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5.74 Introductory Quantum Mechanics II
Spring 2009
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Absorption Lineshape for the Displaced Harmonic Oscillator Model
Investigate the low temperature lineshape for electronic transition coupled to a single harmonic mode.
z := 0 ,
1 .. 2
We will perform calculations for three couplings: weak medium, and strong:
hbar := 1
Lets work in units of
μeg := 1
ωeg := 10
Define the frequency of the electronic transition:
ω0 := 1
and the vibrational frequency:
i := 0 .. 200
Set up displacement grid
m := 1
q := −5 + 0.05⋅
i
i
Set up time grid
t := i⋅
0.1
Set up frequency grid:
ω := −4 + ωeg + 0.05⋅
i
i
i
D :=
z
The unitless displacement of the two harmonic wells is:
0.2
1
2
Ground and excited state potential energy surfaces
1
Vg( qx) :=
2
2
2 ⋅ D⋅hbar ⎞
⎛
Ve( q , D) := ωeg + m⋅ω0 ⋅ ⎜ q −
⎟
2
m⋅ω0 ⎠
⎝
1
2
⋅m⋅ ω0 ⋅qx
2
2
20
V g( q)
V e( q , D 0)
Lineshape function, Dephasing function and dipole
correlation function
⎛
g ( t , D) := −D⋅ ⎝ e
F( t , D) := e
C( t , D) :=
− i⋅ω 0⋅t
15
V e( q , D 1) 10
⎞
V e( q , D 2)
− 1⎠
− g( t , D )
(
μeg
)
5
0
−4
2 − i⋅ω eg⋅t−g( t , D)
⋅e
−2
0
2
4
q
Plot the correlation function and dephasing function for low and high coupling
D = 0.2
D =2
0
2
1
1
0.5
0.5
0
0
− 0.5
− 0.5
Re( C( t , D 0))
Re( F ( t , D0) )
−1
0
10
20
ω eg⋅t
2⋅π
30
−1
0
10
20
30
Absorption lineshape:
δ( g ) := if ( g = 0 , 1 , 0 )
10
σ( ω , D) :=
(
μeg )2⋅e− D⋅ ∑
J=0
⎡ DJ
⎤
⎢ ⋅ ( δ( ω − ωeg − J⋅ω0))⎥
⎣ J!
⎦
Envelope of vibronic progression:
π
Env(ω, D) :=
2
⋅0.25⋅ ( μeg
)
⎡ −( ω − ωeg − D⋅ω0) 2⎤
⎥
⋅exp⎢
⎢
2
⎥
2 ⋅ D⋅ ω0
⎣
⎦
2
D⋅ω0
Plot lineshapes for low, mid and high coupling.
D = 0.2
( a)
1
0
0.8
σ ( ω i , D 0)
0.6
Env( ω i , D 0) 0.4
0.2
0
−4
−2
0
2
4
6
2
4
6
2
4
6
ω i−ω eg
( b)
D = 1
0.5
1
0.4
σ ( ω i , D 1)
0.3
Env( ω i , D 1) 0.2
0.1
0
−4
−2
0
ω i−ω eg
( c)
D =2
0.4
2
σ ( ω i , D 2)
Env( ω i , D 2)
0.3
0.2
0.1
0
−4
−2
0
ω i−ω eg
⎛ 0.2 ⎞
D= ⎜ 1 ⎟
⎜ ⎟
⎝ 2 ⎠
ω0 = 1
2
σ ( ω i , D 2)
1.5
σ( ω i , D1) + 0.4 1
σ( ω i , D0) + 0.8
0.5
0
−4
−2
0
2
ω i−ω eg
4
6