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p. 48
IRREVERSIBLE RELAXATION
†
We want to study the relaxation
of an initially prepared state.
We will show that firstorder
perturbation theory for transfer to a continuum leads to irreversible transfer—an exponential
decay—when you include the depletion of the initial state.
The Golden Rule gives the probability of transfer to a continuum (for a constant perturbation):
+,
2
k
kk
k
0
k
P2
wV
E
E
t
Pw
t
t
P1
P
wS
U
w
0
0
A
AA
A
AA
A
=
P
t
0
1
t
0
k
P
A
P
The probability of being observed in
k
varies linearly in time.
This will clearly only work for
short times, which is no surprise since we said for firstorder P.T.
b
k
t

b
k
0
.
So
!!
w
k
A
represents the tangent to the relaxation behavior at
t
0.
!!
w
k
A
w
P
k
A
w
t
t
0
The problem is we don’t account for depletion of initial state.
What longtime behavior do we expect?
From an exact solution to the twolevel problem, we saw that probability oscillates sinusoidally
between the two states with a frequency given by the coupling:
†
CohenTannoudji, et al.
p. 1344; Merzbacher, p. 510.
MIT Department of Chemistry 5.74,
Spring 2004: Introductory Quantum Mechanics II.
Instructor: Prof. Andrei Tokmakoff
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View Full Documentp. 49
R
/
S:
P
t
1st order P.T.
k
P
A
P
AA
0
1
0
!!
:
R
'
2
.
V
k
A
2
=
But we don’t have a twostate system.
Rather, we are relaxing to a continuum.
Fermi’s Golden Rule says we have a timeindependent rate,
!!
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 Spring '04
 RobertField
 Chemistry

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