Could you please provide a detailed solution for the following question? Likes and comments are rewarded.
A1. Angular momentum in quantum mechanics is given by L = Li + Luj + Lk with components Lx = yp - 2py, Ly = zpx - xpz, and L = xpy - yPx.
a) Show that [Ly, L] = ihL and [L^2, L] = 0, where L^2 = L + L + L. Discuss your results in terms of the concept of simultaneous observables. [Hint: You may use the known commutation rules for y, P, and z.]
b) Explain what is meant by the joint eigenstates l, m of the operators L and Lz, including the definitions of the azimuthal and magnetic angular-momentum quantum numbers l and m.
c) For a system in a joint eigenstate l, m, write down the values of L, Lz, and L^2.
d) (i) For a system in a joint eigenstate l, m, use the commutation relations for the angular-momentum components to show [L^2, L] = 0 and [Lz, L] = ihL. (ii) Hence, determine L.
e) Consider the spherical harmonic Y^(-1) = sin(theta)exp(-i*phi), where theta and phi are the polar and azimuthal angles, respectively. (i) Express Y^(-1) in terms of y and r. (ii) Show that [L, f(r)] = 0 for functions that depend only on the radial coordinate r. (iii) Show that Y^(-1) is an eigenfunction of L.
