# Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

• quasar987
In summary, the hermite polynomial gives you a formula for the eigenfunctions of the harmonic oscillator, and they are all orthogonal.

#### quasar987

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

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

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

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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!

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

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

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quasar987 said:
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.

quasar987 said:
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:

Dr Transport said:
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

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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.

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.