= = = = 1. Operator identities. (15 points) For operators A, B, C, show that (a) (AB)* = Bt At; (b) [A, B]* = [BT, AT];

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1 Operator Identities 15 Points For Operators A B C Show That A Ab Bt At B A B Bt At 1 (36.04 KiB) Viewed 54 times

= = = = 1. Operator identities. (15 points) For operators A, B, C, show that (a) (AB)* = Bt At; (b) [A, B]* = [BT, AT]; (c) [AB,C] = A(B,C] + [A, C]B; (d) [A, Bk] = k Bk-1 [A, B], provided [[A, B], B] = 0; = (e) e4 eB = eBeA e[4,B] and eA+B = A B e-[A,B]/2 , if [[A, B], A] = [[A, B], B] Hint: consider the \-derivative of the functions e\ AB and e(A+B). (f) Prove the Baker-Hausdorff identity +. [A, [A.... [A, B]...]]+... k! В = = 0 = e^ Be-4 = B+[A, B] +3 [4, [A, B] = .+