_{xx}=-k^2\psi$. With zero potential, substitution givesIt is a free-particle momentum eigenstate with momentum $\hslash k$, energy $\hslash \omega$ 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.