Schrodinger equation generalized model

Modem theoretical treatments of defects in semiconductors usually begin with an approximate solution of the Schrodinger equation appropriate to an approximate model of the defect and its environment (Pantelides, 1978 Bachelet, 1986). Both classes of approximation are described in the following subsection as they pertain to the computational studies addressed in this Chapter. If it were not necessary to make approximations, the computational simulation would faithfully reproduce the experimental result. This would be ideal, but unfortunately, it is not possible. As a consequence, contact with experiment is not always so conclusive or satisfying. A successful theory, however, may still extract from the computational results the important essential features that lead to simple and general models for the fundamental phenomena. 

The initial purpose of pioneer quantum mechanics was to provide the theoretical framework to account for the structure of hydrogen and the nuclear model of atoms in general. The final result, a quantum theory of atomic structure can be discussed in terms of the time-independent Schrodinger equation, in its most general form... 

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

Ab Initio Models. The general term used to describe methods seeking approximate solutions to the many-electron Schrodinger Equation, but which do not involve empirical parameters. Ab initio models include Hartree-Fock Models, Moller-Plesset Models and Density Functional Models. 

Coming back to quantum mechanical continuum models, in the most general sense we now seek to solve the non-linear Schrodinger equation... 

Equations (3.23) and (3.24) are valid also for a model space containing several unperturbed energies, e.g. several atomic configurations. These equations will form the basis for our many-body treatment. The generalized Bloch equation is exact and completely equivalent to the Schrodinger equation for the states considered. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

Two general groups of methodologies are used to solve the Schrodinger equation in combination with cluster models, the Hartree-Fock (HF) approach and related methods to include correlation effects like Mpller-Plesset perturbation theory (MP2) or configuration interaction (Cl) [58,59] and the Density Functional Theory (DFT) approach [59,60]. 

The concept can be illustrated with a simple one-dimensional problem, in which a spherically symmetric potential well is surrounded by a barrier (see Fig. 4). One might consider this problem as a model for dissociation of a diatomic molecule. According to the general theory, the spectrum is discrete for < 0 and continuous for E > 0. Let us now look for unbound solutions, x( ) = of fhe time-independent Schrodinger equation which behave like exp(ifci ) at large R. The desired wave functions have the form... 

We illustrate this procedure in detail for a simplified model in the next section so that you will see how energy levels and wave functions are obtained. We also use the model problem to illustrate important general features of quantum mechanics including restrictions imposed on the form of the wave function by the Schrodinger equation and its physical interpretation. 

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