1. Particle in a cylindrically symmetric potential Let p, p, z be the cylindrical coordinates of a spinless particle (x = pcos, y = psin ; p≥ 0, 0 ≤<2π). Assume that the potential energy of this particle depends only on p, and not on y and z. Recall that: a² a² + მე2 მყო 82 = მი2 + - 1 ə 1 02 + a. Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with L, and P. Show that this allows writing the wave functions associated with the stationary states of the particle as: On,m,k(P,4,2)=fn,m(p) eim eikz where the values that can be taken on by the indices m and k are to be specified. b. Write, in cylindrical coordinates, the eigenvalue equation of the Hamiltonian H of the particle. Derive from it the differential equation that fn,m(p) obeys. c. Let Σy be the operator whose action, in the {\r)} representation, is to change y to -y (reflection with respect to the Oz plane). Does Σ, commute with H? Show that Σy anticommutes with L₂, and show that, as a result, Σyn,m,k) is an eigenvector of L. What is the corresponding eigenvalue? What can be concluded concerning the degeneracy of the energy levels of the particle? Could this result be predicted directly from the differential equation established in (b)?