Appendix 1 - Outline of Density Matrix Analysis

Appendix 1
Outline of Density Matrix Analysis
A1.1 DEFINITION OF DENSITY MATRIX AND
EXPECTATION VALUES
The density matrix offers an effective technique for dealing statistically with a
system consisting of many electrons using the quantum theory for an electron.
A mixed state consisting of a statistical distribution of various quantum states
can be specified by a set of probabilities p
j
with which the electron is found
in a quantum state j
j
i. The density operator  is defined by
 ¼
X
j
j
j
ip
j
h
j
jðA1:1Þ
The probability satisfies 0 p
j
1 and
P
j
p
j
¼ 1. The operator  is a Hermite

operator, and the matrix description of  is called the density matrix. Using
a system of eigenstate fjnig, the elements of the density matrix are given by

nn
0
¼hnjjn
0
i¼
X
j
hnj
j
ip
j
h
j
jn
0
iðA1:2Þ
The diagonal elements of the density matrix

nn
¼
X
j
p
j
jhnj
j
ij

2
ðA1:3Þ
give the probability with which the system belongs to the eigenstate jni. The
off-diagonal elements represents the correlation of states jni and jn
0
i. The
expectation value hAi for a physical quantity represented by an operator A,
being the weighted average of the expectation values for states j
j
i, can be
written as
hAi¼
X
j
p
j
h
j
jAj
j
i
¼
X
jnn
0
p
j
h
j
jnihnjAjn

0
ihn
0
j
j
i
Copyright © 2004 Marcel Dekker, Inc.
¼
X
nn
0

n
0
n
A
nn
0
¼ TrfAgðA1:4Þ
Since hAi can be expressed by A and  only, it is possible to calculate the
value of the macroscopic observable hAi without knowing j
j
i and p
j
,
provided that  is obtained.
A1.2 EQUATION OF MOTION FOR THE DENSITY
OPERATOR
The time variation of a state j
j

i can be written by using the system
Hamiltonian H as
j
j
ðtÞi ¼ UðtÞj
j
ð0Þi, UðtÞ¼exp
ÀiHt
hh

ðA1:5Þ
and, if the time dependence of p
j
is omitted, the time variation of  can be
written as
ðtÞ¼
X
j
UðtÞj
j
ð0Þip
j
h
j
ð0ÞjUðtÞ
y
¼ UðtÞð0ÞUðtÞ
y
ðA1:6Þ
Then, calculation of the time derivative of  results in

d
dt
ðtÞ¼
HðtÞðtÞÀðtÞHðtÞ
ihh
¼
1
ihh
½HðtÞ, ðtÞ
ðA1:7Þ
Thus, the equation of motion for  is described by using the commutation
relation between H and . When the initial state (0) is given by a matrix
representation based on an appropriate eigenstate system, solving the above
equation to calculate (t), followed by calculation of hAi by Eq. (A1.4),
clarifies the behavior of the whole system concerning the observation of the
quantity A. The above description is made in the Schro
¨
dinger picture using a
time-dependent operator (t). However, for cases where the Hamiltonian H
can be written as a sum of a Hamiltonian H
0
with the interaction omitted
and an interaction Hamiltonian H
i
, i.e.,
HðtÞ¼H
0
þ H
i
ðtÞðA1:8Þ

converting (t) into a density operator in the interaction picture:

I
ðtÞ¼U
0
ðtÞ
y
ðtÞU
0
ðtÞ, U
0
ðtÞ¼exp
ÀiH
0
t
hh

ðA1:9Þ
286 Appendix 1
Copyright © 2004 Marcel Dekker, Inc.
transforms the equation of motion into that in the interaction picture:
d=dt
I
ðtÞ¼
1
ihh
½H
I
ðtÞ, 
I

ðtÞ ðA1:10aÞ
H
I
ðtÞ¼U
0
ðtÞ
y
H
I
ðtÞU
0
ðtÞðA1:10bÞ
where H
I
(t) is the interaction Hamiltonian in the interaction picture. Let
E
n
¼ hh!
n
be the energy eigenvalues of jni; then the density matrix elements

Inn
0
and 
nn
0
are correlated by

nn
0

ðtÞ¼expðÀi!
nn
0
tÞ 
Inn
0
ðtÞ, !
nn
0
¼ !
n
À !
n
0
ðA1:11Þ
In the interaction picture, the expectation value of A is given by
hAi¼Tr 
I
ðtÞA
I
ðtÞ
ÈÉ
, A
I
ðtÞ¼U
0
ðtÞ
y
AU
0

ðtÞðA1:12Þ
Outline of Density Matrix Analysis 287
Copyright © 2004 Marcel Dekker, Inc.