HW1 - B . 3. Theorem 3 : If the operators a and b satisfy...

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PHY4221 Quantum Mechanics I Fall term of 2004 Problem Set No.1 (Due on October 2, 2004) 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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Unformatted text preview: B . 3. Theorem 3 : If the operators a and b satisfy the commutation relation [ a,b ] = 1, then show that exp(-ξba ) a exp( ξba ) = a exp( ξ ) exp(-ξba ) b exp( ξba ) = b exp(-ξ ) exp ‰-1 2 ξ ‡ b 2-a 2 · ± a exp ‰ 1 2 ξ ‡ b 2-a 2 · ± = a cosh( ξ ) + b sinh( ξ ) exp ‰-1 2 ξ ‡ b 2-a 2 · ± b exp ‰ 1 2 ξ ‡ b 2-a 2 · ± = b cosh( ξ ) + a sinh( ξ ) . —— End ——...
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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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