[UPSC IFoS 2015] Schrodinger’s Wave equation and applications (Q177, 10+5+5=20M)

Solve the Schr"odinger equation for an electron of mass m confined in a one-dimensional potential well of the form \begin{align*} V &= 0 \text{ when } 0 \le x \le L \ &= \infty \text{ when } x < 0; \ x > L \end{align*} Obtain the discrete energy levels and the normalized eigen functions.