# 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?