Quantum Mechanics for Mathematicians

which in the Schr ̈odinger representation is the differential equation

(Q−τP)ψ(q) =

(

q+iτ

d

dq

)

ψ(q) = 0

which has solutions
ψ(q)∝ei

τ

2 |τ|^2 q

2

(26.20)

This will be a normalizable state for Im(τ)>0, again showing the necessity of
the positivity condition.
Eigenstates ofHτforτ =ic,c >0 real, are known as “squeezed states”
in physics. By equation 26.20 the lowest energy state| 0 〉will have spatial
dependence proportional to

e−

21 cq^2

and higher energy eigenstates|n〉will also have such a Gaussian factor in their
position dependence. Forc <1 such states will have narrower spatial width
than conventional quanta (thus the name “squeezed”), but wider width in mo-
mentum space. Forc >1 the opposite will be true. In some sense that we
won’t try to make precise, the limits asc→ ∞andc→0 correspond to the
Schr ̈odinger representations in position and momentum space respectively (with
the distinguished Bargmann-Fock state| 0 〉approaching the constant function
in position or momentum space).
The subgroupR⊂SL(2,R) of equation 24.7 acts non-trivially onJ 0 , by

J 0 =

(

0 1

−1 0

)

→

(

er 0

0 e−r

)(

0 1

−1 0

)(

e−r 0

0 er

)

=

(

0 e^2 r

−e−^2 r 0

)

takingJ 0 to the complex structure with parameterτ=ie^2 r.
Recall from section 24.4 that changing fromq,pcoordinates toz,zcoordi-
nates on the complexified phase space, the groupSL(2,R) becomes the isomor-
phic groupSU(1,1), the group of matrices

(

α β

β α

)

satisfying
|α|^2 −|β|^2 = 1

Looking at equation 24.11 that gives the conjugation relating the two groups,
we see that (
−i 0
0 i

)

∈SU(1,1)↔

(

0 1

−1 0

)

∈SL(2,R)

and as expected, in these coordinatesJ 0 acts onzby multiplication byi, onz
by multiplication by−i.
SU(1,1) matrices can be parametrized in terms oft,θ,θ′by taking

α=eiθ

′

cosht, β=eiθsinht