From the time-dependent Schrödinger equation for $\psi(x, t)$, derive the equation

$$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$$

for $\rho(x, t)=\psi^{*}(x, t) \psi(x, t)$ and some suitable $j(x, t)$.

Show that $\psi(x, t)=e^{i(k x-\omega t)}$ is a solution of the time-dependent Schrödinger equation with zero potential for suitable $\omega(k)$ and calculate $\rho$ and $j$. What is the interpretation of this solution?