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3. Commutator algebra. Prove the following commutator identities. A, B, C are arbitrary operators, a and ß are arbitrary
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3. Commutator algebra. Prove the following commutator identities. A, B, C are arbitrary operators, a and ß are arbitrary complex numbers, and n is an integer. i. [A, B] = [B,A]. ii. [aA, BB] = aß [A, B]. iii. [A, B+C] = [A, B] + [A, C]. iv. [A, A"] = 0. v. [A, f(A)] = 0. (Hint: expand f(A) as f(A) = Σa,A".) vi. [AB,C] = A[B, C] + [A, C]B. (Hint: add and subtract ACB.) vii. [A, BC] = [A, B]C + B[A,C].