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Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

  1. Mar 26, 2007 #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.
     
  2. jcsd
  3. Mar 26, 2007 #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].
     
  4. Mar 26, 2007 #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]
     
    Last edited: Mar 26, 2007
  5. Mar 26, 2007 #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!
     
  6. Mar 26, 2007 #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]
     
    Last edited: Mar 26, 2007
  7. Mar 26, 2007 #6

    nrqed

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    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.
     
  8. Mar 27, 2007 #7

    Hans de Vries

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    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


    Yes,


    Regards, Hans
     
    Last edited: Mar 27, 2007
  9. Mar 27, 2007 #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.
     
  10. Mar 27, 2007 #9
    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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