Cylindrical coordinates of a spinless particle

Question

Discuss the below:

Q: Let p, φ, z be the cylindrical coordinates of a spinless particle (x = p cos φ, y = p sinφ; p ≥ 0, 0 ≤φ < 2∏). Assume that the potential energy of this particle depends only on p, and not on φ and z. Recall that:

∂²/∂x² +∂²/∂y²=∂²/∂ρ²+1/ρ.∂/∂ρ +∂²/∂φ²ρ

a. Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with Lz and P. Show from this that the wave functions associated with the stationary states of the particle can be chosen in the form:

φ_n,m,k(ρ,φ,z)=f_n,m(ρ)e^_imφ e^ikz

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 which yields f_n,m(ρ)