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08_751_hw09 - Physics751Homework#9 1 z =e 2 z/2 z2 z3 0 z 1...

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Physics 751 Homework #9 1. 2 2 3 / 2 0 1 2 3 2! 3! z z z z e z = + + + + . Exercise : Check that this state is correctly normalized, and is an eigenstate of . ˆ a 2. Prove using an algebraic identity that ( ) * ˆ ˆ 0 za z a e is an eigenstate of . Is it also an eigenstate of ? Prove your assertion. ˆ a ˆ a 2. Prove that if 2 2 3 / 2 0 1 2 3 2! 3! z z z z e z = + + + + , the unit operator dxdy I z z π = ∫∫ 3. Prove that [ ] 1 2 , A B A B A B e e e e + = is correct up to terms A 3 and B 3 by expanding the exponentials on both sides and comparing. 4. How does a (position) translation operator affect a wave function expressed in momentum space, ( ) p ψ ? What is the operator that shifts the momentum space wave function ( ) p ψ to ( ) 0 p p ψ ? How does that operator change ( ) x ψ ? 5. Prove: ( ) 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ , , , 2! xA xA x f x e Be B x A B A A B = = + + + by writing the Taylor series for ( ) f x and finding the successive derivatives at the origin.
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