Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

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  • #1
quasar987
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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.
 

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  • #2
jtbell
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I've never seen a completely explicit formula, only ones written in terms of the Hermite polynomials [itex]H_n(x)[/itex].
 
  • #3
quasar987
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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]
 
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  • #4
quasar987
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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!
 
  • #5
Dr Transport
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According to my well worn copy of Schiff, the eigenfunctions for the Harmonic Oscillator is

[tex] \sqrt{\frac{\alpha}{\pi^{1/2} 2^{n} n!}} H_{n}(x)e^{-(\alpha x)^{2} /2} [/tex]
 
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  • #6
nrqed
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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]
Use raising and lowering operators!!!! It's a snap to calculate [itex] <\psi_n| x^a p^b |\psi_m> [/itex] for any value of m,n,a and b (integer, non negative, of course) using raising and lowering operators.
 
  • #7
Hans de Vries
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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. :/
They are all orthogonal. Which gives you a simpler answer....

Some other nice and useful properties:

1) [itex]\phi_k[/itex] is the k-th derivative of [itex]\phi_0[/itex], 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

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

Yes,


Regards, Hans
 
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  • #8
quasar987
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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.
 
  • #9
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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.
 

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