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Physics 751 Homework #9
1.
2
23
/2
01
2
3
2!
3!
z
zz
ze
z
−
⎛⎞
=+
+
+
+
⎜⎟
⎝⎠
…
.
Exercise
: Check that this state is correctly normalized, and is an eigenstate of
.
ˆ
a
2.
Prove using an algebraic identity that
(
)
†*
ˆˆ
0
za
z a
e
−
is an eigenstate of
.
Is it also an eigenstate
of
?
Prove your assertion.
ˆ
a
†
ˆ
a
2.
Prove that if
2
2
3
3!
z
z
−
+
+
+
…
,
the unit operator
dxdy
I
π
=
∫∫
3.
Prove that
[
]
1
2
,
AB
A B
ee
e
e
−
+
=
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,
()
p
ψ
?
What is the operator that shifts the momentum space wave function
p
ψ
to
( )
0
p
p
ψ
−
?
How does
that
operator change
( )
x
ψ
?
5.
Prove:
2
ˆ
ˆ
ˆ
,,
,
xA
xA
x
fx eB
e
B xAB
AAB
−
⎡⎤
==
+
+
⎢⎥
+
⎣
⎦⎣
⎦
⎣
⎦
…
by writing the Taylor series for
( )
f
x
and finding the successive derivatives at the origin.
A unitary squeeze operator is defined by:
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.
 Fall '07
 MichaelFowler
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