Write down expressions for the probability density $\rho(x, t)$ and the probability current $j(x, t)$ for a particle in one dimension with wavefunction $\Psi(x, t)$. If $\Psi(x, t)$ obeys the timedependent Schrödinger equation with a real potential, show that

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

Consider a stationary state, $\Psi(x, t)=\psi(x) e^{-i E t / \hbar}$, with

$$\psi(x) \sim \begin{cases}e^{i k_{1} x}+R e^{-i k_{1} x} & x \rightarrow-\infty \\ T e^{i k_{2} x} & x \rightarrow+\infty\end{cases}$$

where $E, k_{1}, k_{2}$ are real. Evaluate $j(x, t)$ for this state in the regimes $x \rightarrow+\infty$ and $x \rightarrow-\infty$.

Consider a real potential,

$$V(x)=-\alpha \delta(x)+U(x), \quad U(x)= \begin{cases}0 & x<0 \\ V_{0} & x>0\end{cases}$$

where $\delta(x)$ is the Dirac delta function, $V_{0}>0$ and $\alpha>0$. Assuming that $\psi(x)$ is continuous at $x=0$, derive an expression for

$$\lim _{\epsilon \rightarrow 0}\left[\psi^{\prime}(\epsilon)-\psi^{\prime}(-\epsilon)\right]$$

Hence calculate the reflection and transmission probabilities for a particle incident from $x=-\infty$ with energy $E>V_{0} .$