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Wave packets that feel harmonic potential

  1. Jun 6, 2007 #1
    Are there any nice wave packets you could write as a superposition of eigenstate solutions of a one-dimensional harmonic oscillator? The question deals with a situation, where a particle feels a harmonic potential, but is far away from the center and is travelling as a wave packet, probably oscillating like a classical particle before spreading.

    I tried the usual gaussian wave packet, but it lead to an integral

    [tex]
    \int\limits_{-\infty}^{\infty} H_n(x) e^{-Ax^2+Bx}dx
    [/tex]

    which I found too difficult for myself. Do you know if a foolproof technique already exists for integrating this, or if there is other kind of wave packets that are easier?

    [tex]H_n[/tex] is Hermite's polynomial.
     
  2. jcsd
  3. Jun 6, 2007 #2
    Hello,

    The problem you meet seems "how to do the integral [tex]\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx[/tex]"
    I try your problem as far as i can ( with brute force >"< ):
    In the begining, use the generating function of Hermite polynomials
    [tex]e^{-t^2+2tx}=\sum_{n=0}^{\infty}\frac{1}{n!}t^nH_n(x)[/tex]
    Multiply [tex]e^{-Ax^2+Bx}[/tex] both sides and integral over all [tex]x[/tex]:
    [tex]\text{L.H.S.}=\int_{-\infty}^{+\infty}e^{-t^2+2tx}e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}[/tex]
    [tex]\text{R.H.S.}=\sum_{n=0}^{\infty}\frac{1}{n!}t^n\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx[/tex]
    One can calculate the Taylor expansion of the L.H.S. and find the coefficient of the [tex]\frac{t^n}{n!}[/tex] term:
    [tex]\sqrt{\frac{\pi}{A}}\left(\frac{\partial^n}{\partial t^n}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}\right)_{t=0}[/tex]
    The integral you want appeals in the Taylor expansion coefficient of [tex]\frac{t^n}{n!}[/tex] term:
    [tex]\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}\left(\frac{\partial^n}{\partial t^n}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}\right)_{t=0}[/tex]
    Finally, it can be calculated further with Leibniz rule and obtain the result:
    [tex]\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}e^{\frac{B^2}{4A}}\sum_{N=0}^{[\frac{n}{2}]}\frac{n!}{(n-2N)!N!}\left(\frac{1}{A}-1\right)^N\left(\frac{B}{A}\right)^{n-2N}[/tex]
    where [tex][x][/tex] gives the maximum integer that is equal to or less than [tex]x[/tex].


    Best regards
     
    Last edited: Jun 6, 2007
  4. Jun 6, 2007 #3

    jtbell

    User Avatar

    Staff: Mentor

    I'm on the road right now and don't have access to my books, but I do seem to remember that it's possible to construct a Gaussian wave packet for the SHO, and that in fact it doesn't "spread." That is, it's width [itex]\Delta x = \sqrt {<x^2> - <x>^2}[/itex] remains constant as the packet moves back and forth. I remember doing this as an exercise in grad school.
     
  5. Jun 6, 2007 #4
    Nice trickery, variation! :tongue2: I just tried integration by parts and recursion relations of Hermites polynomials, without success. Seems I should add more tricks to my arsenal.
     
  6. Jun 7, 2007 #5

    Hans de Vries

    User Avatar
    Science Advisor

    Here you have them numerically:

    the non-spreading version:

    http://chip-architect.com/physics/gaussian.avi

    A narrow Gaussian cyclically spreading out and contracting back:

    http://chip-architect.com/physics/narrow_gaussian.avi


    I wanted to visualize both the phase as well as the magnitude. It shows
    nicely how the phase change on the x-axis corresponds to the local
    momentum. The one spreading out has momentum going both ways
    while spreading out and contracting back again.



    Regards, Hans
     
    Last edited: Jun 7, 2007
  7. Jun 8, 2007 #6

    Mentz114

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    Gold Member

    Jostpuur, this is nicely done in 'Quantum Theory' by David Bohm (1951) Dover.
    On page 307 he derives the wave packet for the QSHO and gets the non-spreading WP.
     
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