Generalized Fourier expansions

For Bessel functions and other types of special functions to be introduced later, it is possible to construct orthonormal sets of basis functions (/ n(x), which satisfy the orthogonality and normalization conditions  

113) is substituted into Eq. (7.112), with the appropriate use of a dummy variable, we obtain 

The last quantity in square brackets has the same effect as the deltafunction 5(x — x ). The relation known as closure 

Generalized Fourier series find extensive application in mathematical physics, particularly quantum mechanics. 

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In both potentials in eqs. (2.27) and (2.28) the bairier towards linearity is given implicitly by the force constant. A more general expression which allows even quite complicated energy functionals to be fitted is a general Fourier expansion. 

When combined with the Fourier expansion of functions, separation of variables is another powerful method of solutions which is particularly useful for systems of finite dimensions. Regardless of boundary conditions, we decompose the solution C(x, t), where the dependence of C on x and t is temporarily emphasized, to the general one-dimensional diffusion equation with constant diffusion coefficient... 

Such a sine expansion is generally made possible with the condition of zero concentration at x=0 by using an odd function for the initial concentration distribution. A simple example is for initial uniform concentration C0 between 0 and X for which we can assume a fictitious concentration — C0 between — X and 0. Using the results of Chapter 2, the Fourier expansion of the boxcar function which is C0 between 0 and X, and 0 at x=0 and x = X is... 

The function that has such a Fourier expansion while satisfying C(0,f)=C(0,. Y)=0 is the boxcar function. We found in Section 2.6.2 that the a coefficients of this function are 0 for even values and 4/nn for odd values of n. The general solution is therefore... 

As we have discussed previously, any function with two-dimensional periodicity can be expanded into two-dimensional Fourier series. If a function has additional symmetry other than translational, then some of the terms in the Fourier expansion vanish, and some nonvanishing Fourier coefficients equal each other. The number of independent parameters is then reduced. In general, the form of a quantity periodic in x and y would be... 

It is common practice to describe torsional rotations around single bonds and those around multiple bonds with the same type of potential function but with very different force constants. The function must be able to describe multiple minima. Generally, a Fourier expansion of the torsional angle with only cosine terms is used (Eq. 2.23),... 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

In this appendix we provide the generalized equations for the mean-field potential and double layer interaction free energy between two surfaces having distinct but periodic nonuniform distributions. These results were taken from Ref. [78]. We denote by yL(s) the charge distribution on the left (L) surface at z = 0 and yR(s) that on the right (R) surface at z — h. As in the text the variable y represents either surface potential, T (s), or surface charge, periodic distribution represented by the Fourier expansions,... 

To use this general solution, it is only necessary to obtain the distribution of the radiative flux on the reactor wall, say by applying a ray tracing method to the solar collector, and to expand the resulting distribution of radiative flux in a Fourier expansion like (48). Then the solution inside the reactor is obtained. 

The solution of the Schrodinger equation for a particle in the random potential, caused by the QW width fluctuations and the alloy disorder is beyond the scope of the present work (much work has been done in this field, see, e.g. (21) and references therein). We can mention only some general properties that (ry) should have (i) it should be localized within some distance L> Lw (ii) it should be smooth and without nodes. As a consequence, its spatial Fourier expansion should contain mainly the components with wavevectors k < 1/L. 

The geometry and height of the barrier can be derived by fitting the observed rotational transitions to a model for the barrier. The simplest possible model for the rotations of a dumbbell molecule is one of planar reorientation about an axis perpendicular to the midpoint of the H-H bond in a potential of twofold symmetry (Figure 6.1). More generally, terms with higher symmetry than twofold may be included in the Fourier expansion of the rotational potential ... 

This Fourier expansion in terms of the one-dimensional modes propagating in the z direction is the most general form for the shape of a deformed cylinder when overhangs are forbidden. In general, the volume of the deformed cylinder is given by... 

Universal. The parameters in the Universal force field (UFF) [14-16] are calculated using general rules based only upon the element, its hybridization, and its connectivity. For this reason, the UFF has broad applicability but is inherently less accurate than extensively parameterized force fields such as COMPASS. Bond-stretching terms in the UFF are either harmonic or Morse functions. The anglebending and torsion terms are described by a small cosine Fourier expansion. For nonbonded terms, the LJ 6-12 potential and Coulombic terms are used for steric and electrostatic terms, respectively. 

For example, the Universal Force Field (UFF), i which is a molecular mechanics force field, uses the sum of atom-type-specific, single-bond radii with some corrections for bond order and electronegativity as the reference bond length and derives the force constant for this bond based on a generalization of Badger s rules.Bond angle distortions are described by means of a cosine Fourier expansion ... 

From the analytically point of view, the potential periodicity of the Eq. (3.20) allows the effective writing of the potential form in a general manner, known as Fourier expansion ... 

To this end we use the generalized multipole expansion of the intermolecular potential F[22, 57,58], based on the Fourier transform (FT) of the reciprocal of the interparticle distance (Fig. 5)... 

Flory reviewed in 1969 the development and applications of the rotational isomeric state scheme calculations, which allow, by matrix algebra, the statistical mechanical averaging over the rotational states of chain properties which may be expressed as a vector or tensor quantity associated with the chain bonds, and estimations of the probabilities of chosen conformational sequences. The methods were generalized and schemes for reducing the dimensions of certain generator matrices were presented in 1974, when comparisons were also made with an alternative Fourier expansion method, currently in use for atactic polypropylene. These techniques have greatly contributed to an understanding... 

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