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. . Appendix 1 Outline of Density Matrix Analysis A1 .1 DEFINITION OF DENSITY MATRIX AND EXPECTATION VALUES The density matrix offers an effective. picture, the expectation value of A is given by hAi¼Tr  I ðtÞA I ðtÞ ÈÉ , A I ðtÞ¼U 0 ðtÞ y AU 0 ðtÞðA1 :12 Þ Outline of Density Matrix Analysis 287 Copyright ©