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 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?