[UPSC IFoS 2012] Schrodinger’s Wave equation and applications (Q189, 20+10=30M)

Solve the Schrödinger equation to obtain the energy levels and eigenfunctions of a particle in a one dimensional infinitely deep potential well given by \begin{aligned} V(x) &= 0 \text{ for } 0 < x < a \\ &= \infty \text{ for } x < 0 \text{ and for } x > a \end{aligned} Show that they form an orthonormal set of functions. What do you understand by completeness condition ? Show that they form a complete set of functions.