5. Periodic potentials (see B&J4.8) Suppose we have a potential which is periodic with period L, i.e. V(a: + L) = V(93) (a) Show that if v(x) is a stationary state, then v(x + nL) is also a stationary state with the same energy, for any integer n. (b) Given any stationary state v(x), construct a set of stationary states (k (x) with the same energy, with the property that k(x + L) = elkLok(x). Check that you can recover y from the KS. (Hint: use a linear combination Pk(x) = En-.. Cny(x + nL) with appropriately chosen coefficients cn.) (c) Define uk(x) = e-krok(x) and show that ux is periodic, i.e. uk(x + L) = Uk(2). Hence we may choose an energy eigenbasis in which the stationary states are OK () = erkruk (). This is known as Bloch's theorem