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

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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
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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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