Separation of Many-Particle Systems

Abstract

The chapter illustrates a number of steps that are necessary to reduce the many-particle problem to a tractable form. The analysis starts from a system of interacting electrons and nuclei; such a system is not made of identical particles and its Hamiltonian operator is not necessarily separable. Besides that, the number of particles that are present in the typical volume of, e.g., a solid-state device is so huge that the solution of the Schrödinger equation of such a system in the original form is a hopeless task. The first step consists in the application of the adiabatic approximation, by which the system made of the electrons is separated from that of the nuclei. The way in which such a separation is accomplished has the inconvenience that the nuclei are kept fixed in the equilibrium positions; this approximation is too strong, because it prevents the exchange of energy and momentum between the two systems from occurring: in fact, it is used provisionally and is removed at a later stage. The next step deals with the electron system which, despite the separation from the nuclei, is still too complicate to be dealt with directly; using the Ritz method, the Schrödinger equation for the electron system is separated into single-particle equations, in which each electron is subjected to the average field of the others. This step yields the Hartree equations and greatly simplifies the problem; in fact, the equations, besides being separated, are also identical to each other, so that the set of eigenvalues and eigenfunction obtained from one of them is applicable to all electrons. The Hartree equations do not comply with the exclusion principle, which must necessarily be fulfilled because the system under consideration is made of identical particles; a further modification, yielding the Hartree-Fock equations, provides the wave function with the expected antisymmetry property. Finally, the system of nuclei is taken again into consideration to the purpose of eliminating the simplification that the nuclei are fixed in the equilibrium positions: considering the fact that the nuclei are strongly bound together, so that their displacement from the equilibrium position is small, the nuclei are treated as a system of linear harmonic oscillators. In this way, the interaction between an electron and the nuclei is described (in a later chapter) using the quantum-mechanical, first-order perturbation theory applied to the two-particle collision of an electron with a phonon.