p. 72
MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Andrei Tokmakoff
TIME-CORRELATION FUNCTIONS
> T.C.F.s are statistical measures of the time-evolution of an observable under equilibrium
conditions. → They describe an ensemble. > T.C.F.s can be used to describe spectroscopy and other time-dependent phenomena.
> They are generally applicable to any time-dependent process for an ensemble, but are commonly used to describe random or stochastic processes in condensed phases. Qualitatively:
• A correlation function describes how long a given property of a system persists until it is
averaged out by microscopic motions of system.
• It describes how and when a statistical relationship has vanished.
• T.C.F.s describes the dynamics associated with a dynamical variable A(t), averaged over
the ensemble.
A:
Microscopic variable
A : Equilibrium ensemble average
Classical: A ≡ ∫ dp ∫ dq A ( p, q ; t ) f ( p, q )
Quantum: A = ∑ p n
n A n
f: equil. Prob. distribution function
p n = e −βEn / Z
n
Ai (t ): Expectation value of A for a member of ensemble as a function of time.
Ai ( t )
A
t
p. 73
If we look at this behavior there seems to be little information in the random fluctuations of A, but
there are characteristic time scales and amplitudes to these changes. We can characterize these by
comparing the value of A at time t with the value of A at time t’ later.
Correlation functions are defined as a time-dependent quantity, A ( t ) , multiplied by that quantity
at some later time, A ( t ′ ) , and averaged over ensemble: CAA ( t, t ′ ) ≡ A ( t ) A (
t ′ )
← auto-correlation function
CAB ( t, t ′ ) ≡ A ( t ) B ( t ′ )
← cross-correlation function
These are products of a pair of dynamical variables.
Properties of Correlation Functions
A2
CAA(t,t')
A
2
t
1.
CAA ( t, t ) = A ( t ) A ( t ) = A 2 ( t ) ≥ 0
= A2
2.
(mean square value of A – independent of time)
CAA ( t, t ′ ) = CAA ( t + τ, t ′ + τ ) = CAA ( t − t ′ )
for τ = −t ′
Stationary random process.
Time average = Ensemble average
3.
lim
t →∞
CAA ( t ) = A ( t ) A ( 0 ) = A
2
Processes lose all correlation at infinite time separation .
Just express in time separation
C(t)
p. 74
4.
For classical mechanics:
A ( t ) A ( t′) = A ( t′) A ( t )
CAA ( t ) = C AA ( − t )
5.
Even in time
If we are observing fluctuations about an average position:
Redefine δA ≡ A − A
deviation from average
CδAδA ( t ) = δA ( t ) δA ( 0 ) = CAA ( t ) − A
Properties: lim CδAδA ( t ) = 0
2
all correlation lost at t = ∞
t →∞
CδAδA ( 0 ) = δA 2 = A 2 − A
2
δA2
(variance)
e − t / τc
CδAδA(t)
t
6.
τc : Correlation time is characteristic time for decay to zero.
There may be several time scales over which the correlation vanishes, so we can
define
1
τc =
δA 2
∞
∫ dt
0
δA ( t ) δA ( 0 )
p. 75
EXAMPLE 1: Velocity autocorrelation function for gas.
Vx : x̂ Component of molecular velocity
Vx = 0
CVx Vx ( t ) = Vx ( t ) Vx ( 0 )
CVx Vx ( 0 ) = Vx2 ( 0 ) =
kT
m
Ideal gas: no collisions
kT / m
CVx Vx ( t )
t
Dilute gas: infrequent collisions
kT / m
CVx Vx ( t )
Vx ( t ) = Vx ( 0 ) for t < τc
Vx ( t ) = Vx ( 0 ) ± δ for t > τc
τc
t
τc is related to the mean time between collisions.
EXAMPLE 2: µi : dipole moment for diatomic molecule in dilute gas.
µi = 0
(all angles are equally likely: isotropic system)
µi = µ 0 ⋅ û
unit vector along dipole
Cµµ ( t ) = µ ( t ) µ ( 0 )
= µ 02 uˆ ( t ) ⋅û ( 0 )
µ 02
collisional damping
Project time-dependent orientation onto initial orientation oscillation frequency gives
moment of inertia
p. 76
EXAMPLE 3: Displacement of harmonic oscillator.
mq = −κq
Since
→
q 2 ( 0 ) =
q(t )
= q(0 )cosωt
q = −ω2 q
kT
mω2
damping will cause Cqq to decay
kT / mω2
Cqq ( t ) = q ( t ) q ( 0) = q 2 ( 0 ) cos ωt
 kT 
=
cos ωt
2 
 mω 
QUANTUM CORRELATION FUNCTIONS
Equilibrium (thermal) ensemble average over product of Hermetian operators evaluated two times.
→
Heisenberg Representation
CAA ( t, t ′ ) =
A ( t ) A ( t′)
Here I’m temporarily using a double bracket to remind you that this is the equilibrium ensemble
average over the expectation value of the product of operators. It is also sometimes written
A ( t ) A ( 0 ) eq
Referring back to our notation for mixed states, we can represent the state of the system through
ψ k = ∑ a nk n
n
CAA ( t, t ′ ) = ∑ ψ k A ( t ) A ( t ′ ) ψ k p k
k
=
∑ p (a )
k
k *
n
a mk n A ( t ) A ( t ′ ) m
k,n ,m
= ∑ a *n a m
n,m
n A ( t ) A ( t′) m
p. 77
The equilibrium ensemble average of the expansion coefficients is equivalent to phase averaging
over the expansion coefficients, since at equilibrium all phases are equally probable:
*
n
a am
1 2π *
1
=
a n a m dφ =
an am
∫
0
2π
2π
φnm
∫
2π
0
e
− i ( φn −φm )
where a n = a n eiφn
dφnm
The integral is quite clearly zero unless φn = φm , giving
a *
n a m = p n δ n ,m =
e −βEm
δ m,n
Z
So
CAA ( t, t ′ ) = ∑ p n n A ( t ) A ( t ′ ) n
Heisenberg representation
n
More commonly, we will be writing correlation functions in the interaction picture operators.
We can also write this in the Schrödinger picture:
U = exp ( −iH t
)
CAA ( t, t ′ ) = ∑ p n n U † ( t ) A U ( t ) U † ( t ′ ) A U ( t ′ ) A n
n
= ∑ pn n A U ( t − t′) A n e
+ iωn ( t − t ′ )
n
insert
= ∑ p n
n A j
jAn e
− iω jn ( t − t ′ )
n, j
= ∑ p n
A jn e
2
− iω jn ( t − t ′ )
n, j
Notice that CAA ( t, t ′ ) = CAA ( t − t ′ )
CAA ( − t ) = C*AA ( t )
⇒ CAA ( t )
∑
j
j j
p. 78
Properties of Quantum Correlation Functions
There are various properties of quantum correlation functions that can be obtained using the
properties of the time-evolution operator.
A ( 0 ) A ( t ) = A ( 0 ) U † A U
= U A U† A
= A ( −t ) A ( 0)
A ( t ) A ( 0 ) = U† A U A
*
*
= U A U† A
= A ( 0) A ( t )
∴
A ( −t ) A ( 0 ) = A ( t ) A ( 0 ) = A ( 0 ) A ( t )
*
CAA ( t ) = CAA ( −t )
*
A ( t ) A ( t ′ ) = U † ( t ) A ( 0 ) U ( t ) U † ( t ′ ) A ( 0 ) U (
t ′ )
= U ( t ′ ) U † ( t ) A U ( t ) U
† ( t ′ ) A
= U† ( t − t′) A U ( t − t′) A = A ( t − t′) A ( 0 )
Note that CAA ( t ) is complex. You cannot directly measure a quantum correlation function, but
observables are often related to the real or imaginary part of correlation functions, or other
combinations of correlation functions.
CAA ( t ) = C′AA ( t ) + i C′′AA ( t )
C′AA ( t ) =
1
1
1
CAA ( t ) + C*AA ( t )  =  A ( t ) A ( 0 ) + A ( 0 ) A ( t )  = A
( t ) , A ( 0 ) +
2
2
2 
C′′AA ( t) =
1
1
1
 CAA ( t ) − C*AA ( t )  =  A ( t ) A ( 0 ) + A ( 0 ) A ( t )  =
A ( t ) , A ( 0 ) 

2i
2i
2i 
p. 79
Density Matrix
Earlier we showed that:
A ( t ) = ∑ a *n ( t ) a m ( t ) n A m = Tr A
 ρ ( t ) 
n,m
We also showed that
ρ( t ) = U ρ (0 )U †
A(t ) = Tr[A(t )ρ (0)]. ρ0 (or ρ( 0) or ρ( −∞
) or ρeq ) is the equilibrium canonical density matrix.
ρ0 =
e− βH
Z
Z = Tr ( e−βH )
for H n = E n n
Z = ∑ n e−βH n = ∑ e−βEn
n
n
( ρ0 )nm = Z1
n e−βH m = Z1 e−βEn δ n ,m
= p n δn ,m
So, the ensemble averaged expectation value (at equilibrium) is
A(t ) = Tr[A(t )ρ 0 ]
or for correlation functions:
CAA = ∑ p n n A ( t ) A ( 0 ) n
n
= Tr ( ρ0 A ( t ) A ( 0 ) )
= Tr ( A ( t ) A ( 0
) ρ0 )
p n = n ρ0 n
p. 80
More on Stationary Processes1
We stated that correlation functions are stationary: they do not depend on the absolute point of
observation, but rather the time-interval between observations.
Let’s look at this a bit more: the ensemble average value of A can be expressed as a time-average
or an ensemble average. For an equilibrium system:
A=
lim
1
2T
T→∞
A =∑
n
e−βE
n
Z
∫
T
−T
dt A i ( t )
… = time − average
… = equil. ensemble average
nAn
These quantities are equal for an ergodic system A = A . We assume this property for our
correlation functions.
So, the correlation of fluctuations can be written:
A ( t ) A ( 0) =
lim
T→∞
1
T
∫
T
0
dτ A i ( t + τ ) A i ( τ )
e −βEn
A ( t ) A ( 0) = ∑
n A ( t ) A ( 0) n
Z
n
1
See McQuarrie, p. 553