Consider a particle of mass m which moves under the influence of gravity; the particle's Hamiltonian is H^=P^z2/(2m)−mgZ^, where g is the acceleration due to gravity, g=9.8 m s−2.
(a) Calculate d⟨Z^⟩/dt, d⟨P^z⟩/dt, d⟨H^⟩/dt.
(b) Solve the equation d⟨Z^⟩/dt and obtain ⟨Z^⟩(t), such that ⟨Z^⟩(0)=h and ⟨P^z⟩(0)=0. Compare the result with the classical relation z(t)=−21gt2+h.