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Physics 512
Winter 2003
Homework Set #9 – Solutions
Note:
The homework set posted on the web had a second page (part
c
) of problem #3
and problem #4). I mistakenly omitted the second page on the set I handed out in class,
so only the Frst three problems are required. What should have been problem #4 is now
part of Homework #10.
1. We consider Frst order time dependent perturbation theory for a Hamiltonian
H
(
t
)=
H
0
+
V
(
t
). Suppose that a system is initially in an eigenstate

s
i
of
H
0
at time
t
0
. Let
O
be some observable of the system. Show that the expectation value of
O
at time
t
is given to Frst order in
V
(
t
)by
h
ψ
(
t
)
O
ψ
(
t
)
i
=
h
s
O
s
i−
i
¯
h
Z
t
t
0
h
s

[
e
O
(
t
)
,
e
V
(
t
0
)]

s
i
dt
0
where
e
O
(
t
) and
e
V
(
t
) are in the interaction picture.
Since we are interested in a system at a later time
t
, we transform to the interaction
picture and use the time evolution operator. Going from the Schr¨
odinger picture to
the interaction picture is simple (we just put tilde’s over everything)
h
ψ
(
t
)
O
ψ
(
t
)
i
=
h
e
ψ
(
t
)

e
O
(
t
)

e
ψ
(
t
)
i
To Frst order, the time evolution operator in the interaction picture gives

e
ψ
(
t
)
i
=
e
T
(
t, t
0
)

e
s
i
=
±
1
−
i
¯
h
Z
t
t
0
e
V
(
t
0
)
dt
0
²

e
s
i
where

e
s
i
is the initial state, converted to the interaction picture. Since we assume
that
V
is Hermitian (so that
˜
V
is also Hermitian), the bra state evolves similarly
h
e
ψ
(
t
)

=
h
e
s

±
1+
i
¯
h
Z
t
t
0
e
V
(
t
0
)
dt
0
²
We may combine these expressions to obtain
h
ψ
(
t
)
O
ψ
(
t
)
i
=
h
e
s

±
i
¯
h
Z
t
t
0
e
V
(
t
0
)
dt
0
²
e
O
(
t
)
±
1
−
i
¯
h
Z
t
t
0
e
V
(
t
0
)
dt
0
²

e
s
i
=
h
e
s

e
O
(
t
)

e
s
i
+
i
¯
h
Z
t
t
0
³
h
e
s

e
V
(
t
0
)
e
O
(
t
)

e
s
i−h
e
s

e
O
(
t
)
e
V
(
t
0
)

e
s
i
´
dt
0
=
h
s
O
s
i
¯
h
Z
t
t
0
h
e
s

[
e
O
(
t
)
e
V
(
t
0
)]

e
s
i
dt
0
1
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View Full Document(where we have ignored the higher order term where
e
V
appears twice). Note that in the
frst term we have converted back to the Schr¨odinger picture by removing the tilde’s.
Finally, since

s
i
is an eigenstate o±
H
0
, it is easily converted into the interaction
picture (at time
t
0
) according to

e
s
i
=
e
iH
0
t
0
/
¯
h

s
i
=
e
iE
s
t
0
/
¯
h

s
i
Since this is just a phase relating

s
i
to
e
s
i
, it cancels ±rom the braket
h
e
s
···
e
s
i
. This
proves the result
h
ψ
(
t
)
O
ψ
(
t
)
i
=
h
s
O
s
i−
i
¯
h
Z
t
t
0
h
s

[
e
O
(
t
)
,
e
V
(
t
0
)]

s
i
dt
0
2. Apply frstorder time dependent perturbation theory to a Forced harmonic oscillator
H
=(
a
†
a
+
1
2
)¯
hω
+
f
(
t
)
a
+
f
∗
(
t
)
a
†
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 Winter '03
 Unknow
 Physics, Work

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