Harmonic oscillator Hermite polynomials

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... 

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

Some of the Hermite polynomials and the corresponding harmonic-oscillator wave functions are presented in Thble 1. The importance of the parity of these functions under the inversion operation, cannot be overemphasized. 

This evolution of a complex set of numbers from something very simple is rather like a recursion rule. For example, the wave function for a harmonic oscillator contains the Hermite polynomial, Hb(t/), which satisfies the recursion relation ... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, harmonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the trajectory, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

The polynomials defined here are different from the Hermite polynomials which occur in the solutions of the Schrodinger equation for the harmonic oscillator. 

Now consider a one-dimensional harmonic oscillator of charge q. We must evaluate q(m x n). The harmonic-oscillator wave functions are given by (1.133). Using (1.133) and the Hermite-polynomial identity (1.138) with z = a 2x, we have... 

They are related to the Hermite polynomials HN(x), which are the solutions to the Schrodingen equation for a harmonic oscilator by... 

A comparison of equation (G.17) with (D.10) shows that the solutions a(f) are the Hermite polynomials, whose properties are discussed in Appendix D. Thus, the functions [Pg.323]

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. 

Hermite polynomials form part of the solution for the quantum mechanical treatment of the harmonic oscillator. One of these polynomials is defined by... 

To show that the functions we have defined above are the same as those used in the solution of the harmonic oscillator problem, we look for the differential equation satisfied by // ( ). It is first convenient to derive certain relations between successive Hermite polynomials and their derivatives. We note that since S = its partial derivative with respect... 

The harmonic potential is a model of last resort for diatomic molecules. Its behavior at R = 0 and R = oo is unphysical, as is the sign of ae. Exact diatomic molecule vibrational wavefunctions for levels above v = 0, except for their number of nodes, differ from harmonic oscillator eigenfunctions (Hermite polynomials with an exponential factor) in that they are not symmetric about Re and, increasingly so at high v, are skewed toward the outer turning point. 

Such a possible feature can be found, as an example, within a typical set of solutions of the Schrodinger equation. The harmonic oscillator provides an obvious particular case of such an EH space. It is well known that harmonic oscillator solutions constitute the set of Hermite polynomials [73], weighted by a gaussian function [65]. These polynomials can be considered related to the GTO basis functions most widely used in contemporary Quantum Chemistry. First derivatives of Hermite polynomials are always well defined, producing another polynomial of the same kind. 

The polynomial factors in the harmonic-oscillator wave functions are well known in mathematics and are called Hermite polynomials, after a French mathematician. (See Problem 4.19.)... 

The harmonic-oscillator wave functions are given by a Hermite polynomial times an exponential factor (Problem 4.19b). By virtue of the expansion postulate, any well-behaved function/(x) can be expanded as a linear combination of harmonic-oscillator energy eigenfunctions ... 

In the q-coordinate system, the vibrational normal coordinates, the SA atom-dimensional Schrodinger equation can be separated into SA atom one-dimensional Schrodinger equations, which are just in the form of a standard harmonic oscillator, with the solutions being Hermite polynomials in the q-coordinates. The eigenvectors of the F G matrix are the (mass-weighted) vibrational normal coordinates, and the eigenvalues ( are related to the vibrational frequencies as shown in eq. (16.42) (analogous to eq. (13.31)). 

Back to the quantum harmonic oscillator, the Hermite polynomials helps in writing its eigen-functions as ... 

Using the properties of harmonic oscillator wave functions (the Hermite polynomials) that... 

HERMITE POLYNOMIALS AND SOME INTEGRALS INVOLVING THE HARMONIC OSCILLATOR WAVE FUNCTIONS... 

The following equations give several of the Hermite polynomials, H z), z = y Q, in terms of which the harmonic oscillator wave functions are expressed in Eq. (1), Sec. 3-3. Additional explicit forms may be readily evaluated by use of the recursion formula, Eq. (4), Sec. 3-3. 

At very large hyperradii the surface functions are highly localized into small regions of the 0,x) domain. Also, for molecular problems the vibrational motion of the surface hinc-tion can be approximated as a harmonic oscillator (a Gaussian centered at the equilbrium position cf the isolated diatom times a Hermite polynomial). Hence, it becomes difficult to represent this exponential function accurately with a polynomial basis. By factoring out this non-polynomial behavior we can obtain accurate solutions with less computational expense. To do this we substitute = gi into Eq. (55) giving... 

TABLE 8.1 The harmonic oscillator wavefunctions. The Hermite polynomials and normalization constants are given for the first six harmonic oscillator wavefunctions rjvly) =, where the... 

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