Integral over squared Hermite polynomial

Question

I would like to calculate the uncertainty of the nth Eigenstate of a 1-dim harmonic oscillator. To obtain the result I have to compute the integral

$$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx\,,$$ with $$\psi(n,x)=\frac{e^{-\frac{x^2}{2}} H_n(x)}{\pi^{\frac{1}{4}}\sqrt{2^n n!}}\,,$$ where $H_n(x)$ is the Hermite polynomial (degree n) in the physicist version (as implemented in Mathematica). In Mathematica this equals to the Integral over

(2^-n E^-x^2 x^2 HermiteH[n, x]^2)/(Sqrt[π] n!)

Doing this manually gives $1/2+n$, but i can't get Mathematica to solve this integral without specifying $n$. I used `

Assuptions=n ∈ Integers && n >= 0

Is there anyway to compute similar integrals with Mathematica?

Edit: Thanks for you answer, but I should have mentioned, that I'm looking for a way to let Mathematica solve such problems analytical.

Closely related or a duplicate How do I evaluate a symbolic integral involving Hermite polynomials?. — Artes, Dec 25 '13 at 20:00

score 7 · Accepted Answer · answered Dec 25 '13 at 18:42

7

An easy way without struggling with the integral:

FindSequenceFunction[
 Table[Integrate[(2^-n E^-x^2 x^2 HermiteH[n, x]^2)/(Sqrt[π] n!), {x, -Infinity, Infinity}],
       {n, 1, 5}], n]
(*
 1/2 (1 + 2 n)
*)

answered Dec 25 '13 at 18:42

Dr. belisarius

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Is there any way to let Mathematica solve this problem analytical? – NicolasW Dec 25 '13 at 23:17
@Gebbo Not AFAIK – Dr. belisarius Dec 26 '13 at 02:24

Integral over squared Hermite polynomial

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