I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and a shift operator matrix, I thought it'd be better to post it here.
I have to calculate the integral
12nn!√π∫+∞−∞Hn(x)e−x2+kxHl(x)dx
where Hn(x) is the nth Hermite polynomial and prove that it equals
\sqrt{\frac{m_}{m_!}}\left(\frac{k}{\sqrt{2}}\right)^{|n-l|}L_{m_<}^{|n-l|}\left(-\frac{k^2}{2}\right)\exp\left(\frac{k^2}{4}\right)
where m< and m> denote the smaller and the larger respectively of the two indices n and l and where Lmn are the associated Laguerre polynomials.
The last term is exp(k2/4), hence I suppose that I begin with
12nn!√π∫+∞−∞Hn(x)e−x2+kx−k24ek24Hl(x)dx
but here I'm stuck... No matter what or how I can't go further.
Thanks for your help!
No comments:
Post a Comment