Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

[quasar987]

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.

 

[jtbell]

I've never seen a completely explicit formula, only ones written in terms of the Hermite polynomials $H_n(x)$.

 

[quasar987]

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

$$<\phi_k|x|\phi_0>$$

and also

$$<\phi_2|x|\phi_k>$$

for all k. :/


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

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

where

$$a=\frac{m\omega}{\hbar}$$

 

[quasar987]

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!

 

[Dr Transport]

According to my well worn copy of Schiff, the eigenfunctions for the Harmonic Oscillator is

$$\sqrt{\frac{\alpha}{\pi^{1/2} 2^{n} n!}} H_{n}(x)e^{-(\alpha x)^{2} /2}$$

 

[nrqed]

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

$$<\phi_k|x|\phi_0>$$

and also

$$<\phi_2|x|\phi_k>$$

for all k. :/


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

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

where

$$a=\frac{m\omega}{\hbar}$$

Use raising and lowering operators! It's a snap to calculate $<\psi_n| x^a p^b |\psi_m>$ for any value of m,n,a and b (integer, non negative, of course) using raising and lowering operators.

 

[Hans de Vries]

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

$$<\phi_k|x|\phi_0>$$

and also

$$<\phi_2|x|\phi_k>$$

for all k. :/

They are all orthogonal. Which gives you a simpler answer...

Some other nice and useful properties:

1) $\phi_k$ is the k-th derivative of $\phi_0$, the Gaussian.
2) They are the eigenfunctions of the Fourier transform.

I did made a number of 3D animations a few years back here:

https://www.physicsforums.com/showthread.php?t=62227

According to my well worn copy of Schiff, the eigenfunctions for the Harmonic Oscillator is

$$\sqrt{\frac{\alpha}{\pi^{1/2} 2^{n} n!}} H_{n}(x)e^{-(\alpha x)^{2} /2}$$

Yes,


Regards, Hans

 

[quasar987]

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.

 

[christianjb]

H.O. with angular freq. w, mass m, has matrix elements

<i|x|j>= delta(j,i-1) sqrt[(j+1)hbar/2mw]+delta(j,i+1)sqrt[j hbar/2mw]

Sorry I couldn't find an online reference right now- but it's easy enough to find.