Hermite function

Hermite Functions. A differential equation closely related to Her mite s equation is... 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

This is the important property of the Gaussian function, namely that its transform has the identical functional form as the original. This is called the property of self-reciprocity. All its derivatives also share the same property. The derivatives of the Gaussian function produce the well known Hermite functions. 

The absorption spectrum consists of sequences of transitions from v" = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

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

Collocation using piecewise bicubic Hermite functions... 

The use of piecewise bicubic Hermite functions in collocation schemes for the solution of elliptic partial differential equations has been described by Prenter (13,17) a short outline is presented here. 

The application of the above method is facilitated by the definition of suitable notation and the use of some simple subroutines to produce the cubic Hermite functions and their derivatives. These are described in detail elsewhere (20), the approach used imitates that of Villadsen and Stewart (10). 

The Franck-Condon factor is given by the squared overlap integral of displaced harmonic oscillator functions (Hermite functions). It can be related [154, p. 113] to the so-called Huang-Rhys parameter (or factor) S according to... 

Certain special cases of (45) arise sufficiently often that it is useful to introduce more concise notations for them. In particular, a Hermite function on center P is denoted by... 

Alternatively, the McMurchie-Davidson scheme [133] can be used for the computation of the spatial part of the matrix elements of Hpy (see [106]). The central concept of this method is to expand the product of two Gaussians (the so-called overlap distribution) in terms of Hermite functions according to... 

In order to determine the normalization constant the ortho-normalization property will be used along the Hermite functions property ... 

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... 

Use again the functions x" to construct (i) polynomials orthonormal in the interval (0, ) with weight factor e" and (ii) polynomials orthonormal in (-00, +oo) with weight factor These are the polynomials of Laguerre and Hermite respectively. Now define the Laguerre and Hermite functions, which are orthonormal in the usual sense but for the infinite intervals. 

Also, generalizations of the H x) to nonintegral complex n exist, called the Hermite functions. The lowest-degree Hermite polynomials are listed in Table 6.7. 

On the other hand, in the limit as a tends to infinity, we may in (9.11.16) integrate over the fiiU set of real numbers and the Rys polynomials may then be related to the Hermite functions as... 

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