A particle moving in one space dimension with wavefunction $\Psi(x, t)$ obeys the timedependent Schrödinger equation. Write down the probability density $\rho$ and current density $j$ in terms of the wavefunction and show that they obey the equation

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

Evaluate $j(x, t)$ in the case that

$$\Psi(x, t)=\left(A e^{i k x}+B e^{-i k x}\right) e^{-i E t / \hbar}$$

where $E=\hbar^{2} k^{2} / 2 m$, and $A$ and $B$ are constants, which may be complex.