请输入您要查询的字词:

 

单词 HermitePolynomials
释义

Hermite polynomials


The polynomialPlanetmathPlanetmath solutions of the Hermite differential equationMathworldPlanetmath, with n a non-negative integer, are usually normed so that the highest degree (http://planetmath.org/PolynomialRing) is (2z)n and called the Hermite polynomialsDlmfDlmfDlmfMathworldPlanetmath Hn(z).  The Hermite polynomials may be defined explicitly by

Hn(z):=(-1)nez2dndzne-z2,(1)

since this is a polynomial having the highest (2z)n and satisfying the Hermite equation.  The equation (1) is the Rodrigues’s formula for Hermite polynomials.  Using the Faà di Bruno’s formula, one gets from (1) also

Hn(x)=(-1)nm1+2m2=nn!m1!m2!(-1)m1+m2(2x)m1.

The first six Hermite polynomials are

H0(z) 1,
H1(z) 2z,
H2(z) 4z2-2,
H3(z) 8z3-12z,
H4(z) 16z4-48z2+12,
H5(z) 32z5-160z3+120z,

and the general is

Hn(z)(2z)n-n(n-1)1!(2z)n-2+n(n-1)(n-2)(n-3)2!(2z)n-4-+

Differentiating this termwise gives

Hn(z)= 2n[(2z)n-1-(n-1)(n-2)1!(2z)n-3+(n-1)(n-2)(n-3)(n-4)2!(2z)n-5-+],

i.e.

Hn(z)= 2nHn-1(z).(2)

The Hermite polynomials are sometimes scaled to such ones Hen which obey the differentiation rule

Hen(z)=nHen-1(z).(3)

Such Hermite polynomials form an Appell sequence.

We shall now show that the Hermite polynomials form an orthogonal set (http://planetmath.org/OrthogonalPolynomials) on the interval  (-,)  with the weight factor (http://planetmath.org/OrthogonalPolynomials) e-x2.  Let m<n;  using (1) and integrating by parts (http://planetmath.org/IntegrationByParts) we get

(-1)n-Hm(x)Hn(x)e-x2𝑑x=-Hm(x)dne-x2dxn𝑑x
=/-Hm(x)dn-1e-x2dxn-1--Hm(x)dn-1e-x2dxn-1𝑑x.

The substitution portion here equals to zero because e-x2 and its derivatives vanish at ±.  Using then (2) we obtain

-Hm(x)Hn(x)e-x2𝑑x= 2(-1)1+nm-Hm-1(x)dn-1e-x2dxn-1𝑑x.

Repeating the integration by parts gives the result

-Hm(x)Hn(x)e-x2𝑑x= 2m(-1)m+nm!-H0(x)dn-me-x2dxn-m𝑑x
= 2m(-1)m+nm!/-dn-m-1e-x2dxn-m-1= 0,

whereas in the case  m=n  the result

-(Hn(x))2e-x2𝑑x= 2n(-1)2nn!-e-x2𝑑x= 2nn!π

(see area under Gaussian curve).The results that the functionsMathworldPlanetmathxHn(x)2nn!πe-x22  form an orthonormal set on  (-,).

The Hermite polynomials are used in the quantum mechanical treatment of a harmonic oscillator, the wave functions of which have the form

ξΨn(ξ)=CnHn(ξ)e-ξ22.
TitleHermite polynomials
Canonical nameHermitePolynomials
Date of creation2013-03-22 15:16:25
Last modified on2013-03-22 15:16:25
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id28
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 33E30
Classificationmsc 33B99
Classificationmsc 26C05
Classificationmsc 26A09
Classificationmsc 12D99
Related topicSubstitutionNotation
Related topicAppellSequence
Related topicLaguerrePolynomial
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 10:34:24