What is the probability current for a particle of mass $m$, wavefunction $\psi$, moving in one dimension?

A particle of energy $E$ is incident from $x<0$ on a barrier given by

$$V(x)=\left\{\begin{array}{cc} 0 & x \leqslant 0 \\ V_{1} & 0<x<a \\ V_{0} & x \geqslant a \end{array}\right.$$

where $V_{1}>V_{0}>0$. What are the conditions satisfied by $\psi$ at $x=0$ and $x=a$ ? Write down the form taken by the wavefunction in the regions $x \leqslant 0$ and $x \geqslant a$ distinguishing between the cases $E>V_{0}$ and $E<V_{0}$. For both cases, use your expressions for $\psi$ to calculate the probability currents in these two regions.

Define the reflection and transmission coefficients, $R$ and $T$. Using current conservation, show that the expressions you have derived satisfy $R+T=1$. Show that $T=0$ if $0<E<V_{0}$.