The state of the art of semiconductor technology allows one to manufacture hardware in which the electrons move freely in two dimensions (e.g., the yz-plane). In the third dimension (x-axis), by means of appropriate coatings of material -- even single-atom monolayers are ``under control'' -- very narrow, high potential wells can be achieved. These lead to a sharp quantization of the motion in this direction. In this problem, a single, infinitely high potential well (this almost exists) is to be treated. Imagine that the free motion in the yz-plane has been separated off and solve the following subproblems.
(a) Determine the eigenfunctions and energy eigenvalues for motion in the infinitely high potential box
V(x)={0∞for x∈(−a,a)otherwise.
(b) Show that the system of eigenfunctions is complete.
(c) Determine the momentum distribution for a particle in the n-th energy eigenstate.