(a) In (one-dimensional) quantum mechanics, the differentiation operator P[f(x)] = f′(x) represents the momentum of a particle, while the operator Q[f(x)] = x f(x) of multiplication by the function x represents its position. Prove that the position and momentum operators satisfy the Heisenberg Commutation Relations [P, Q] = P ○ Q - Q ○ P = I.
(b) Prove that there are no matrices P, Q that satisfy the Heisenberg Commutation Relations.