HW1Add - There is another method to prove the Question 1 Let the L.H.S of the relation to be a function of as f = exp A B exp A(1 then we could

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There is another method to prove the Question 1: Let the L.H.S. of the relation to be a function of ξ as () ( ) ( ) exp exp f AB A ξ ξξ =− , (1) then we could calculate that () ( ) ( ) ( ) ( ) ( ) [] () ( ) 1 exp exp exp exp exp exp exp exp exp exp exp , exp exp exp d fA B A d A A ABA A A BA A A AB A A A AC A ⎡⎤ ⎣⎦ =− ⋅ + ⋅ ⋅ −− ≡− , (2) where we have defined C 1 = [B, A] and used the fact that A commutes with exp (-ξA) or exp (ξA). Similarly, we have ( ) ( ) ( ) ( ) 1 11 1 2 exp exp exp exp exp exp exp , exp xp , , exp exp exp d C A d A CA A A AC A ACA A A A A A ′′ =−⋅ , (3) in which C 2 = [C 1 , A] = [[B, A], A]. Go on as this, we will easily have ( ) ( ) 1 exp exp exp exp n n n d C A d A (4) where C n = [C n-1 , A] = […[[B, A], A], …A] (there are n A`s).
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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