[UPSC IFoS 2011] Schrodinger’s Wave equation and applications (Q192, 20+10=30M)

Solve the Schrodinger equation for a potential step function given by \begin{aligned} v(x) &= 0 \text{ for } x < 0 \\ &= v_0 \text{ for } x > 0 \end{aligned} and calculate the reflection and transmission coefficients. Show that for E < v_0 there is a finite probability of finding the particle in a classically forbidden region.