[UPSC CSE 1989] Schrodinger’s Wave equation and applications (Q238, 60M)

Write down Schrodinger's equation in one-dimension and explain the significance of the eigenvalues and eigenfunction of this equation. Solve the Schrodinger's equation if the potential function V is given as V(x) = V_e(\text{a constant}) for |x| < a/2 = 0 for |x| > a/2 State the boundary conditions you use along with their justification (In the solution consider particles incident from one side only.) If he total energy F < V_0 show that there is an important difference with classical mechanics. Explain with the help of the uncertainty principle that E < V_0 does not mean that the kinetic energy is negative.