Problem 10.3: Landau levels. (35 points) Consider a uniform magnetic field pointing in the z-direction, B = B2, and a particle of charge q moving in the z-y plane in the presence of this field. The particle is constrained not to move in the z-direction, so we may drop all z-derivatives. a) To write down the Schrodinger equation we need to choose a vector potential A. (Since E = 0, we will always choose the scalar potential , = 0.) One choice for the uniform magnetic field that we have already used in class is ' = (1/2)B x F = (B/2)(xy - yi). However, we will find a different choice of gauge to be more useful here: consider the gauge choice A = Bry. Find the gauge transformation parameter A relating this choice of gauge to A', thereby proving that they lead to the same B. b) Consider the time-independent Schrodinger equation for a charged particle moving in the I-y plane in the presence of the vector potential A = Bry, and write a trial stationary state in the variable-separated form, (I, y) = f(x)gly) . (10) Which of p, and py commutes with the Hamiltonian? Choose * to be an eigenvector of the commuting momentum, and thus determine g(y) (it will depend on a single quantum number) - don't bother normalizing g(y). Substitute the resulting form of + back into the TISE to find an equation for f(I). c) Verify that with a constant shift in the z-coordinate, (11) where zo is a constant you need to determine in terms of other quantities, the TISE equation from the previous part can be cast in the form of a simple one-dimensional Schrodinger equation with a familiar potential. Indicate what kind of potential it is, and what ro needs to be for this to work. d) What are the resulting energy eigenvalues of the full system, and what are their degen- eracies? These states are called Landau levels. 3 e) The degeneracy found above can be reduced by confining the particle to a finite region on the z-y plane. Say it is confined to a square defined by 0