# Physics 751 Homework #9

1. Exercise: Check that this state is correctly normalized, and is an eigenstate of .

2.  Prove using an algebraic identity that is an eigenstate of .  Is it also an eigenstate of ?  Prove your assertion.

2.  Prove that if ,

the unit operator 3.  Prove that is correct up to terms A3 and B3 by expanding the exponentials on both sides and comparing.

4.  How does a (position) translation operator affect a wave function expressed in momentum space, ?   What is the operator that shifts the momentum space wave function to ?  How does that operator change ?

5.  Prove: by writing the Taylor series for and finding the successive derivatives at the origin.

A unitary squeeze operator is defined by: Use the result for above to prove that: Deduce that so for positive θ, the wave function is scaled down—squeezed—in x space, but simultaneously expanded in p space, as it must be, since it was a minimum uncertainty packet.

Is it still a minimum uncertainty packet?   Is it still an eigenstate of the annihilation operator?   If not, what is it an eigenstate of?  How do you think it develops in time?