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
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(1) |
and the general polynomial form is
Differentiating this termwise gives
i.e.
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(2) |
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
The substitution portion here equals to zero because
Repeating the integration by parts gives the result
whereas in the case
(see the area under Gaussian curve). The results mean that the functions
The Hermite polynomials are used in the quantum mechanical treatment of a harmonic oscillator, the wave functions of which have the form
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