Density Matrix
State of a system at time t:
t   Cn (t ) n
 n  orthonormal
n
 Cn (t )  1
2
Contain time dependent
phase factors.

t
normalized
n
Density Operator
 (t )  t t
We’ve seen this before, as a “projection operator”
Can find density matrix in terms of the basis set
n
Matrix elements of density matrix:
 ij ( t )  i  ( t ) j
 it t j
Copyright – Michael D. Fayer, 2009
Two state system:
t  C1 ( t ) 1  C 2 ( t ) 2
t  C1* ( t ) 1  C 2* ( t ) 2
Calculate matrix elements of 2×2 density matrix:
11  1 t t 1
 1  C1 1  C 2 2   C1* 1  C 2* 2  1
11  C1C1*
12  1 t t 2
 21  2 t t 1
12  C1C
 21  C 2C1*
*
2
Time dependent phase
factors cancel
cancel. Always
have ket with its complex
conjugate bra.
 22  2 t t 2
 22  C 2C 2*
Copyright – Michael D. Fayer, 2009
In general:
t   Ci (t ) i
i
t t   C i ( t ) i
i
j C *j ( t )
j
 ij ( t )  i t t j


*
 i   C k ( t ) k l C l ( t )  j
 k l

 ij ( t )  C i C *j
ij density matrix element
Copyright – Michael D. Fayer, 2009
2×2 Density Matrix:
 C1C1*
 (t )  
*
C
C
2
1

C1C 2* 

C 2C 2* 
C1C1*  Probability of finding system in state 1
C 2C 2*  Probability of finding system in state 2
Diagonal density matrix elements  probs. of finding system in various states
Off Diagonal Elements  “coherences”
Since
 Cn (t )  1
2
n
T  (t )  1
Tr
And
trace = 1 for any
dimension
 ij   *ji
Copyright – Michael D. Fayer, 2009
Time dependence of  ( t )


d  (t )
dt
d  (t )
dt
 d

 d


t  t  t 
t 
dt 
dt 
product rule
Using Schrödinger Eq. for time derivatives of t & t
d t
i
H t
dt
d t
i
 t H
dt
d t
1
 H t
dt
i
d t
1

t H
i
dt
Copyright – Michael D. Fayer, 2009
Substituting:
d  (t )
dt
d  (t )
dt
1
1
 H (t ) t t 
t t H (t )
 i
i

1
 H ( t ) t t  t t H ( t ) 
i
density operator

1
 H ( t ),  ( t ) 
i
Therefore:

i   ( t )   H ( t ),  ( t ) 
The fundamental equation of the density matrix representation.
Copyright – Michael D. Fayer, 2009
Density Matrix Equations of Motion
i
 ( t )    H ( t ),  ( t ) 


since  ij  C i C *j
 dC *j 
*  dC i 
 ij  C i 
  Cj 


dt
dt





 *
j
i

 C C  C Ci
*
j
time derivative of
density matrix elements
by product rule
For 2×2 case, the equation of motion is:

  11

  21



 12 
i  H
    11


   H 21
 22 
H 12   11
H 22    21
12   11


 22    21
12   H 11
 22   H 21
H 12  

H 22  
i

 11   [( H 11 11  H 12  21 )  ( 11 H 11  12 H 21 )]

i

 11   ( H 12  21  H 21 12 )
Copyright – Michael D. Fayer, 2009

  11

  21


 12 
i   H 11
  


   H 21
 22 

 12  

 12  
H 12   11
H 22    21
12   11


 22    21
12   H 11
 22   H 21
H 12  

H 22  
i
( H 11 12  H 12  22 )  ( 11 H 12  12 H 22 )


i
( H 11  H 22 ) 12  (  22  11 ) H 12 


Equations of Motion – from multiplying of matrices:


i

 11    22   ( H 12  21  H 21 12 )


 12   *21  
i
( H 11  H 22 ) 12  (  22  11 ) H 12 


11   22  1 (trace of  = 1)
*
12   21
Copyright – Michael D. Fayer, 2009
In many problems:
H  H 0  H I (t )
time
independent
time
dependent
e.g., Molecule in a radiation field:
H 0  molecular Hamiltonian
H I ( t )  radiation field interaction (I) with molecule
Natural to use basis set of H 0
H 0 n  En n
Write t as:
(orthonormal)
(time dependent phase factors)
t   Cn (t ) n
n
Eigenkets of H 0
Copyright – Michael D. Fayer, 2009
For this situation:
i
 ( t )    HI ( t ),  ( t ) 



time evolution of density matrix elements
elements, Cij(t),
(t) depends only on H I ( t )
 time dependent interaction term
See d
S
derivation
i ti in
i book
b k – will
ill prove later.
l t
Like first steps in time dependent perturbation theory
before any approximations.
In absence of H I, only time dependence from time dependent phase factors
from H 0 . No changes in magnitude of coefficients Cij .
Copyright – Michael D. Fayer, 2009
Time Dependent Two State Problem Revisited:
Previously treated in Chapter 8 with Schrödinger Equation.
Basis set  1 , 2   degenerate eigenkets of H 0
No H I
H 0 1  E 1   0 1
H 0 2  E 2   0 2
Interaction H I
H I 1   2
   of Ch. 8
H I 2   1
The matrix H I
0
HI  


0 
Because degenerate states
states, time dependent phase factors cancel
in off-diagonal matrix elements – special case.
In general, the off-diagonal elements have time dependent phase factors.
Copyright – Michael D. Fayer, 2009
Use

i
 ( t )    H I ( t ),  ( t )


   11 12   11 12   0




0   2 1  22   21  22   
i  0
    
  

0
HI  

 

0  

0 

 11   i[(0 11   21 )  ( 11 0  12  )]
  i  (  21  12 )  i  ( 12   21 )
Multiplying matrices and subtracting gives
( 11   22 ) 
 ( 12   21 )
  i 


(



)

(



)
11
22
12
21 

Equations of motion of density matrix elements:


 11  i  ( 12   21 )

 22   i  ( 12   21 )
Probabilities

 12  i  ( 11   22 )

 21   i  ( 11   22 )
Coherences
Copyright – Michael D. Fayer, 2009

Using
 11  i  (  12  21 )
T k time
Take
i
derivative
d i i
 11  i  (  12  21 )


Substitute  12 &  21




 12  i  ( 11   22 )

 21   i  ( 11   22 )

 11  2  2 ( 11   22 )
11   22  1
U i Tr
Using
T  = 1,
1 i.e.,
i
Then  22  1  11
and

 11  2  2  4  2 11
For initial condition 11  1 at t = 0.
d cos 2 (  t )ddt  2  cos((  t ))sin(
i ( t )
d 2cos 2 (  t )dt 2  2  2 (sin 2 (  t )  cos 2 (  t ))
but sin 2 (  t )  1  cos 2 (  t )
then d 2cos 2 (  t )dt 2  2  2 (1  2cos 2 (  t ))
 2  2  4  2 cos 2 (  t )
11  cos 2 (  t )
   /
 22  sin 2 (  t )
From Ch. 8
Same result as Chapter 8 except obtained probabilities directly.
No probability amplitudes.
Copyright – Michael D. Fayer, 2009
Can get off-diagonal elements

 12  i  ( 11   22 )
Substituting:

 12  i  (cos 2  t  sin 2  t )
12  i    cos 2  t  sin 2  t  dt
i
2
12  sin(2  t )
2
cos
 xdx  (1/2) x  (1/ 4)sin 2 x
2
( 4)sin
) 2x
sin
 xdx  ((1/2)) x  (1/
Since  ij   *ji
i
2
 21   sin(2
(  t)
Copyright – Michael D. Fayer, 2009
Density matrix elements have no time dependent phase factors.
t   Ci (t ) i
i
t t   C i ( t ) i
i
j C *j ( t )
j
 ij ( t )  i t t j


*
 i   C k ( t ) k l C l ( t )  j
 k l

time dependent phase factor in ket, but
 C i ( t ) i i C *j ( t ) j j
its complex conjugate is in bra
bra. Product
*
 C i ( t )C j ( t )
is 1. Kets and bras normalized, closed
bracket gives 1.
 ij ( t )  C i ( t )C *j ( t )
ij density matrix element
Time dependent coefficient,
coefficient but no phase factors.
factors
Copyright – Michael D. Fayer, 2009
Can be time dependent phase factors in density matrix equation of motion.
i
 ( t )    H ( t ),  ( t ) 


For two levels, but the same in any dimension.
 H11
H  
 H 21
H12 

H 22 
1  1s e  iE1t / 
2  2s e
 iE2 t / 
s – spatial
H 11  1 H 1  1s H 1s e iE1t /  e  iE1t /   1s H 1s
no time dependent phase
factor
H 12  1 H 2  1s H 2 s e iE1t /  e  iE2t /   1s H 2 s e i ( E1  E2 ) t / 
time dependent phase factor
if E1 ≠ E2.
Therefore, in general, the commutator matrix,
 H ( t ),  ( t ) 


when yyou multiply
p y it out,,
will have time dependent phase factors if E1 ≠ E2.
Copyright – Michael D. Fayer, 2009
Expectation Value of an Operator
A  t At
Complete orthonormal basis set
j
t   C j (t ) j
j
Matrix elements of A
Aij  i A j
Derivation in book and see lecture slides

A  Tr  ( t ) A

Expectation value of A is trace of the product of density matrix with
the operator matrix A .
Important:  ( t ) carries time dependence of coefficients.
coefficients
Time dependent phase factors may occur in off-diagonal matrix elements of A.
Copyright – Michael D. Fayer, 2009
E  
Example: Average E for two state problem
E  
E  H  Tr  H
E=0
E
H  H0  H I  
 
 11
T  H  Tr
Tr
T 
  21
E
 
E 
12   E
 22   
Time dependent phase factors cancel
because degenerate.
g
Special
p
case.
In general have time dependent phase factors.
 
E 
Only
y need to calculate the
diagonal matrix elements.
 11 E  12    21    22 E
 E ( 11   22 )   ( 12   21 )
i
2
 22  sin
i 2 ( t )
12  sin(2
i (2  t )
H  E (cos 2  t  sin 2  t ) 
i 
(sin 2 t  sin 2  t )
2
11  cos 2 (  t )
E
i
2
 21   sin(2  t )
Copyright – Michael D. Fayer, 2009
Coherent Coupling by of Energy Levels by Radiation Field
Two state problem
2
E =   0
1
  0/2
radiation
field
Zero of energy half
way between states.
-  0/2
2 electronic states S0 
 S1
2 vibrational states V0 
V1
NMR – 2 spin state, magnetic transition dipole
In general, if radiation field frequency  is near E, and other transitions
are far off resonance, can treat as a 2 state system.
Copyright – Michael D. Fayer, 2009
Molecular Eigenstates as Basis

H0 1   0 1
2
 0
H0 2 
2
2
Zero of energy half way between states.
Interaction due to application of optical field (light) on or near resonance.
H I ( t )  e x 12 E0 cos(( t )
e x 12  transition dipole
p operator
p
x polarized light
E0  amplitude
Copyright – Michael D. Fayer, 2009
H I couples states
1 H I ( t ) 2  E0 cos( t ) 1 e x 12 2
  E0 cos( t ) e
 i 0 t / 2
   E0 cos( t ) e
take out time dependent phase factors
1 e x12 2 e  i 0t / 2
 i0 t
 is value of
transition dipole bracket
bracket,
Note – time independent
kets. No phase
factors. Have taken
phase factors out.
2 H I ( t ) 1   * E0 cos( t )e i 0t
T k  reall (doesn’t
Take
(d
’t change
h
results)
lt )
Define Rabi Frequency, 1
1   E0
Then
1 H I ( t ) 2   1 cos( t )e  i 0t
2 H I ( t ) 1   1 cos( t )e i 0t
H I ( t ) matrix:

0
H I (t )   
i 0 t
1 cos( t )e
1 cos( t )e  i t 
0
0


Copyright – Michael D. Fayer, 2009
General state of system
t  C1 ( t ) 1  C 2 ( t ) 2

i
 ( t )    H I ( t )),  ( t ) 
Use


0
i  
    
  1 cos( t )e i 0 t
1 cos( t )e  i t   11 12 




0
22 
 21
0

 11 12  
0
 

i 0 t


 21
22   1 cos( t )e

  11

  21

0
0

 

 12 

1 cos( t )e  i t  


 22 

 i cos( t ) e i 0t   e  i 0t 
12
21
 1

i 0 t

i

co
s(

t
)
e
( 11   22 )
1


i1cos( t )e  i0t ( 11   22 )

 i1cos( t ) e
i 0 t
12  e  i 0t  21





Blue diagonal
Red off-diagonal
Copyright – Michael D. Fayer, 2009
Equations of Motion of Density Matrix Elements


 11  i1cos( t ) e i 0t 12  e  i 0t  21



 22   i1cos( t ) e i 0t 12  e  i 0t  21


 12  i1cos( t )e  i 0t ( 11   22 )

 21   i1cos( t )e i 0t ( 11   22 )
11   22  1
*
12   21


  11    22


  12   *21
T
Treatment
exact to this
hi point
i
Copyright – Michael D. Fayer, 2009
Rotating Wave Approximation

1 i t
cos(( t )  e  e  i t
2

Put this into equations of motion
Will have terms like
e
 i ( 0  ) t
and
e
 i (  0  ) t
But  0  
Terms with ( 0   )  2 0 off resonance  Don
Don’tt cause transitions
Looks like high frequency Stark Effect
 Bloch – Siegert
g Shift
Small but sometimes measurable shift in energy.
Drop these terms!
Copyright – Michael D. Fayer, 2009
With Rotating Wave Approximation
Equations of motion of density matrix

 11  i
e

2
1

 22   i

 12  i

e

2
1
1
2
 21   i
i ( 0  ) t
12  e  i ( 0  )t  21
i ( 0  ) t

12  e  i (0  ) t  21

e  i ( 0  ) t ( 11   22 )
1
2
e i ( 0  ) t ( 11   22 )
These are the
Optical Bloch Equations for optical transitions
or just
j st the Bloch Eq
Equations
ations for NMR.
NMR
(NMR - 1   m H1 )
Copyright – Michael D. Fayer, 2009
Consider on resonance case  = 0

 11  i
2

 22   i

 12  i


1
1
2
1
2
 21   i
e i ( 0  ) t 12  e  i ( 0  ) t  21
e
1
2


e i ( 0  ) t 12  e  i ( 0  ) t  21
 i ( 0  ) t
e
( 11   22 )
i ( 0  ) t

E
Equations
ti
reduce
d
to
t

 11  i
1
2

( 11   22 )
All of the phase factors = 1.
 22   i

 12  i

1
2
1
2
 21   i
 12   21 
 12   21 
( 11   22 )
1
2
( 11   22 )
These are IDENTICAL to the degenerate time dependent 2 state problem
with  = 1/2.
Copyright – Michael D. Fayer, 2009
On resonance coupling to time dependent radiation field induces transitions.
2
Looks identical to time independent
coupling of two degenerate states.
E    0
In effect, the on resonance radiation
field “removes” energy differences and
time dependence of field.
Start in ground state, 1
Th
Then
11  cos (
2
 22  sin 2 (
1 t
2
1 t
2
populations
  0
1
11  1;  22  0, 12  0,  21  0
at t = 0.
i
2
)
12  sin(1t )
)
 21   sin(1t )
i
2
coherences
Copyright – Michael D. Fayer, 2009
11  cos 2 (
Recall
1 t
2
 22  sin 2 (
)
1 t
2
)
populations
1   E0  Rabi Frequency
1t  

11  0,
 22  1
This is called a  pulse  inversion, all population in excited state.
1t 

2

11  0.5,
 22  0.5
As t is increased, population
oscillates between ground
and excited state at Rabi frequency.
Transient Nutation
Coherent Coupling
22 – excited staate prob.
This is called a /2 pulse  Maximizes off diagonal elements 12, 21
 pulse
1
0.8
0.6
0.4
0.2
2
2 pulse
4
t
6
8
Copyright – Michael D. Fayer, 2009
Off Resonance Coherent Coupling
   0  
Amount radiation field frequency is off resonance from transition frequency.
Define
 e     
2
For same initial conditions:
2
1

1/ 2
 Effective Field 1 = E0 - Rabi frequency
11  1;  22  0, 12  0,  21  0
Solutions of Optical Bloch Equations
12 2
11  1  2 sin (e t /2)
e
12 
12 2
 22  2 sin (e t /2)
e
1  i e

 21  2  
sin( e t )   sin 2 ( e t /2)  e i  t
e  2

Oscillations Faster  e
Max excited state probability:
1  i e
  i  t
2
sin(

t
)



sin
(

t
/2)
e
e
e

2 
e  2


max
22
12
 2
e
(Like non-degenerate time dependent 2-state problem)
Copyright – Michael D. Fayer, 2009
Near Resonance Case - Important
1  
Then
 e     
2
2
1

1/ 2
 e  1
11, 22 reduce to on resonance case.
i
12  sin(1t )e  i t
2
i
 21   sin(1t )e i t
2
Same as resonance case
p for p
phase factor
except
For /2 pulse, maximizes 12, 21
1t =  /2
But
 t << /2  0
because 1  
Then, 12, 21 virtually identical to on resonance case and
11, 22 same as on resonance case.
This is the basis of Fourier Transform NMR. Although spins have
different chemical shifts, make ω1 big enough, all look like on resonance.
Copyright – Michael D. Fayer, 2009
Free Precession
 = 1t
Aft pulse
After
l off
(fli angle)
(flip
l )
On or near resonance
i
12  sin
2
i
 21   sin
2
11  cos 2 ( /2)
 22  sin
i 2 ( /2)
After pulse – no radiation field.
Hamiltonian is H0

i
   H 0, 


0 
  0 /2
H0  
 0 /2 
 0
Copyright – Michael D. Fayer, 2009
i
   H 0, 



0 
  0 /2
H0  

0
/2

0


0   11 12 
i    0 /2
    
 0 /2   2 1  22 
  0

0 
 11 12    0 /2
 

 0

/2


0
22  
 21


 11  0

 22  0

 12  i 0 12
t = 0 is at end of pulse

 21   i 0  21
Solutions
11 = a constant = 11((0))
12  12 (0)e
(0)e i 0t
22 = a constant = 22(0)
 21   21 (0)e  i0t
Populations don’t change.
Off-diagonal density matrix elements
 Only time dependent phase factor
Copyright – Michael D. Fayer, 2009
Off-diagonal density matrix elements after pulse ends (t = 0).
 . Recall   e x 12
p
value of transition dipole
p
Consider expectation
  Tr  
1
1 0
 
2 
 11 (0)
Tr    Tr 
  21 (0)e  i 0t
2

0 
No time dependent phase factors
because because matrix elements
involve time independent kets
kets.
Phase factors were taken out as part
of the derivation.
12 (0)e i 0t   0  


 22 (0)    0 
    12 (0)e i 0t   21 (0)e  i 0t 
For a flip angle 
    sin sin( 0 t )
Oscillating electric dipole (magnetic dipole - NMR)
att frequency
f
0,  Oscillating
O ill ti E
E-field
fi ld ((magnett fi
field)
ld)
I E
2
for ensemble, coherent emission
t = 0, end of pulse
Free precession.

Rot. wave approx.
Tip of vector goes in circle.
Copyright – Michael D. Fayer, 2009
Pure and Mixed Density Matrix
Up to this point - pure density matrix.
One system or many identical systems.
systems
Mixed density matrix 
Describes nature of a collection of sub-ensembles each with different properties.
The subensembles are not interacting.
Pk  probability of having kth sub-ensemble with density matrix, k.
0  P1 , P2 ,    Pk ,    1
and
P
k
1
Sum of probabilities is unity.
k
Density matrix for mixed systems
 ( t )   Pk  k ( t )
k
or integral if continuous distribution
Total density matrix is
the sum of the individual density matrices times their probabilities.
Because density matrix is at probability level, can sum (see Errata and Addenda).
Copyright – Michael D. Fayer, 2009
Example: Light coupled to two different transitions – free precession
 01
 02
Light frequency in the
middle of 01 & 02.
Difference between 01 & 02 small compared to 1
and both near resonance.
Equal probabilities  P1=0.5 and P2=0.5
For a given pulse of radiation field,
both sub-ensembles will have same flip angle .
Calculate
  Tr  ( t )
  Pk Tr  k 
k
Copyright – Michael D. Fayer, 2009
Pure density matrix result for flip angle :
    sin  sin( 0 t )
For 2 transitions - P1=0.5 and P2=0.5
1
    sin   sin( 01t )  sin(02 t )
2
 1

1

   sin sin  ( 01  02 )t  cos  ( 01  02 )t  

2

 2
from trig.
identities
Call: center frequency  0,
shift from the center  
then, 01 = 0 + 
and 02 = 0 -  ,
with  << 0
Therefore,
    sin   sin( 0 t )cos( t )
high freq
freq. oscillation
low freq
freq. oscillation
oscillation, beat
Beat gives transition frequencies – FT-NMR
Copyright – Michael D. Fayer, 2009
Equal amplitudes – 100% modulation, ω01 = 20.5; ω01 = 19.5
2
1
5
10
15
20
-1
-2
Copyright – Michael D. Fayer, 2009
Amplitudes 2:1 – not 100% modulation , ω01 = 20.5; ω01 = 19.5
1
0.5
5
10
15
20
-0.5
-1
Copyright – Michael D. Fayer, 2009
Amplitudes 9:1 – not 100% modulation , ω01 = 20.5; ω01 = 19.5
1
0.5
5
10
15
20
-0.5
-1
Copyright – Michael D. Fayer, 2009
Equal amplitudes – 100% modulation, ω01 = 21; ω01 = 19
2
1
5
10
15
20
-1
-2
Copyright – Michael D. Fayer, 2009
center freq
Free Induction Decay
0
Identical molecules have
range of transition frequencies.
Different solvent environments.
Doppler shifts, etc.
h  frequency
of particular molecule
Gaussian envelope
 
Frequently, distribution is a Gaussian probability of finding a molecule at a particular frequency.
1
 ( h  0 )2 / 2 2
e
2
standard deviation
2
normalization
constant
Then
 (t ) 

1
2
2
e
 ( h  0 )2 / 2 2
 ( h , t )d h

probability, Ph
pure density matrix
Copyright – Michael D. Fayer, 2009
Radiation field at  = 0 line center
1 >>  – all transitions near resonance
Apply pulse with flip angle 
Calculate  , transition dipole expectation value.
Following pulse, each sub-ensemble will undergo free precession at h
  Tr  ( t )


1
2
2
e
 ( h  0 )2 / 2 2
Tr  ( h , t ) d h

U i result
Using
l for
f single
i l frequency
f
h and
d fli
flip angle
l 
 
 sin
2 2

e
 ( h  0 )2 / 2 2
sin( h t )d h

Copyright – Michael D. Fayer, 2009
Substituting  = (h – 0),
frequency of a molecule as difference from center frequency (light frequency).
Then h = ( +0) and dωh = d.
With the trig identity: sin( x  y )  sin( x )cos( y )  cos( x )sin( y )
 
 sin 

cos( 0 t )  e
2 

2
2
 / 2
2

sin( t ) d   sin( 0 t )  e
 2 / 2 2


cos( t ) d 

First integral zero; integral of an even function multiplying an odd function.
    sin sin( 0 t )e
 t 2 / 2( 1 /  )2
Oscillation at 0; decaying amplitude 
Gaussian decay with standard deviation in time  1/ (Free Induction Decay)
Phase relationships lost  Coherent Emission Decays
Off-diagonal density matrix elements – coherence;
diagonal - magnitude
Copyright – Michael D. Fayer, 2009
    sin sin( 0 t )e
flip angle
 t 2 / 2( 1 /  )2
light frequency
free induction decay
1

0.8
0.6
Decay of oscillating macroscopic dipole.
Free induction decay.
E 
0.4
2
I E  
0.2
0.5
t = 0

1
1.5
2
2.5
3
t = t'

2
Coherent emission
of light.
higher frequencies
rotating frame at
center freq., 0
lower frequencies
Copyright – Michael D. Fayer, 2009
Proof that only need consider

i
 ( t )    HI ( t ),  ( t ) 


when working in basis set of eigenvectors of H.
Working with basis set of eigenkets of time independent piece of Hamiltonian,
Hamiltonian
H0, the time dependence of the density matrix depends only on the time
dependent piece of the Hamiltonian, HI.
Total Hamiltonian
H  H 0  H I (t )
H 0 time independent
H 0 n  En n
Use
 n  as basis set.
t   Cn (t ) n
n
Copyright – Michael D. Fayer, 2009
Time derivative of density operator (using chain rule)
 d

 d

dt t  t  t dt t 




(A)
Use Schrödinger Equation

1
1
H (t ) t t 
t t H (t )
i
i 

1
1
1
1
H0 t t  HI t t 
t t H0 
t t HI
i
i
i 
i 
(B)
Substitute expansion t   C n ( t ) n into derivative terms in eq. (A).
n
 d

 d

*
C
n
t

t
C
n
dt n 
dt n 
n
n








d
d




   C n n  t    C n
n  t  t   C n* n   t   C n*
n
dt 
dt 
 n

 n

 n
 n
(B) = (C)
(C)
Copyright – Michael D. Fayer, 2009
Using Schrödinger Equation

 1
d
C
n
  n d t   i H 0 t
 n

Right
g multiply
p y top
p eq.
q by
y t .


1
* d
C
n

  n d t  i t H 0
 n

Left multiply bottom equation by t .
Gives


d
1
C
n
t
H0 t t

 n d t 
i
 n



1
* d
t   Cn
n 
t t H0
d
t

i

 n

Using these see that the 1st and 3rd terms
in (B) cancel the 2nd and 4th terms in (C).
1
1
1
1
H0 t t 
HI t t 
t t H0 
t t HI
i
i
i 
i 
(B)




d
d




   C n n  t    C n
n  t  t   C n* n   t   C n*
n
dt 
dt 
 n

 n

 n
 n
(C)
Copyright – Michael D. Fayer, 2009
After canceling terms, (B) = (C) becomes

 1

 *


  C n n  t  t   C n n   i   H I ,  
 n

 n

Consider the ij matrix element of this expression.
expression
The matrix elements of the left hand side are

 *
Cn i n t j  i t
Cn n j

 *
n
n
 C i C *j  C i C j

  ij

i
 ( t )    H I ( t ),  ( t ) 


In the basis set of the eigenvectors of H0,
H0 cancels out of equation of motion of density matrix.
Copyright – Michael D. Fayer, 2009

Proof that t A t = A  Tr  ( t ) A

Expectation value
A  t At
j
complete orthonormal basis set.
t   C j (t ) j
j
Matrix elements of A
Aij  i A j


 
t A t    C i* i  A   C j j 
 i
  j

  C i* ( t )C j ( t ) i A j
i, j
Copyright – Michael D. Fayer, 2009
 ji  j  ( t ) i
note
t order
d
 jt t i
 C i* ( t )C j ( t )
Then
t A t   j  (t ) i i A j
i, j
Matrix multiplication, Chapter 13 (13.18)
N
ckj   bki aij
i 1
t At
like matrix multiplication but only diagonal elements –
j on both sides.
g
elements.
Also, double sum. Sum over j – sum diagonal
Therefore,

A  Tr  ( t ) A

Copyright – Michael D. Fayer, 2009