Wave equations Periodic functions

Each reflection is the result of diffraction from complicated objects, the molecules in the unit cell, so the resulting wave is complicated also. Before considering how the computer represents such an intricate wave, let us consider mathematical descriptions of the simplest waves. 

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 

The constant h in a simple wave equation specifies the frequency or wavelength of the wave. For example, the wave/(x) = cos 2tt(5x) has five times the frequency (or one-fifth the wavelength) of the wavef(x) = cos 2 irx (compare c with a in Fig. 2.13). (In the wave equations used in this book, h takes on integral values only.) 

Finally, the constant a specifies the phase of the wave, that is, the position of the wave with respect to the origin of the coordinate system on which the wave is plotted. For example, the position of the wave/(x) = cos 2ir(x + V4) is shifted by one-quarter of 2tt radians (or one-quarter of a wavelength, or 90°) from the position of the wave/(x) = cos 2ttx (compare Fig. 2.13d with Fig. 2.13a). Because the wave is repetitive, with a repeat distance of one wavelength or 2tt radians, a phase of V4 is the same as a phase of 1l/4, or 2l/4, or 31/4, and so on. In radians, a phase of 0 is the same as a phase of 2tt, or 4tt, or 6-tt, and so on. 

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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. 

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. 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

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... 

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 ... 

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],... 

Any well-behaved periodic function (such as a spectrum) can be represented by a Fourier series of sine and cosine waves of varying amplitudes and harmonically related frequencies. A typical NIR spectrum may be defined mathematically by a series of sines and cosines in the following equation ... 

We have thus found the formalism, according to which any mechanical problem can be treated. What we have to do is to find the one-valued and finite solutions of the wave equation for the problem. If in particular we wish to find the stationary solutions, i.e. those in which the wave function consists of an amplitude function independent of the time and a factor periodic in the time (standing vibrations), we make the assumption that ijj involves the time only in the form of the factor Schrodinger s equation, we find... 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

This solution describes periodic waves traveling in opposite directions along the front. The as yet undetermined complex amplitudes Ri and Si of these waves are bounded functions of the slow variables and Hi can grow linearly in time on the slow time scales. Solvability conditions which are described below will provide equations for these amplitudes. 

What is the form of Bloch functions Equation [8] implies that a Bloch function can be written as the product of a plane wave and a periodic function u(r k) with the same periodicity of the lattice ... 

That is, asymptotically the pitch of the spiral is unity. (See DeSimone, Beil and Scriven, 1973, for a discussion of the geometry of spiral waves in the context of reaction-diffusion equations.) In real space, r = p/k, the wavelength of the spiral pattern tends asymptotically to 2n/k, which fixes the length scale introduced earlier. Finally, as we go around a large circle from (p, (p)) back again, we should see a 27T-periodic function that is,... 

The black curve in Figure 6-4a shows the application of Equation 6-6 to two waves of identical frequency but somewhat different amplitude and phase angle. The resultant is a periodic function with the same frequency but larger amplitude than cither of the component waves. Figure 6-4b differs from 6-4a in that the phase difference is greater here, the resultant amplitude is smaller than the amplitudes of the component waves. A maximum amplitude occurs when the... 

Third, careful comparison of Eqs. (15-15) and (15-4) shows that they are not exactly the same. Equation (15-15) instructs us to find a periodic function Uj (p) and multiply it by exp(z j p) at every point in p. Think of the sine or cosine related to the exponential and imagine what this means as we multiply it times a 2p j on some carbon. Say the cosine is increasing in value as it sweeps clockwise past the carbon nucleus at 2 00 on a clock face. This produces a product of cosine and 2p j that is unbalanced—smaller toward 1 00 than toward 3 00, because the cosine wave modulates Ujip) everywhere. But Eq. (15-4) is different. It instructs us to take the value of the cosine at 2 00 and simply multiply the 2p r AO on that atom by that number. The 2p AO is not caused to become unbalanced. Only its size in the MO is determined by the cosine. Equation (15 ) is called a Bloch sum. Such sums are approximations to Bloch functions, but any errors inherent in this form are likely to be quite small if the basis functions and unit cell are sensibly chosen. (Using Bloch sums is similar in spirit to the familiar procedure of approximating a molecular wavefunction as a linear combination of basis functions.)... 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

Figure 2.9 illustrates the approximate dependence of the energy on the wave vector. The picture is very similar to the parabolic form of a free electron (see Eq. (2.23)) however, there are deviations (see the thick lines) as a result of the obstacles we have inserted (a,2a,3a etc.) . We remember that the Schrodinger equation is a wave equation. We expect diffraction effects at the relevant positions in the reciprocal space (k space) marked in Fig. 2.9 . In the case of a small box, it is true that e quadratically depends on a, but there are only a few discrete points. For a large box the function becomes continuous. Since we imagine our periodic soUd as composed (cf. Fig. 2.2) of small boxes forming a large box , we expect a behaviour according to Fig. 2.9.

The traveling wave equation develops thereby a double periodicity on the coordinate x and on time t. One can, for instance, fix a particle coordinate (jc = const.) and consider its displacement as a function of time. Alternatively, one can fix a moment of time (t = const.) and consider particle displacement as a function of coordinates. So, standing on a pier one can take a picture of the surface of the sea at time instant t, or having thrown an object into the sea (i.e., having fixed a coordinate jc), one can check its oscillation in time. Both these cases are given as graphs in Figure 2.18. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

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