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Harmonic Oscillator & Schrodinger

  1. Aug 23, 2008 #1
    I wish to graph a couple of the waveforms of a harmonic oscillator. I have consulted several resources and have found two that I like but the final equation differs even though they are both labeled normalized harmonic oscillator wavefunction.

    The first reference explains how the harmonic oscillator function is determined and the harmonic equation is at the very bottom of this 3 page document.
    http://physchem.ox.ac.uk/~hill/tutorials/qm2_tutorial/sho_series.pdf

    The second reference only presents the harmonic equation after a very brief discussion and it too is at the bottom of a 1 page display.
    http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html#c1

    The difference is the first term after the equate which is: (alpha/pi)^1/4.
    This term does not appear in the first reference but does appear in the second.

    Also the constants just before the Hermite function differ.


    Does anybody know why there is a difference between these two equations and how the difference occured?
     
  2. jcsd
  3. Aug 23, 2008 #2

    Dr Transport

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    Their definitions of y are slightly different, this is where you are getting confused.
     
  4. Aug 23, 2008 #3
    It would appear the definitions of y are the same; however, omega differs.

    Reference 1 defines y as

    y = sqrt(m*omega/hbar)*x where omega = sqrt(k/m)
    This is at equation (2)

    The reference 2 defines y as:

    y = sqrt(alpha)*x where alpha = (m*omega)/hbar and omega = sqrt(k/m) OR omega = 2pi
    This definition occurs just above the harmonic equation at the bottom of the page and omega definition at the top of the page.

    I can not tell which definition for omega was used. I suspect that since pi occurs in the final form that omega was defined as 2pi * (frequency?).

    Thus are the two omegas equal? Gut feeling thinks not.
    sqrt(k/m) = 2*pi*(frequency?)
     
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