The classical probability distribution function for a particle in a one-dimensional box of length L is P = 1/L.

(a) Show that the classical expectation value of x² for a particle in a one-dimensional box of length L centered at the origin (Problem 32) is L²/12.

(b) Find the quantum expectation value of x² for the nth state of a particle in the one-dimensional box of Problem 32 and show that it approaches the classical limit L²/12 for n >> 1.