PHY4221Fall05_Prob1

PHY4221Fall05_Prob1 - PHY4221 Quantum Mechanics I Fall term...

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PHY4221 Quantum Mechanics I Fall term of 2005 Problem Set No.1 (Due on September 27, 2005) Prove the following theorems: 1. Theorem 1 : If A and B are two fixed noncommuting operators and ξ is a parameter, then exp( - ξA ) B exp( ξA ) = B + ξ 1! [ B,A ] + ξ 2 2! [[ B,A ] ,A ] + ξ 3 3! [[[ B,A ] ,A ] ,A ] + ······ . [ Hint : Expand exp ( - ξA ) B exp ( ξA ) in a power series of ξ .] 2. Theorem 2 : If A and B are two noncommuting operators and ξ is a parameter, then exp( - ξA ) B n exp( ξA ) = [exp( - ξA ) B exp( ξA )] n for some integer n , and exp( - ξA ) F ( B )exp ( ξA ) = F (exp( - ξA ) B exp( ξA )) where F ( B ) is an arbitrary function of
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