A particle is confined to move in a one-dimensional box of length L.

(a) If the particle is classical, show that the average value of x is 1/2 L and that the root-mean square value is L/3^1/2.

(b) Show that for large values of n, a quantum particle approaches the classical values. This result is an example of the correspondence principle, which states that, for very large values of the quantum numbers, the predictions of quantum mechanics approach those of classical mechanics.