Polynomials, Hermite

The Hermite polynomials // ( ) are defined by means of an infinite series expansion of the generating function g( , ), 

Applying this property with x = s and y = — to the nth-order partial derivative in equation (D.2), we obtain 

Another expression for the Hermite polynomials may be obtained by expanding g(, s) using equation (A.l) 

We note that 77 ( ) is an odd or even polynomial in according to whether n is odd or even and that the coefficient of the highest power of in // ( ) is 2 . 

Expression (D.4) is useful for obtaining the series of Hermite polynomials, the first 

The quantum-mechanical harmonic oscillator satisfies the Schrodinger equation  

To reduce the problem to its essentials, simplify the constants with h = jx = k = 1, or alternatively, replace x by x. Correspondingly, 

We must now solve a second-order differential equation with 

A useful first step is to determine the asymptotic solution to this equation, giving the form V (x) as x oo. For sufficiently large values of x, i, so that the differential equation can be approximated by 

Differentiating this equation (n +1) times using Leibniz s formula (12.29), we obtain 

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An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, 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 Hermite polynomials are well known in science and engineering. 

In practice, even the determination of a fourth-order correlation function requires a large amount of calculation. However, this procedure may be standardized by a graphical method, which performs the integration in Eq. (1.54) by using properties of Hermite polynomials [37], Without going into details, we give the result... 

Heisenberg representation 267 helium see nitrogen-helium system Hermite polynomials 24, 25 Hubbard dependence 8 Hubbard relation... 

To understand the criteria for basis set choice, then, we need consider only the behavior of the primitive integrals. The primitive integrals over the basis functions can be expressed in terms of Hermite polynomials... 

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)... 

Therefore, we may express the eigenvector n) in terms of the Hermite polynomial i7 ( ) by the relation... 

For reference, the Hermite polynomials for = 0 to = 10 are listed in Table 4.1. When needed, higher-order Hermite polynomials are most easily obtained from the recurrence relation (D.5). If only a single Hermite polynomial is wanted and the neighboring polynomials are not available, then equation (D.4) may be used. 

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... 

This recurrence relation may be used to obtain a Hermite polynomial when the two preceding polynomials are Imown. 

With this reeurrenee relation, a Hermite polynomial may be obtained from the preceding polynomial. By applying the relation (D.7) to /f ( ) k times, we have... 

To find the differential equation that is satisfied by the Hermite polynomials, we first differentiate the second recurrence relation (D.6) and then substitute (D.6) with n replaeed by n — 1 to eliminate the first derivative of i ( )... 

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 ... 

The Hermite polynomials Hn ) form an orthogonal set over the range —oo oo with a weighting factor e . If we equate coefficients of stY on each side of equation (D.12), we obtain... 

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

The Hermite polynomials introduced above represent an example of special functions which arise as solutions to various second-order differential equations. After a summary of some of the properties of these polynomials, a brief description of a few others will be presented. The choice is based on their importance in certain problems in physics and chemistry. 

While the Hermite polynomials can be developed with the use of the recursion formula [Eq. (90)], it is more convenient to employ one of their fundamental... 

It is analogous to die generating function for the Hermite polynomials % fEq. (94)], although somewhat mote complicated. It can foe used to obtain die useful recursion relations... 

The matrix elements of x4 can be evaluated with the use of the relation developed in Section 5.5.1 for the Hermite polynomials (See Appendix IX). In the notation employed here Eq. (5-99) becomes... 

Hermite polynomials 104-107 integrals 99-100 matrix methods 172-175 operators 151-153 particle in a box 96-100,122, 309-311... 

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