[UPSC CSE 1998] Schrodinger’s Wave equation and applications (Q228, 60M)

An infinite well potential is V(x) = \begin{cases} 0, & -a/2 < x < a/2 \\ \infty, & x < -a/2, \quad x > a/2 \end{cases} Solve the time independent Schrodinger equation and find the closed form expressions for the eigenvalues and eigenfunction of potential. Give a schematic representation of the first three energy levels and corresponding wave functions and probability distribution functions. Show that the zero-point energy is in accord with the uncertainty principle.