Introduction to Quantum Mechanics, Spring 2015
Problem Set 9
Due Thursday, February 5
Problem 1: Consider ther group SU (1, 1), defined as the group of two by two
complex matrices g of determinant one, satisfying
0
1 0
† 1
g
g=
0 −1
0 −1
• Show that
α
SU (1, 1) = {
β
β
α
1
T =√
2
: |α|2 − |β|2 = 1}
• Show that, if
1
i
i
1
then
g ∈ SU (1, 1) ↔ T gT −1 ∈ SL(2, R)
and that this gives an isomorphism of the groups SL(2, R) and SU (1, 1).
• Under this isomorphism, what element of SU (1, 1) corresponds to the
standard complex structure J0 in SL(2, R)?
• Which subgroups of SL(2, R) and SU (1, 1) commute with the standard
complex structure? In other words, give explicitly the group of matrices
g such that gJ0 = J0 g in both cases.
Problem 2: Show that the linear transformation of annihilation and creation
operators
a → a0 = αa + βa†
a† → (a0 )† = βa + αa†
preserves the commutation relations iff
|α|2 − |β|2 = 1
Such transformations are known as “Bogoliubov transformations”.
For which Bogoliubov transformations does a|0i = 0 imply a0 |0i = 0?
Problem 3: For the coherent state |αi, compute
hα|Q|αi
1
and
hα|P |αi
Show that coherent states are not eigenstates of the number operator N = a† a
and compute
hα|N |αi
Similarly, for squeezed states, compute
τ h0|N |0iτ
as a function of τ .
Problem 4: Find the coherent state |αi in the Schrödinger representation. In
other words, find a wavefunction ψ(q) that satisfies
1
√ (Q + iP )ψ(q) = αψ(q)
2
2