Classically, if a particle is not observed, the probability per unit length of finding it in a box is a constant 1/L along the entire length of the box.
(a) Show that the classical expectation value of the position is (1/2)L, that the expectation value of the square of the position is (1/3)L^2 and that the uncertainty in the position is L/(sqrt (12)).
(b) Treat this system quantum mechanically and show that the uncertainty in a particle's position in an infinite well in the general case of an arbitrary n is given by L [1/12 - 1/(2(n^2)(pi^2)]^(1/2).
In what circumstance does this quantity agree with the classical uncertainty?