(a) Write down the expressions for the probability density $\rho$ and associated current density $j$ of a quantum particle in one dimension with wavefunction $\psi(x, t)$. Show that if $\psi$ is a stationary state then the function $j$ is constant.

For the non-normalisable free particle wavefunction $\psi(x, t)=A e^{i k x-i E t / \hbar}$ (where $E$ and $k$ are real constants and $A$ is a complex constant) compute the functions $\rho$ and $j$, and briefly give a physical interpretation of the functions $\psi, \rho$ and $j$ in this case.

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

$$V(x)=\left\{\begin{array}{cl} -V_{0} & 0 \leqslant x \leqslant a \\ 0 & x<0 \text { or } x>a \end{array}\right.$$

where $a$ and $V_{0}$ are positive constants. Write down the form of the wavefunction in the regions $x<0,0 \leqslant x \leqslant a$ and $x>a$.

Suppose now that $V_{0}=3 E$. Show that the probability $T$ of transmission of the particle into the region $x>a$ is given by

$$T=\frac{16}{16+9 \sin ^{2}\left(\frac{a \sqrt{8 m E}}{\hbar}\right)}$$