08_751_hw09

08_751_hw09 - Physics751Homework#9 1 z =e 2 z/2 z2 z3 0 z 1 2 3 2 3 Exercise Check that this state is correctly normalized and is an eigenstate of

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Physics 751 Homework #9 1. 2 23 /2 01 2 3 2! 3! z zz ze z ⎛⎞ =+ + + + ⎜⎟ ⎝⎠ . Exercise : Check that this state is correctly normalized, and is an eigenstate of . ˆ a 2. Prove using an algebraic identity that ( ) †* ˆˆ 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 3! z z + + + , the unit operator dxdy I π = ∫∫ 3. Prove that [ ] 1 2 , AB A B ee 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 ,, , xA xA x fx eB e B xAB AAB ⎡⎤ == + + ⎢⎥ + ⎦⎣ by writing the Taylor series for ( ) f x and finding the successive derivatives at the origin. A unitary squeeze operator is defined by:
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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08_751_hw09 - Physics751Homework#9 1 z =e 2 z/2 z2 z3 0 z 1 2 3 2 3 Exercise Check that this state is correctly normalized and is an eigenstate of

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