Periodic functions, waves

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

Based on the same underlying principles as the molecular-based quantum methods, solid-state DFT represents the bulk material using periodic boundary conditions. The imposition of these boundary conditions means that it becomes more efficient to expand the electron density in periodic functions such as plane waves, rather than atom-based functions as in the molecular case. The efficiency of the calculations is further enhanced by the use of pseudo-potentials to represent the core electrons and to make the changes in the electron density... 

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

The number of times that an event or a periodic function repeats during a unit value of an independent variable, usually time. 2. For any periodic motion, the frequency, symbolized by r, is the number of repetitions (e.g., number of revolutions or cycles or oscillations of an electromagnetic wave) of some process occurring in a unit period of time. The SI unit for frequency, reciprocal second, is often referred to as a hertz (Hz). However, the term hertz should only be used in reference to cycles per second and not for radial (circular) frequency or angular velocity symbolized by (p (= 2777 ) and having SI units of rad-s h... 

Spatially localized functions are an extremely useful framework for thinking about the quantum chemistry of isolated molecules because the wave functions of isolated molecules really do decay to zero far away from the molecule. But what if we are interested in a bulk material such as the atoms in solid silicon or the atoms beneath the surface of a metal catalyst We could still use spatially localized functions to describe each atom and add up these functions to describe the overall material, but this is certainly not the only way forward. A useful alternative is to use periodic functions to describe the wave functions or electron densities. Figure 1.2 shows a simple example of this idea by plotting... 

Theoretically, the perturbation can be an arbitrary, more or less complex, function of time. However, only a limited number of functions have been shown to be of practical importance. These are known as step, pulse, double-step, double-pulse, periodic square wave and periodic sine wave. A survey of the most common techniques is found in Table 2. 

A simple wave, like that of visible light or X rays, can be described by a periodic function, for instance, an equation of the form... 

Because the electron density we seek is a complicated periodic function, it can be described as a Fourier series. Do the many structure-factor equations, each a sum of wave equations describing one reflection in the diffraction pattern, have any connection with the Fourier series that describes the electron density As mentioned earlier, each structure-factor equation can be written as a sum in which each term describes diffraction from one atom in the unit cell. But this is only one of many ways to write a structure-factor equation. Another way is to imagine dividing the electron density in the unit cell into many small volume elements by inserting planes parallel to the cell edges (Fig. 2.16). 

Recall from Chapter 2, Section VI.A, that waves are described by periodic functions, and that simple wave equations can be written in the form... 

According to Fourier theory, any complicated periodic function can be approximated by this series, by putting the proper values of h, Fh, and ah in each term. Think of the cosine terms as basic wave forms that can be used to build any other waveform. Also according to Fourier theory, we can use the sine function or, for that matter, any periodic function in the same way as the basic wave for building any other periodic function. 

The Fourier series that the crystallographer seeks is p(x,y,z), the three-dimensional electron density of the molecules under study. This function is a wave equation or periodic function because it repeats itself in every unit cell. The waves described in the preceeding equations are one-dimensional they represent a numerical value/(x) that varies in one direction, along the x-axis. How do we write the equations of two-dimensional and three-dimensional waves First, what do the graphs of such waves look like ... 

The indices hkl of the reflection give the three frequencies necessary to describe the Fourier term as a simple wave in three dimensions. Recall from Chapter 2, Section VI.B, that any periodic function can be approximated by a Fourier series, and that the approximation improves as more terms are added to the series (see Fig. 2.14). The low-frequency terms in Eq. (5.18) determine gross features of the periodic function p(x,y,z), whereas the high-frequency terms improve the approximation by filling in fine details. You can also see in Eq. (5.18) that the low-frequency terms in the Fourier series that describes our desired function p(x,y,z) are given by reflections with low indices, that is, by reflections near the center of the diffraction pattern (Fig. 5.2). 

The Fourier expression in Eq. (4.3) has been simplified to clarify the concept behind harmonic frequency components in a nonlinear periodic function. For the purist, the following more precise expression is offered. For a periodic voltage wave with fundamental frequency of to = 2jif,... 

SWV is one of the most popular electrochemical techniques, mainly in electroanalysis, due to its great sensitivity, discrimination of background currents, and short experimental times [6, 9, 12]. It was introduced by Baker [13-15] and later developed by the Osteryoungs and coworkers by using a combination of a staircase potential modulation and a periodic square wave potential function [16-19]. In the SWV technique, the potential sequence can be described as (see Scheme 7.3 and [9]) ... 

The amplitude of the composite wave is no longer a periodic function because of the factor defined by the second cosine term. The total wave packet moves along without change in shape providing the component waves have the same velocity. The only instance where this is known to apply is for electromagnetic photons in vacuum. In all other cases, for instance electrons, the velocity depends on wavelength (and k). 

Molecular structure is theoretically intimately related to electron-density distribution functions. In quantum-chemical analysis this density is synthesized as a molecular orbital, by a linear combination of real atomic orbitals, and minimized as a function of total energy. Crystallographically the unit cell density is represented by a Fourier sum over periodic electron wave functions... 

On a plot of Z/N vs Z for all stable nuclides the field of stability is outlined very well by a profile, defined by the special points of the periodic table derived from 4. Furthermore, hem lines that divide the 264 nuclides into 11 groups of 24 intersect the convergence line, Z/N = r, at most of the points that define the periodic function. If the hem lines are extended to intersect the line Z/N = 0.58, a different set of points are projected out and found to match the periodicity, derived from the wave-mechanical model. 

Have the correct form to be a solution of Equation 1.8. As a result, the Bloch theorem affirms that the solution to the Schrodinger equation may be a plane wave multiplied by a periodic function, that is [5,6],... 

Virtually all periodic functions of time can be represented by a Fourier series. Thus, a general periodic voltage can be thought of as a series combination of sinusoidal voltages. For instance, a square wave voltage is defined by a series [ii, iii] ... 

The most general expression for a periodic function is the plane wave, e , in which d is a parameter equal to the vector dot product k R. Hence, the wave function, ip(r), of Eq. 5.24 is of the form ... 

Another experimental approach is to modulate the potential of the electrode by a periodic function usually a sine or square wave and observe the synchronous spectral response by means of a phase-sensitive detector (Aylmer-... 

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