MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Robert Field
5.74 RWF Lecture #8
8–1
Resonance Operators: Equation of Motion
Last time: tools for describing a pump/probe wavepacket experiment
* ρ(t ≤ 0) incoherent, populations on diagonal
* E(0) excitation matrix. Transforms initial state into coherent superposition: for electronic
transition get g, v g′′ translated vertically onto upper potential surface
ρ(t ≤ 0)E† is expressed
* U(t,0) = e–iHt/ time evolution of t = 0 prepared wavepacket. If ρ(0) = E(0)ρ
in the eigenbasis, U has simple diagonal form. Otherwise need to express everything in zeroorder basis and transform.
Why? The pluck is almost always a simple zero-order non-eigenstate.
h
ρ(t) = U(t,0)E(0)ρ
ρ(t ≤ 0)E†(0)U†(t,0)
* Now describe the specific nature of the detection operation
ρ)
I(t) = Trace(Dρ
If you want to sample the ρij(t) element of ρ(t), you want a detection matrix with non-zero Dji element.
Often we observe something in an experiment like
a †i a i
Qi
, Pi
Pi2
2m
V
2
Ψ( t) Ψ
and we would like to understand what this implies about the dynamical mechanism
where does it start?
how fast does it leave?
why?
where does it go next?
why?
how fast does it go there? why?
what fraction gets to the target state? why?
We build a toy model Heff with the goal of reproducing the early time dynamics. We want to be able to look
at the toy model and identify the most important dynamical features of that model.
dynamical feature↔model parameter, dynamical mechanism
Tools for computing dynamics from Hmodel! or constructing Hmodel from expt.
5.74 RWF Lecture #8
8–2
Today: useful ways of monitoring and describing the mechanism of the dynamics encoded in ρ(t).
What do I mean by encoded? N × N ρ(t) matrix. Since ρ is Hermitian, there are
N2 − N
N ( N + 1)
+N =
2
2
independent t-dependent elements. Too much information.
Some useful relationships.
A
t = Trace(Aρ
ρ( t))
if A = ab, then A t = Trace(abρρ) = ∑ ∑ aij b jk ρki
i
j,k
= ∑ b jk ρki aij
ijk
= Trace(bρa )
d
∂A
ih A t = [A, H] t +
dt
∂t
ihρ˙ ( t) = [H,ρρ( t)]
t
Ehrenfest’s Theorem There are many ways to
d
replace
A by something that gives more
dt
insight.
* opposite sign of commutator with
respect to Ehrenfest
* This is a relationship between matrix
elements, not expectation values
Ehrenfest’s Theorem gives us a quantum mechanical form of classical equations of motion
Newton’s
Hamilton’s
r
dr 1 r
= p
dt m
r
r
dp
= −∇V ( r )
dt
∂H
Q̇i
=
∂Pi
∂H
Ṗi = −
∂Qi
We say that the
where Pi, Qi are conjugate observables (page 718, of
HLB-RWF. Poisson’s bracket) Return to these at
the end of 5.74.
center of a wavepacket follows Newton’s equations.
5.74 RWF Lecture #8
8–3
This is not quite true.
We would like to say
d
1
q =
p
dt
m
d
p = −∇V (q) q =
dt
q
but this is not quite true
p2
H=
+ V(q)
2m
Ehrenfest:
d
ih
q = [q,
H] t
dt t
1
2ih
2
[q,H] = 2m q,p = 2m p
[
]
1
d
qt=
p t . This is what we wanted. This relates the motion of the coordinate-center of the
dt
m
wavepacket to the momentum-center of the wavepacket. Recall that even if we write Ψ(0) as ∑ a i ψ i (q) ,
Thus
i
we still specify ⟨p⟩.
d
p = [p, H] t
dt
∂
h
[p, H] = i  ∂q ,V(q) = −ih∇V(q)


ih
Thus
d
p = − ∇V(q) ≠ −∇V( q
dt
)
This is a gradient of V at the
center of the wavepacket
5.74 RWF Lecture #8
8–4
How large an error do we make if we pretend that
d
p = −∇V( q ) ?
dt
Expand V(q) in power series:
( t)
V (q) = V q
dV
+
dq
q= q
1 d 2V
(q − q ) + 2 2
dq
t
q= q
Thus
( t)
V(q) = V q
dV
+
dq
)t 2
t
q= q t
2
( q t −
q t )
(
( t)
q
0
1d V
2
+
−2 q
q
2
t
2 dq
V(q) − V q
(q =
1 d 2V
=
2 dq 2
q= q
(q
2
2
t
+ q
− q
2
t
2
t
)
)
σq
t
The error one makes in predicting ⟨V(q)⟩ by pretending that Newton’s equations apply to the center of the
wavepacket is proportional to
d 2V
dq 2
σq
q= q
curvature of potential
(no curvature, no error)
width of wavepacket
5.74 RWF Lecture #8
8–5
So there are many situations where we can make this useful particle-like approximation even when |Ψ(q)|2
has lost any resemblance to a particle-like state.
Similarly, by expanding
dV
∇V(q)⟩)about q = ⟨q⟩
(which is what we need for ⟨∇
dq
dV
dq
dV
−
dq
t
q= q
1 d 3V
=
2 dq 3
t
σq
q= q
t
Ehrenfest’s Theorem also allows us to describe (and simplify) the dynamics in “state space” rather than
coordinate-momentum space. This can be very useful.
Suppose we can write H = ∑ h j + ∑ h jk
j≠k
j
hj describes isolated sub-systems. It is the basis for a choice of H ( 0 ) = ∑ h j that defines basis states
j
as 1,n1⟩2,n2⟩ …N,nN⟩. Simple (and convenient) product basis (there are often several
useful choices, e.g. normal vs. local modes of vibration).
hjk describes couplings between subsystems. It is responsible for dynamics if the pluck produces a
single product basis (non-eigen) state.
For example,
H=
h
(
)
(
ω1 †
hω 2
a 1 a 1 + a 1a 1† +
a †2a 2 + a 2a †2
2
2
(
+ k122 a †1 a 2a 2 + a 1a †2a †2
)
a 1:2 anharmonically coupled pair of harmonic oscillators.
Recall:
1/ 2
a 1 n1 = n1
n1 − 1
a †
1 n1
= ( n1 + 1)
N1 = a 1† a 1
1/ 2
[ai ,ai† ] = 1
annihilation
n1 + 1
creation
[a ,a ] = 0
i
†
j
)
5.74 RWF Lecture #8
8–6
Ehrenfest tells us to evaluate some interesting commutators:
[h12,H] = … do this second
[N1
,H] = [a1†a1,h1] + [a1†a 1,h2 ]
[
+ a 1†
a 1, h12
]
any operator
commutes with
itself
0
(
)
 † hω 1 †
†  hω 1 †
†
=
a
a
,
a
a
a
a
,
a
a
a
+
1 1
1 1 
1 1 1a1
 1 1 2
2

hω 1
+
a †1 a 1, a 1a 1†
2
[
[
]
[
]
]
a 1a †1 = a 1, a †1 + a 1† a 1
Thus
[
]= 0
[a†1a1,h2 ] = 0
=1
1 commutes
with anything
a †1 a 1, h1
commutes with itself
a 1a 1† do not operate on h2
Finally,
[a†1a1,k122a†1a 2a 2 ] + [a†1a1,k122a1a†2a†2 ]
= k122 (a 2a 2 [a 1† a 1, a 1† ] + a †2a †2 [a 1† a 1,
a 1 ])
= k122 (a 1† a 2a 2 − a 1a †2a †2 )
a †1
Now look at the other interesting commutator
–a1
5.74 RWF Lecture #8
8–7
0
ω1
hω 2
,
,
h
H
=
h
N
+
[ 12 ] 2 [ 12 1] 2 [h12,N2 ] + [h12,h12 ]
hω 1k122
a 1a †2a †2 , a †1 a 1 + a †1 a 2a 2 , a 1† a 1
=
2
hω k
+ 2 122 a 1a †2a †2 , a †2a 2 + a †1 a 2a 2 , a †2a 2
2
hω 1k122
a 1a †2a †2 − a †1 a 2a 2
=
2
hω k
+ 2 122 a 1a †2 a †2a †2 , a 2 + a 1† a 2 a 2a 2 , a †2
2
h
([
([
(
( [
−2a †2
=
])
])
] [
] [
)
]
[
2a 2
(
 hω 1 − 2 hω 2 
k122 a 1a †2a †2 − a †1 a 2a 2


2
)
Change notation
Ω = k122a 1† a 2a 2
Ω† = k122a 1a †2a †2
h12 = Ω + Ω†
resonance operator. h12 is Hermitian so ⟨Ω+Ω†⟩ is real
[N1, H] = Ω – Ω†
rate of change operator
(ω1 – 2ω2)
(Ω† – Ω)
[h12,H] =
2
h
ih
d
N = Ω − Ω†
dt 1
pure imaginary
real
ih
d
h = h(ω1 − 2ω 2 ) Ω† − Ω
dt 12
Ω + Ω†
see
this
])
5.74 RWF Lecture #8
d
N =
Thus
dt 1
8–8
d
h
dt 12
.
h(ω 1 − 2ω 2 ) 2
−
It is also easy to derive
d
1 d
N1 = −
N .
dt
2 dt 2
Conserved quantity: 2⟨N1⟩t + ⟨N2⟩t = 2⟨N1⟩0 + ⟨N2⟩0 (polyad quantum number)
The rate of change of population in mode 1 is proportional to
*
k122
strength of coupling
*
1
. closeness to “resonance”
ω1 − 2ω 2
This algebra is especially useful when there are several different resonance operators.
This algebra allows us to ask what is the fractional importance of each resonance operator for the dynamics
of any conceivable pluck. See page 649 of HLB-RWF.