PlanetPhysics/Hermite Polynomials

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

since this is a polynomial having the highest degree term and satisfying the Hermite equation.\, The first six Hermite polynomials are

\\ \\ \\ \\ \\

and the general polynomial form is


Differentiating this termwise gives i.e.

We shall now show that the Hermite polynomials form an orthogonal set on the interval\, \, with the weight factor .\, Let\, ;\, using (1) and integrating by parts we get Failed to parse (unknown function "\sijoitus"): {\displaystyle = \sijoitus{-\infty}{\quad\infty}H_m(x)\frac{d^{n-1}e^{-x^2}}{dx^{n-1}} -\int_{-\infty}^\infty H'_m(x)\frac{d^{n-1}e^{-x^2}}{dx^{n-1}}\, dx.} The substitution portion here equals to zero because and its derivatives vanish at .\, Using then (2) we obtain Repeating the integration by parts gives the result Failed to parse (unknown function "\sijoitus"): {\displaystyle = 2^m(-1)^{m+n}m!\sijoitus{-\infty}{\quad\infty}\frac{d^{n-m-1}e^{-x^2}}{dx^{n-m-1}} = 0,} whereas in the case\, \, the result (see the area under Gaussian curve). The results mean that the functions\, \, 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