3. Proof of the Hadamard lemma (10 points] Prove that for two operators A and B, we have et Be= B+ [A, B] + - [A, [A, B] + - [A, [A, [A, B]] + .. . . (1) Define f(t) = et Be tA and calculate the first few derivatives of f(t) evaluated at t =0. Then use Taylor expansions. Calculating explicitly the first three derivatives suffices to obtain (1). Do things to all orders by finding the form of the (n + 1)-th term in the right-hand side of (1). To write the answer in a neat form we define the operator ad A that acts on operators A to give operators via the commutator ad A(X) = [A, X] . Confirm that with this notation, the complete version of equation (1) becomes ed Be A = padA ( B)