Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

Hi,

Is there an explicit formula for the eigenfunctions of the harmonic oscillator? By explicit, I mean "not written as the nth power of the operator (ax-d/dx) acting on the ground state".

Thanks.

 

I don'T think this would help. In the context of finding the time dependant perturbation to second order, I need to find the matrix elements

[tex]<\phi_k|x|\phi_0>[/tex]

and also

[tex]<\phi_2|x|\phi_k>[/tex]

for all k. :/


Actually, there seems to be a noticable patern for the k-th eigenfunction. So far I'm sure of this much:

[tex]\phi_k(x)=\left(\frac{a^k}{2^kk!}\right)^{1/2}\left(\frac{a}{\pi}\right)^{1/4}\left((2ax)^k-2^{k-2}ka^{k-1}x^{k-2}+???\right)e^{-ax^2/2}[/tex]

where

[tex]a=\frac{m\omega}{\hbar}[/tex]

 

Actually, I take that first sentence back. There is a formula for the integral of the product of two hermite polynomial, so maybe this can be used!

 

Thanks y'all, but don't sweat anymore on this, I suceeded in showing, using the recursion relation on the hermite polynomials that only for k=1 are these matrice elements non zero.