Lecture 15
Time-dependent perturbation theory
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Time-dependent perturbation theory

The theory described here is essential for
understanding the optical transitions from
one state to another and thus
spectroscopies. It gives the formula for
intensities of spectral bands.

The probability of transition (intensity of a
spectral band for the transition n ← 0) is
wn¬0 µ ò Y
(0)*
n
2
ˆzY0(0) dx E 2
Zeroth order

Before the perturbation is turned on, the
system is in time-independent ground state.
(0)
(0)
(0)
(0)
ˆ
H 0  E0 0

Let us remember that a “time-independent”
wave function does have hidden time
dependence of the form:
Y
(0)
0
(0)
(0) -iE0 t/
0
®Y e
Perturbation = light

Next, we shine light on the system. Light (or a
photon) is a time-oscillating electromagnetic
field. It is very weak as compared to
electrostatic interactions between electrons
and nuclei in atoms and molecules. It can be
treated as a time-dependent perturbation:
(
(0)
(1)
(0)
iw t
-iw t
ˆ
ˆ
ˆ
ˆ
H = H + H (t) = H + zˆE e + e
)
Oscillating real electrostatic field in z direction
with intensity E and angular frequency ω = 2πν.
Perturbation = light
(
(0)
(1)
(0)
iw t
-iw t
ˆ
ˆ
ˆ
ˆ
H = H + H (t) = H + zˆE e + e
)
Oscillating real electrostatic field in z direction with
intensity E and angular frequency ω = 2πν.
First order

Once we turn on the time-dependent
perturbation, the wave function will have
time-dependent response. We consider the
first-order response. We expand the firstorder correction to the zeroth-order wave
function by eigenfunctions of the known
(initial, time-independent) system.
¥
Y (t) = å ck (t)Y e
(1)
0
k
(0)
(0) -iEk t/
k
Light absorption
First order

Substituting into the time-dependent
Schrödinger equation:
¶
ˆ
HY = i
YÛ
¶t
¥
(0)
(0)
æ
ö
(0)
(1)
(0) - iE0 t/
(0) - iEk t/
ˆ
ˆ
H + l H (t) ç Y 0 e
+ l å ck (t)Y k e
+… ÷
è
ø
k
(
)
¥
(0)
ö
¶ æ (0) - iE0( 0 )t/
(0) - iEk t/
=i
Y0 e
+ l å ck (t)Y k e
+… ÷
ç
¶t è
ø
k
First order

Collecting the terms in first order in λ
Hˆ (1) (t)Y (0)
e
0
-iE0( 0 )t/
¥
+ Hˆ (0) å ck (t)Y (0)
e
k
k
-iEk( 0 )t/
(0)
¶ ¥
(0) -iEk t/
=i
ck (t)Y k e
å
¶t k
First order

(0)
Y
Using that k are eigenfunctions of H(0)
Hˆ (1) (t)Y (0)
e
0
Hˆ (t)Y e
(1)
(0)
(0) -iE0 t/
0
-iE0( 0 )t/
¥
+ Hˆ (0) å ck (t)Y (0)
e
k
k
¥
+ å ck (t)E Y e
k
(0)
k
(0)
(0) -iEk t/
k
-iEk( 0 )t/
(0)
¶ ¥
(0) -iEk t/
=i
ck (t)Y k e
å
¶t k
¥
(0)
¶ck (t) (0) -iEk( 0 )t/
(0) (0) -iEk t/
=i å
Yk e
+ å ck (t)Ek Y k e
¶t
k
k
¥
Cancel
First order

Multiplying Y (0)*
from the left and integrating
n
Hˆ (1) (t)Y (0)
e
0
-iE0( 0 )t/
+ Hˆ (0) å ck (t)Y (0)
e
k
k
¥
+ å ck (t)E Y e
Hˆ (t)Y e
(1)
¥
(0)
(0) -iE0 t/
0
(0)
k
k
(0)
(0) -iEk t/
k
-iEk( 0 )t/
(0)
¶ ¥
(0) -iEk t/
=i
ck (t)Y k e
å
¶t k
¥
(0)
¶ck (t) (0) -iEk( 0 )t/
(0) (0) -iEk t/
=i å
Yk e
+ å ck (t)Ek Y k e
¶t
k
k
¥
Cancel
òY
¶c
(t)
-i é Ek( 0 ) -E0( 0 ) ùt/
k
û
Hˆ (t)Y dx = i å
e ë
¶t
k
¥
(0)*
n
(1)
(0)
0
(0)* (0)
Y
ò n Y k dx
(
)
1 only if n = k
Hˆ (1) (t) = zˆE eiw t + e -iw t
First order

Simplifying
òY
é E ( 0 ) -E ( 0 ) ùt/
¶c
(t)
-i
Hˆ (t)Y dx = i å k e ë k 0 û
¶t
k
¥
(0)*
n
(1)
(0)
0
(
)
Hˆ (1) (t) = zˆE eiw t + e -iw t
(
E e
iw t
+e
-iw t
(0)* (0)
Y
ò n Y k dx
1 only if n = k
)òY
(0)*
n
¶cn (t) -iéë En( 0 ) -E0(0 ) ùût/
zˆY dx = i
e
¶t
(0)
0
Equal
Energy
conservation!
E
(0)
n
-E
(0)
0
= w or - w
First order
E
(0)
n
-E
(0)
0
= w or - w
Excitation or deexcitation
First order

Removing the time-oscillating factors
Eò Y
(0)*
n
1
cn (t) =
i
¶cn (t)
zˆY dx = i
¶t
(0)
0
t
é E Y (0)* zˆY (0) dx ùdt
ò0 ë ò n 0 û
Fermi’s golden rule

The probability of transition (intensity of a
spectral band for the transition n ← 0) is
wn¬0 µ cn (t) µ ò Y
2

(0)*
n
2
zˆY dx E
(0)
0
2
When the dipole moment interacts with the
oscillating electric field (light), the transition
probability is proportional to the square of
the transition dipole moment (in the
direction of polarization) and square of the
intensity of light.
Fermi’s golden rule
wn¬0 µ ò Y
(0)*
n
2
zˆY dx E
(0)
0
2
Summary




Spectroscopic transitions can be explained
by the time-dependent Schrödinger equation.
One-photon transition can be understood by
first-order time-dependent perturbation
theory.
Energy conservation
Transition dipole moment
Intensity of light