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5.74 Introductory Quantum Mechanics II
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View Full Document Andrei Tokmakoff, MIT Department of Chemistry, 5/10/2007
p. 1144
11.5. CHARACTERIZING FLUCTUATIONS
Eigenstate vs. system/bath perspectives
From our earlier work on electronic spectroscopy, we found that there are two equivalent ways
of describing spectroscopic problems, which can be classified as the eigenstate and system/bath
perspectives.
Let’s summarize these before turning back to nonlinear spectroscopy, using
electronic spectroscopy as the example:
1) Eigenstate: The interaction of light and matter is treated with the interaction picture
Hamiltonian
HH
=
0
+
V
t
()
.
H
0
is the full material Hamiltonian, expressed as a
function of nuclear and electronic coordinates, and is characterized by eigenstates which
are the solution to
0
Hn
=
E n
. In the electronic case
n
=
e
n
1
2
,,
K
represent
n
labels for a particular vibronic state. The dipole operator in
Vt
( )
couples these states.
Given that we have such detailed knowledge of the matter, we can obtain an absorption
spectrum in two ways.
In the time domain, we know
2
e
−
i
ω
mn
t
(1)
Ct
=
∑
p
n
n
μ
t
( )
0
n
=
∑
p
n
μμ
nm
n
,
n
m
The absorption lineshape is then related to the Fourier transform of
,
2
1
σω
=
∑
p
n
(2)
nm
ωω
−Γ
i
−
nm
,
nm
nm
where the phenomenological damping constant
Γ
nm
was first added into eq. (1).
This
approach works well if you have an intimate knowledge of the Hamiltonian if your
spectrum is highly structured and if irreversible relaxation processes are of minor
importance.
2) System/Bath: In condensed phases, irreversible dynamics and featureless lineshapes
suggest a different approach. In the system/bath or energy gap representation, we separate
our Hamiltonian into two parts: the system
H
s
contains a few degrees of freedom
Q
which we treat in detail, and the remaining degrees of freedom (
q
) are in the bath
H
B
.
Ideally, the interaction between the two sets
H
SB
(
q
Q
)
is weak.
H
0
=
H
S
+
H
B
+
H
S
B
.
(3)
Andrei Tokmakoff, MIT Department of Chemistry, 5/10/2007
p. 1145
Spectroscopically we usually think of the dipole operator as acting on the system state,
i.e. the dipole operator is a function of
Q
. If we then know the eigenstates of
H
S
,
Hn
=
E n
where
n
=
g
or
e
for the electronic case, the dipole correlation
S
n
function is
2
−
i
ω
t
eg
Ct
() =
μ
e
exp
⎡
⎢
⎣
−
i
∫
0
t
H
SB
()
t
′
dt
′
⎤
⎥
⎦
(4)
μμ
eg
The influence of the dark states in
H
B
is to modulate or change the spectroscopic energy
gap
eg
in a form dictated by the timedependent systembath interaction. The system
bath approach is a natural way of treating condensed phase problems where you can’t
treat all of the nuclear motions (liquid/lattice) explicitly. Also, you can imagine hybrid
approaches if there are several system states that you wish to investigate
spectroscopically.
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
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 Chemistry

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