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MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
p. 33
Instructor: Prof. Andrei Tokmakoff
PERTURBATION THEORY
Given a Hamiltonian
Ht
( )
( )
=
H
0
+
Vt
where we know the eigenkets for
H
0
n
n
=
E
n
H
0
we often want to calculate changes in the amplitudes of
n
induced by
( )
:
ψ
t
( )
=
∑
c
n
n
( )
t
where
n
()
=
k
=
( )
t
kU t
,
t
0
(
)
(
)
t
0
c
k
t
In the interaction picture, we defined
=
e
+
i
ω
k
r
c
k
t
( )
=
k
( )
b
k
t
I
which contains all the relevant dynamics.
The changes in amplitude can be calculated by solving
the coupled differential equations:
i
∂
b
k
=
−
i
∑
e
−ω
nk
t
V
k
n
b
n
t
t
∂
t
=
n
For a complex system or a system with many states to be considered, solving these equations isn’t
practical.
Alternatively, we can choose to work directly with
U
I
t
,
t
0
as:
(
)
, and we can calculate
b
k
t
b
k
=
kU
I
t
,
t
0
(
)
t
0
(
)
where
Ut
,
t
)
=
e
x
p
−
i
∫
t
V
d
I
(
0
+
=
I
ττ
t
0
Now we can truncate the expansion after a few terms.
This is perturbation theory, where the
dynamics under
H
0
are treated exactly, but the influence of
)
on
b
n
is truncated.
This works
(
well for small changes in amplitude of the quantum states with small coupling matrix elements
relative to the energy splittings involved.
≈
; V
±
E
k
−
E
b
k
(
t
)
b
k
(
0
)
n
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View Full Documentp. 34
Transition Probability
Let’s take the specific case where we have a system prepared in
probability of observing the system in
A
, and we want to know the
k
at time
t
, due to
Vt
)
.
(
2
P
k
() =
t
bt
k
U
I
(
t
,
t
0
)
A
k
()
k
t
i
bt k
τ
τ
e
x
p
+
−
∫
d
V
I
A
k
t
0
=
i
t
=
k
A
d
τ
k
τ
A
V
I
−
=
∫
t
0
−
i
2
t
τ
2
+
∫
t
0
d
τ
2
∫
t
0
d
τ
1
k
A
+
…
V
I
V
(τ
1
)
τ
2
I
=
using
−
i
ω
A
k
t
V
k
A
t
kV
I
t
A
=
kU
0
†
U
0
A
=
e
−
i
A
k
τ
b
k
t
“first order”
=
δ
k
A
−
i
∫
t
t
0
d
1
e
1
V
k
A
1
=
t
+
∑
−
i
2
∫
t
0
d
2
∫
t
0
V
km
2
−
i
A
m
1
+
…
“second order”
2
d
1
e
−
i
mk
2
e
V
m
A
1
=
m
This expression is usually truncated at the appropriate order.
Including only the first integral is
firstorder perturbation theory.
If
is not an eigenstate, we only need to express it as a superposition of eigenstates, but
ψ
0
remember to convert to
c
k
t
=
e
−
k
t
b
k
(
t
)
.
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 Spring '04
 RobertField
 Chemistry

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