Appendix 1 - Outline of Density Matrix Analysis

Appendix 1 Outline of Density Matrix Analysis EXPECTATION VALUES The density matrix offers an effective technique for dealing statistically with a system consisting of many electrons usi

Appendix 1 Outline of Density Matrix Analysis

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 pjwith which the electron is found

in a quantum state j ji The density operator  is defined by

j

The probability satisfies 0  pj1 andP

jpj¼1 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 0¼ hnjjn0i ¼X

j

The diagonal elements of the density matrix

j

give the probability with which the system belongs to the eigenstate jni The off-diagonal elements represents the correlation of states jni and jn0i The expectation value hAi for a physical quantity represented by an operator A, being the weighted average of the expectation values for states j ji, can be written as

j

pjh jjAj ji

jnn 0

pjh jjnihnjAjn0ihn0j ji

nn 0

n 0 nAnn 0

Since hAi can be expressed by A and  only, it is possible to calculate the

provided that  is obtained

OPERATOR

Hamiltonian H as

 h

ðA1:5Þ

written as

j

Then, calculation of the time derivative of  results in d

dtðtÞ ¼

HðtÞðtÞ
ðtÞHðtÞ

ih

ih½HðtÞ, 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

and an interaction Hamiltonian Hi, i.e.,

converting (t) into a density operator in the interaction picture:

IðtÞ ¼ U0ðtÞyðtÞU0ðtÞ, U0ðtÞ ¼exp
iH0t

 h

ðA1:9Þ

transforms the equation of motion into that in the interaction picture: d=dtIðtÞ ¼ 1

En¼hh!nbe the energy eigenvalues of jni; then the density matrix elements

Inn 0 and nn0 are correlated by

In the interaction picture, the expectation value of A is given by hAi ¼Tr  IðtÞAIðtÞ