⍺ is a constant, L⃗ is the angular momentum operator. Hamiltonian operator of a system is given by:
H=⍺(L_x+L_y).
a) The expected value of the angular momentum vector over Ψ(r, t), which satisfies the Schrödinger equation, ⟨L⃗⟩(t) = ∫d3r Ψ*(r, t) L⃗ Ψ(r, t), is a function of time. Find the following for this system:
d⟨L_z⟩(t)/dt
d^2⟨L_z⟩(t)/dt^2
b) Solve the equation d^2⟨L_z⟩(t)/dt^2 you found in terms of functions that are clearly dependent on time. Express the coefficients in the solution in terms of initial values of ⟨L⃗⟩(0).