Consider a free electron near a boundary surface. (a) If øk (r)'s are the electron eigenfunctions, show that the function u(a, t) = Eox(=)¢x (0) cxp() satisfies a diffusion-type cquation. Ideutify the corresponding diffusion co- efficient. (b) Froum the thoory of diffusion how would you expect u(0, t) to be influenced by the prescnce of a boundary at a distance l fron the origin? Would the boundary be felt immediately or only after an clapse of tine? (c:) Exainine the expression for u(0, t) as a sum over k as given in (a). What is the range of Ek which contribute significantly to u(0, t) at the tiume when the influence of the boundary is felt by the clectron?

Consider a free electron near a boundary surface. (a) If øk (r)'s are the electron eigenfunctions, show that the function u(a, t) = Eox(=)¢x (0) cxp() satisfies a diffusion-type cquation. Ideutify the corresponding diffusion co- efficient. (b) Froum the thoory of diffusion how would you expect u(0, t) to be influenced by the prescnce of a boundary at a distance l fron the origin? Would the boundary be felt immediately or only after an clapse of tine? (c:) Exainine the expression for u(0, t) as a sum over k as given in (a). What is the range of Ek which contribute significantly to u(0, t) at the tiume when the influence of the boundary is felt by the clectron?