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[parent] Hermite polynomials (Definition)

The polynomial solutions of the Hermite differential equation, with n a non-negative integer, are usually normed so that the highest degree term is (2z)n and called the Hermite polynomials H n(z). The Hermite polynomials may be defined explicitly by

Hn(z) := (−1)nez2 -dn- dzne−z2 , (1)

since this is a polynomial having the highest degree term (2z)n and satisfying the Hermite equation. 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 polynomial form 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 H′n(z) = 2n[ n− 1 (n−1)(n−2) n− 3 (n−1)(n−2)(n−3)(n−4)- n−5 ] (2z ) − 1! (2z ) + 2! (2z) − + ⋅⋅⋅, i.e.

H′n(z) = 2nHn−1(z). (2)

We shall now show that the Hermite polynomials form an orthogonal set on the interval (−∞, ∞) with the weight factor e−x2. Let m < n; using (1) and integrating by parts we get

∫ ∞ ∫ ∞ n −x2 n −x2 d-e---- (− 1) −∞ Hm (x)Hn (x)e dx = −∞ Hm (x ) dxn dx =

∫ ∞ dn−1e− x2 ∞ ′ dn−1e−x2 = / Hm (x)----n−1--− H m(x )----n−-1--dx. −∞ dx −∞ dx

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

∫ ∞ 2 ∫ ∞ dn− 1e−x2 Hm (x)Hn (x)e−x dx = 2(− 1)1+nm Hm −1(x)----n−1--dx. −∞ − ∞ dx

Repeating the integration by parts gives the result

∫ ∞ ∫ ∞ n− m −x2 Hm (x)Hn (x )e−x2 dx = 2m(− 1)m+nm! H0 (x)d----e----dx = −∞ −∞ dxn−m

2 m m+n ∞ dn−-m−1e−x-- = 2 (− 1) m! −/∞ dxn−m −1 = 0,

whereas in the case m = n the result

∫ ∫ ∞ 2 −x2 n 2n ∞ −x2 n √ -- (Hn (x)) e dx = 2 (− 1) n! e dx = 2 n! π −∞ −∞

(see the area under Gaussian curve). The results mean that the functions x↦→√-Hn(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

− ξ2- ξ ↦→ Ψn (ξ) = CnHn (ξ)e 2 .

"Hermite polynomials" is owned by pahio.
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Cross-references: wave, functions, Hermite equation, differential equation
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This is version 1 of Hermite polynomials, born on 2009-04-19.
Object id is 684, canonical name is HermitePolynomials.
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Classification:
Physics Classification: 02.30.Hq (Ordinary differential equations)
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