Consider a particle-on-a-sphere with a spherical harmonic wave function ψt, mt(θ,φ). a. Suppose that l= 2 and ml= 2. Compute the following observables (numerically):

i. magnitude of the angular momentum vector of the particle |L|.
ii. projection of the angular momentum vector of the particle on the z-axis Lz.

b. Now suppose the particle is in the state withl= 1 and ml= 1. What are the values of the observables |L| and Lz(as multiples of~)? Sketch the angular momentum vector of the particle in thex-zplane, labeling its length |L| and z-component. What is the angle that the angular momentum vector makes with the zaxis?