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

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

The wave-function is

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

where $E=\hbar^{2} k^{2} / 2 m$ and $R$ is a constant, which may be complex. Evaluate $j$.