The density matrix offers an effective technique for dealing statistically with a system consisting of many electrons using the quantum theory for an electron. Amixed state consisting of a statistical distribution of various quantum states can be specified by a set of probabilities pj with which the electron is found in a quantum state j ji. | Appendix 1 Outline of Density Matrix Analysis 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 pj with which the electron is found in a quantum state I Vj . The density operator p is defined by P X I Vj Pj h Vj I J The probability satisfies 0 pj 1 and Pj pj 1. The operator p is a Hermite operator and the matrix description of p is called the density matrix. Using a system of eigenstate In the elements of the density matrix are given by Pnn p X n VjiPj hVj In j The diagonal elements of the density matrix Pnn X Pj Ih I Vj 2 j give the probability with which the system belongs to the eigenstate In . The off-diagonal elements represents the correlation of states In and In . The expectation value A for a physical quantity represented by an operator A being the weighted average of the expectation values for states I Vj can be written as A X Pj h Vj IAI Vj j X Pj h Vj In nIAIn n I Vj jnn Copyright 2004 Marcel Dekker Inc. E pnlnAnn. nn WA Since A can be expressed by A and p only it is possible to calculate the value of the macroscopic observable A without knowing j and pj provided that p is obtained. EQUATION OF MOTION FOR THE DENSITY OPERATOR The time variation of a state can be written by using the system Hamiltonian H as t U t j 0 U t ex -H and if the time dependence of pj is omitted the time variation of p can be written as p t X U t j 0 pj j 0 U t y U t p 0 U t y j Then calculation of the time derivative of p results in d_ H t p t p t H t dt 1 ih 1 H t p t in Thus the equation of motion for p is described by using the commutation relation between H and p. When the initial state p 0 is given by a matrix representation based on an appropriate .