Write down the expressions for the probability density $\rho$ and the associated current density $j$ for a particle with wavefunction $\psi(x, t)$ moving in one dimension. If $\psi(x, t)$ obeys the time-dependent Schrödinger equation show that $\rho$ and $j$ satisfy

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

Give an interpretation of $\psi(x, t)$ in the case that

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

and show that $E=\frac{\hbar^{2} k^{2}}{2 m}$ and $\frac{\partial \rho}{\partial t}=0$.

A particle of mass $m$ and energy $E>0$ moving in one dimension is incident from the left on a potential $V(x)$ given by

$$V(x)=\left\{\begin{array}{rl} -V_{0} & 0<x<a \\ 0 & x<0, x>a \quad, \end{array}\right.$$

where $V_{0}$ is a positive constant. What conditions must be imposed on the wavefunction at $x=0$ and $x=a$ ? Show that when $3 E=V_{0}$ the probability of transmission is

$$\left[1+\frac{9}{16} \sin ^{2} \frac{a \sqrt{8 m E}}{\hbar}\right]^{-1}$$

For what values of $a$ does this agree with the classical result?