[UPSC CSE 1997] Schrodinger’s Wave equation and applications (Q229, 60M)

Write down the time independent Schrodinger equation for the x < 0 and x > 0 in case of a step potential V(x) = \begin{cases} 0, & x < 0 \\ V_o, & x > 0 \end{cases} Discuss the solutions thus obtained for the case E > V_o. Obtain the following relations. |R|^2 = \left(\frac{1-\mu}{1+\mu}\right)^2 \quad \text{and} \quad \frac{k}{k_o}|T|^2 = \mu \left(\frac{2}{1+\mu}\right)^2 There \mu = \sqrt{1 - \frac{V_o}{E}}, R and T are reflection and transmission coefficients respectively. Other symbols have their usual meaning. Plot the behaviour of |R|^2 and k/k_o |T|^2 with \mu.