Quiz1 - PHY4221 Quantum Mechanics I (Fall 2004) Quiz 1...

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PHY4221 Quantum Mechanics I (Fall 2004) Quiz 1 (Answer) Step 1: Pre-proof: [ ] [ ] [ ] ( 29 [ ] [ ] [ ] [ ] [ ] 1 1 1 1 2 2 1 0 2 1 3 2 1 1 1 0 , , , , , , , ...... , , , ...... , , n n n n n n n n n n n n n n m n m m A B A A B A A B A B A A A B A A B A B A A A A B A A A B A A A B A A A B A A A B A - - - - - - - - - - - - - - = = = + = + + = = + + + + = ; Also we could make the Taylor expansion as following: ( 29 0 exp ! n n n x A xA n = = , then you see that exp(xA) is still an operator which is not commutative with B. Step 2: Proof: Consider a function ( 29 xA xB f x e e = and it is easy to obtain its first order derivative: ( 29 ( 29 , xA xB xA xB xA xB xA xB xA xB xA xB xA xB xA xB xA xB xA xB df x d d e e e e dx dx dx Ae e e Be Ae e Be e Be e e Be A B e e e B e = + ÷ ÷ = + = + - + = + + ; Now consider the term , xA e B :
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[ ] [ ] [ ] { } [ ] ( 29 [ ] 0 0 0 0 1 1 0 0 1
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Quiz1 - PHY4221 Quantum Mechanics I (Fall 2004) Quiz 1...

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