Integral over squared Hermite polynomial

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

∫∞−∞ψ∗x2ψdx\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx with ψ(n,x)=e−x22Hn(x)π14√2nn!\psi(n,x)=\frac{e^{-\frac{x^2}{2}} H_n(x)}{\pi^{\frac{1}{4}}\sqrt{2^n n!}}, where Hn(x)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[Ï€] n!)

Doing this manually gives 1/2+n1/2+n, but i can’t get Mathematica to solve this integral without specifying nn. 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.




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


1 Answer


An easy way without struggling with the integral:

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



Is there any way to let Mathematica solve this problem analytical?
– Gebbo
Dec 25 ’13 at 23:17



@Gebbo Not AFAIK
– Dr. belisarius
Dec 26 ’13 at 2:24