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

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.

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

Homework Helper
Gold Member
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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Homework Helper
Gold Member
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
Gold Member
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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nrqed
Homework Helper
Gold Member
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
Gold Member
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:

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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