Multiply periodic functions

Before we can proceed to apply our results to systems of several degrees of freedom we must introduce the conception of multiply periodic functions, and examine some of their properties. 

We will now prove that the variables wk, 3k, introduced in this way, have similar properties to w and J for one degree of freedom, namely, that the qks are multiply periodic functions of the wh s with the fundamental period system... 

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. 

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],... 

Here we shall l>ri( iiy cnnsidor the mcichanics of multiply periodic motions a.nd the corres]lending ( ua,ntum conditions. According to Hamilton, the motion of a, syst( . m is (h serihed completely by stating the energy as a function of t m co-ordinates and the momenta the so-called Hamiltonian function H q, . , . , p2 0- (1 ... 

PERIODIC AND MULTIPLY PERIODIC MOTIONS 81 Written as a function of t ... 

The condition (B) introduces a fresh restriction. Considered as functions of the time, the wk s, as well as the wk s, must be linear from (6) it follows that tpk s are likewise linear functions of the time, but if they vary at all with the time they must be multiply periodic, as has just be shown they must therefore be constant. This means, however, that, in the exponent of the Fourier series, the only combinations of the wk which can occur are such as make... 

Since the functions O. in (5) depend only on the relative positions of the particles of the system with respect to one another and to the fixed axis, these relative positions will be determined also by wx. . . wf 1 while fixes the absolute position of the system. According to (6), 2irwf can be regarded as the mean value of the azimuth of the arbitrarily selected particle of the system over the motions of the relative angle variables w,. . . wf v The motions can therefore be considered as a multiply periodic relative one on which is superposed a uniform precession about the fixed axis. If H, regarded as a function of the Jk, does not depend on this precession is zero the system is then degenerate. 

A multiply periodic degenerate system may frequently be changed into a non-degenerate one by means of slight influences or variation of the conditions. We shall consider, in particular, the simple case where the Hamiltonian function involves a parameter A and the system is degenerate for A=0. We imagine the energy function H expanded in powers of A for sufficiently small values of A we can break off this series after the term linear in A and write... 

The only solutions which arc of importance from the point of view of the quantum theory are those of a multiply periodic nature. We assume, therefore, that the perturbed motion has a principal function of the form... 

The standard recipe for constructing such a Lagrangian, L, is to regard the state equations as side conditions to be included in the functional by means of Lagrange multipliers (52). Since the state equations apply at all times within the control period, the Lagrange multiplier is a generalized multiplier, a function of time, and the addition of all the side conditions becomes an integral over the time. Thus, we add to the characteristic to be made least, the side conditions in the form... 

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.)... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

To compute this value requires survivor functions for relevant time periods as well as values for v(f). The survivor functions were computed from male and female life tables available for 1982 and 1997 from Statistics Canada. Economic values for additional life years were computed based on Murphy and Topel (2005) and converted to Canadian dollars using the average per capita ratios of Canadian to U.S. income for 1994-2003. These income-adjusted life year values were then multiplied by 1.267 the purchasing power parity (PPP) rate between Canada and the United States in 2004, expressed in 2004 dollars. 

Let us instead turn our attention to the consequences of sampling the function at evenly spaced intervals of x. Consider the A function and its transform, a sine function squared, shown in Fig. 3. Suppose that we wish to compute that transform numerically. First, let us replicate the A by convolving it with a low-frequency III function. Now multiply it by a high-frequency III function to simulate sampling. We see a periodically replicated and sampled A. The value of each sample is represented as the scaled area under a Dirac <5 function. 

Fig. 31 Period of kink lattice A = 2Tt/q multiplied by the velocity v of the quench interface as a function of v for = 1 (solid circles with a solid line as a guide to the eye) v = 1.622 from (71). The dashed line corresponds to (72) [126]...

The autocorrelation (or correlation) function is obtained by multiplying each y (f) by y (t — t°), where t° is a time delay, and summing the products over all points [43]. Examination of the sum plotted as a function of t° reveals the level of dependency of data points on their neighbors. The correlation time is the value of t° for which the value of the correlation function falls to exp (—1). When the correlation function falls abruptly to zero, that indicates that the data are without a deterministic component a slow fall to zero is a sign of stochastic or deterministic behavior when the data slowly drop to zero and show periodic behavior, then the data are highly correlated and are either periodic or chaotic in nature [37,43]. 

Figure 14. Plot, as a function of the absolute energy E, of the frequencies of the classical periodic orbits belonging to various famihes. The frequencies of the [y]-type POs are divided by two, whereas the frequencies of the [RIA] family are multiplied by two. The energy scale is shifted to higher energies by 0.23 eV— that is, the zero-point energy of the OH stretch mode. See the text for more details.

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