Find the probability distributions of the orbital angular momentum
variables L2 and Lz for the following orbital state functions:
(a) Ψ(x) = f(r) sinθ cos φ,
(b) Ψ(x) = f(r)(cos θ)^2,
(c) Ψ(x) = f(r) sinθ cos θ sin φ.
Here r, θ, φ are the usual spherical coordinates, and f(r) is an arbitrary
radial function (not necessarily the same in each case) into which the
normalization constant has been absorbed.