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

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Closely related or a duplicate How do I evaluate a symbolic integral involving Hermite polynomials?. –  Artes Dec 25 '13 at 20:00
    
You could code up (50) or (51) from here. –  b.gatessucks Dec 26 '13 at 9:36
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1 Answer

up vote 5 down vote accepted

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)
*)
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Is there any way to let Mathematica solve this problem analytical? –  Gebbo Dec 25 '13 at 23:17
    
@Gebbo Not AFAIK –  belisarius Dec 26 '13 at 2:24
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