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 ¼hnjjn 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 TrfAgðA1:4Þ 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 UðtÞð0ÞUðtÞ y ðA1:6Þ Then, calculation of the time derivative of results in d dt ðtÞ¼ HðtÞðtÞÀðtÞHðtÞ ihh ¼ 1 ihh 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 ihh 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 ©