MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Robert Field
5.74 RWF Lecture #7
7–1
Motion of Center of Wavepacket
Last time: Discussed experiments to monitor the central quantity
⟨Ψ(t)|Ψ(0)⟩
in Heller’s picture: the absorption spectrum, I(ω), is the FT of the autocorrelation function. Ψ(0) in
the autocorrelation function is the ground state, Vg′′( R) , vibrational wavefunction g, v g′′ transferred
vertically onto the excited state potential, Ve′( R) .
Is there a way to monitor ⟨Ψ(t)|Ψ(0)⟩ directly in the time domain? Many suggested schemes.
We need tools to examine various excitation/detection experimental schemes.
Excitation at t = 0, E(0)
Evolution, U(t,0) = e–iHt/
Detection, D
h
ρ(0)E†(0)U†(t,0)
ρ(t) = U(t,0)E(0)ρ
Observation: Trace (D ρ(t))
Consider the simplest 3 level system first
2
1
0
5.74 RWF Lecture #7
7–2
Short excitation pulse
Ψ( t) = β 0 e − iE 0 t / h + α 1 1 e − iE1t / h + α 2 2 e − iE 2 t / h
both eigenstates |1⟩ and |2⟩ are bright
β = [1 – |α1|2 + |α2|2]1/2
αi = ciµ i0
ci describes the intensity, spectral distribution,
phase, and duration of the excitation pulse.
µ i0 is the electric dipole transition moment
2

β

ρ( t) = Ψ Ψ =  β* α 1e +iω 01t
β* α e +iω 02 t

2
βα 1* e − iω 01t
α1
2
α 2α 1* e +iω12 t
βα *2e − iω 02 t 

α *2α 1e − iω12 t 
2

α2

This ρ(t) is obtained by two transformations of ρ(0)
ρ(t) = U(t,0)E(0)ρ
ρ(0)E†(0)U†(t,0)
E(0) is the excitation matrix, operating at t = 0, on ρ(0).
β

E(0) =  α 1

α2
α *1
0
0
α *2 

0

0
5.74 RWF Lecture #7
7–3
β

E(0)ρ(0)E† (0) =  α 1

α2
α 1*
β

=  α1

α2
α *1
 β2

=  α 1β*
 α β*
 2
0
0
0
0
α *2   1 0 0  β*


0   0 0 0  α 1


0   0 0 0  α 2
α *2  β*

0  0

0  0
βα 1*
α1
2
α 2α 1*
α 1*
0
0
α *2 

0

0
α *2 

0

0
α 1*
0
0
βα *2 

α 1α *2 
2
α 2 
U(t,0) is the time evolution matrix. If ρ(0) is expressed in the eigen-basis
 e − iE 0 t / h

U( t, 0) = e − iHt / h =  0

 0
0
0
e − iE1t / h
0
2

β

ρ( t) = U( t, 0)E(0)ρ(0)E† (0)U† ( t, 0) =  β* α 1e +iω 01t
β* α e +iω 02 t

2
as required from |Ψ(t)⟩⟨Ψ(t)|



− iE 2 t / h 
e

0
βα 1* e − iω 01t
α1
2
α 2α 1* e +iω12 t
βα *2e − iω 02 t 

α *2α 1e − iω12 t 
2

α2

5.74 RWF Lecture #7
7–4
If the bright state is not an eigenstate, it is often convenient to set up ρ(0), E(0), and H in the zero-order
basis set. Then find the transformation that diagonalizes H and apply it to ρ(( 00 )) .
 E0
T HT =  0

 0
†
0
E1
0
0
0

E 2
T† ρ((00))T = ρ(0)
Now, we have a choice of several detection schemes.
Detection could involve:
(i)
modification of a beam of probe radiation;
(ii)
detection of emitted radiation through a filter or monochromator.
Let us consider the latter possibility.
Now there are several more possibilities:
(a)
the detector is blind to radiation at ω10 (and ω12);
(b)
the detector is sensitive to radiation at both ω10 and ω20 (but not ω12), and both ω10 and ω20
radiation are detected with the same phase;
(c)
same as (b) but ω10 is detected with phase opposite that at ω20.
This could be based on a polarization trick. The 1←0 transition is ∆M = 0 (z-polarized) and
2←0 is ∆M = ±1 (x or y-polarized). Detection with polarizer at +π/4 and –π/4 would
correspond to cases (b) and (c).
Detection: I(t) = Trace (D ρ(t))
5.74 RWF Lecture #7
7–5
For detection of radiation in transition back to |0⟩
D = ∑ i µ i 0µ 0 j j
i, j
Trace(Dρ( t)) = D22ρ22 + D11ρ11 + D12ρ21 + D21ρ12
(a)
If we set µ10 = 0, µ20 ≠ 0
(blind to ω10)
I(t) = D22ρ22 = |µ20|2 |α2|2
If we set µ20 = 0, µ10 ≠ 0
(blind to ω20)
I(t) = |µ 10|2 |α1|2
(b)
If we set µ10 = µ20 = µ, α 1 = α2 = α
I ( t) = µ 20 α 2 + µ 10 α 1 + µ 10µ 02α *1 α 2eiω12 t + µ 20µ 01α 1α *2e − iω12 t
2
2
2
2
2
2
2
2
= 2 µ α + 2 µ α cos ω12 t
= 2 µ α [1 + cos ω12 t]
2
2
“phased up” at t = 0
Quantum Beats. 100% amplitude, modulation.
(c)
if we set µ 10 = –µ20 = µ, α 1 = α2 = α
I(t) = 2|µ| 2 |α|2[1–cos ω12t] “phased out” at t = 0
If, instead of both eigenstates being bright, we excite a system with one bright state and one dark state, at
t = 0 we form Ψ(0) = ψbright = cos θψ1 + sin θψ2
eigenstates
Then E and D could be expressed in terms of bright states rather than eigenstates. In that case, α1 and α2
include cos θ and sin θ factors, and the phase of the Quantum Beats could depend on θ through α1, α2.
5.74 RWF Lecture #7
7–6
Zewail experiment
e+
e1
e0
 ee1
He = 
 H e1e 0
H e1e 0 

E e0 
e–
µ 00
µ 11
g1
g0
µ00 and µ11 ≠ 0 in zero-order basis
µ10 = µ01 = 0 in zero-order basis
because of He1e0 ≠ 0, both e+ and e– are bright from both g0 and g1.
But
e1 is bright from g1 and dark wrt g0
e0 is bright from g0 and dark wrt g1
as a result, detecting at ωe+,g1 is phased up at t = 0 but at ωe+,g0 is phased out at t = 0. Vice versa for ωe–,g1 and
ωe–,g0.
5.74 RWF Lecture #7
7–7
Figure removed due to copyright reasons.
We can use this ρ, E, U, D formalism to describe much more complicated experiments.
*
Another sudden perturbation between t = 0 and time of detection.
*
Detection could be using a beam of coherent radiation. Then one would integrate over t. Off
resonance? Spectrally not a simple δ-function.
*
Include elements of D that correspond to detection via ε molecule ( t) + ε local oscillator ( t) cross term.
2