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
*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, _{}? What is the
operator that shifts the momentum space wave function _{} to _{}? How does *that* operator change _{}?

5. Prove:

_{}

by writing the _{} 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?