(a) Define the probability density $\rho$ and probability current $j$ for the wavefunction $\Psi(x, t)$ of a particle of mass $m$. Show that

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

and deduce that $j=0$ for a normalizable, stationary state wavefunction. Give an example of a non-normalizable, stationary state wavefunction for which $j$ is non-zero, and calculate the value of $j$.

(b) A particle has the instantaneous, normalized wavefunction

$$\Psi(x, 0)=\left(\frac{2 \alpha}{\pi}\right)^{1 / 4} e^{-\alpha x^{2}+i k x}$$

where $\alpha$ is positive and $k$ is real. Calculate the expectation value of the momentum for this wavefunction.