Wave equation three dimensions

The simplest type of wave in three dimensions is the plane wave in which, by definition, the disturbance at any instant of time has the same value at all points in any given plane that is perpendicular to the direction of propagation. Such a wave traveling along the x direction is described by the equation... 

It is possible to generalize equations 1.27 and 1.28 to deal with plane waves in three dimensions. We then obtain... 

Such a theory is embodied in the wave equation of Schrodinger. The propagation of a wave in three dimensions is represented by the expression g ... 

Up to this point we have considered particle motion only in the jc-direetion. The generalization of Schrodinger wave mechanics to three dimensions is straightforward. In this section we summarize the basic ideas and equations of wave mechanics as expressed in their three-dimensional form. 

In principle, the calculation of bonding in two or three dimensions follows the same scheme as outlined for the chain extended in one dimension. Instead of one lattice constant a, two or three lattice constants a, b and c have to be considered, and instead of one sequential number k, two or three numbers kx, ky and k- are needed. The triplet of numbers k = (kx, ky, kz) is called wave vector. This term expresses the relation with the momentum of the electron. The momentum has vectorial character, its direction coincides with the direction of k the magnitudes of both are related by the de Broglie relation [equation (10.5)]. In the directions a, b and c the components of k run from 0 to nja, njb and n/c, respectively. As the direction of motion and the momentum of an electron can be reversed, we also allow for negative values of kx, ky and kz, with values that run from 0 to —nja etc. However, for the calculation of the energy states the positive values are sufficient, since according to equation (10.4) the energy of a wave function is E(k) = E(—k). 

G is the reduced Green function of the Schrodinger equation and B = (Us)-Action of the operator O2 on the wave function can be checked not to produce functions more singular than G2 or c2. Therefore, in contrast to the second iteration of the original perturbation, Eq.(12), that of the operator 02 delivers a result which is finite in three dimensions. 

Another method recently developed for manipulating small particles uses the forces created by a two- or three-dimensional sound field that is excited by a vibrating plate, the surfaces of which move sinusoidally and emit an acoustic wave into a layer of fluid. Such a wave is reflected by a rigid surface and generates a standing sound field in the fluid, the forces of which act on particles by displacing them in one, two or three dimensions. In this way, particles of sizes between one and several hundred microns can be simultaneously manipulated in a contactless manner. Equations describing this behaviour have been reported [63]. 

The particle in a box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the three-dimensional Schrodinger equations. The same methods used for the one-dimensional box can be expanded to three dimensions for atoms. These orbital equations include three... 

Lane equations Equations that, like the Bragg equation, express the conditions for diffraction in terms of the path difference of scattered waves. Laue considered the path length differences of waves that are diffracted by two atoms one lattice translation apart. These path differences must be an integral number of wavelengths for diffraction (that is, reinforcement) to occur. This condition must be true simultaneously in all three dimensions. 

Now, electron waves arc described by a wave equation of the same general form as that for string waves. The wave functions that are acceptable solutions to this equation again give the amplitude, <, this time as a function, not of a single coordinate, but of the three coordinates necessary to describe motion in three dimensions. It is these electron wave functions that we call orbitals. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

The probabilistic interpretation requires that any function must meet three mathematical conditions before it can be used as a wave function. The next section illustrates how these conditions are extremely helpful in solving the Schrodinger equation. To keep the equations simple, we will state these conditions for systems moving in only one dimension. All the conditions extend immediately to three dimensions when proper coordinates and notation are used. (You should read Appendix A6, which reviews probability concepts and language, before proceeding further with this chapter.)... 

We present quantitative, computer-generated plots of the solutions to the particle-in-a-box models in two and three dimensions and use these examples to introduce contour plots and three-dimensional isosurfaces as tools for visual representation of wave functions. We show our students how to obtain physical insight into quantum behavior from these plots without relying on equations. In the succeeding chapters we expect them to use this skill repeatedly to interpret quantitative plots for more complex cases. 

In crystalline semiconductors, it is relatively easy to understand the formation of gaps in energy states of electrons and hence of the valence and conduction bands using band theory (see Ziman, 1972). Band structure arises as a consequence of the translational periodicity in the crystalline materials. For a typical crystalline material which is a periodic array of atoms in three dimensions, the crystal hamiltonian is represented by a periodic array of potential wells, v(r), and therefore is of the form, 7/crystai = ip l2m) + v(r), where the first term p l2rri) represents the kinetic energy. It imposes the eondition that the electron wave functions, which are solutions to the hamiltonian equation, H V i = E, Y, are of the form... 

This is a wave equation in three-dimensional space, whose solutions we shall investigate later. It may make matters easier for the reader if we begin with corresponding problems in one and two dimensions and for the sake of perspicuity we shall take our examples from classical mechanics (acoustics). 

The extension of the nonstandard finite-difference method in three dimensions requires the definition of three Laplacian operators for the calculation of spatial derivatives in the wave equation. Keeping the notation of the 2-D case and assuming a uniform grid, L2[.] is now depicted by... 

THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES IN THREE DIMENSIONS... 

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