Hermite orthogonal functions

The most useful sets of orthogonal functions for our purposes are the wave functions belonging to a given wave equation. In preceding chapters we have shown that the solutions of certain wave equations form sets of normalized orthogonal functions, such as for example the Hermite orthogonal functions which... 

On application of this equation it is found that the momentum wave functions for the harmonic oscillator have the same form (Hermite orthogonal functions) as the coordinate wave functions (Prob. 64-1), whereas those for the hydrogen atom afe quite different.1... 

The solutions of the harmonic oscillator equation, Eq. (6), Sec. 3-1, have been described in many places. They are called the Hermite orthogonal functions and are of the form... 

The orthogonal functions s (( 0x,(ay) are given in terms of finite series of products of Hermite polynomials Hn [51] in the components of the angular velocity as... 

To obtain the orthogonality and normalization relations for the Hermite polynomials, we multiply together the generating functions g(, 5) and g( , t), both obtained from equation (D.l), and the factor e and then integrate over ... 

Here, the permutations of j, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). 

The coefficients c in this probability distribution are referred to as the quasimoments of the distribution. Because of the orthogonality of Hermite polynomials, the quasimoments of a function are obtained by integration of the product of the function and the related Hermite polynomial over all space. For the one-dimensional case,... 

The choice of the specific orthogonal polynomial is determined by the convergence. If the signal to be approximated is a bell-shaped function, it is evident to use a polynomial derived from the Gauss function, i.e. one of the so-called classical polynomials, the Hermite polynomial. Widely used is the Chebyschev polynomial one of the special features of this polynomial is that the error will be spread evenly over the whole interval. 

Two major forms of the OCFE procedure are common and differ only in the trial functions used. One uses the Lagrangian functions and adds conditions to make the first derivatives continuous across the element boundaries, and the other uses Hermite polynomials, which automatically have continuous first derivatives between elements. Difficulties in the numerical integration of the resulting system of equations occur with the use of both types of trial functions, and personal preference must then dictate which is to be used. The final equations that need to be integrated after application of the OCFE method in the axial dimension to the reactor equations (radial collocation is performed using simple orthogonal collocation) can be expressed in the form... 

Solution of Packed Bed Heat-Exchanger Models by Orthogonal Collocation Using Piecewise Cubic Hermite Functions... 

Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonal properties and known recurrence relations [51] for the associated Legendre functions Pf cosi ) and the Hermite polynomials H (z) then yields the infinite hierarchy of differential recurrence relations for the clnm(t) governing the orientational relaxation of the system, namely,... 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

However, for finding the associate eigen-ftinetions, i.e., the wave-functions of the harmonie oscillator, one needs to solve the Hermite s orthogonal equation above yet there is very interesting that it may be further simplified by the substitution(s)... 

Obviously any basis set method is heavily reliant on the choice of appropriate expansion functions. Conventional vibrational basis set have usually been constructed from products of one-dimensional expansions of orthogonal polynomials. In particular Hermite or associated Laguerre... 

The idea behind the DVR method [8-11] is to use a representation in terms of localized functions obtained by transformation from a global basis [12], Usually, bases constructed from orthogonal polynomials, noted F x), which are solution of one dimensional problems such as the particle in a box (Chebyshev polynomials) or the harmonic oscillator (Hermite polynomials), are used. These polynomial bases verify the general relationship... 

Recognize the polynomials as being Hermite polynomials, and utilize some of the known properties of these functions to establish orthogonality and normalization constants for the wavefunctions. 

Dixon, A.G, Solution of Packed-Bed Heat-Exchanger Models by Orthogonal Collocation using Piecewise Cubic Hermite Functions (MRC Technical Summary Report 2116, Mathematics Research Center, Univ. of Wisconsin-Madison, 1980). 

Cameron RH, Martin WT (1947) The orthogonal development of nonlinear functionals in series of Fouiier-Hermite functionals. Ann Math 48 385—392 Cramer H (1966) On the intersectirais between the trajectories of a normal stationary stochastic process and a high level. Ark Math 6 337-349 Desai A, Sarkar S (2010) Analysis of a nonlinear aero-elastic system with parametric uncertainties using polynomial chaos expansion. Math Probl Eng, pages Article ID 379472. doi 10.1155/2010/379472 Evans M, Swartz T (2000) Approximating integrals via Monte Carlo and deterministic methods. Oxford University Press, Oxford... 

The Hermite polynomials of degree n, H x), that appear in the HO functions in Cartesian form fulfil the orthogonality relation [11,12]... 

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