Appendix A: Advanced Formalism

Overview

In developing the general matrix model depicted in Fig. 1.6.5b, our starting point was the one-electron Schr dinger equation

(A.1)

describing the time evolution of the one-electron wavefunction ? from which the electron density is obtained by summing ?* ? for different electrons. Although it is a little tricky explaining exactly what one means by different electrons, this procedure is adequate for dealing with simple problems involving coherent interactions where the background remains unaffected by the flow of electrons. But to go beyond such phenomena onto more complex processes involving phase-breaking interactions or strong electron electron interactions it is desirable to use a more comprehensive viewpoint that describes the electrons in terms of field operators c. For non-interacting electrons these second quantized operators obey differential equations

(A.2)

that look much like the one describing the one-electron wavefunction (see Eq. (A.1)). But unlike ?* ?, which can take on any value, the number operator c ⁺ c can only take on one of two values, 0 or 1, thereby reflecting a particulate aspect that is missing from the Schr dinger equation. The two values of c+ c indicate whether a state is full or empty. At equilibrium, the average value of c ⁺ c for a one-electron state with energy ? is given by the corresponding Fermi function c ⁺ c = f ₀( ? ? ). However, this is true only if our channel consists of non-interacting electrons...