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Simple Harmonic Oscillator - Normalization Constant

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine the normalization constants for the harmonic oscillator wavefunctions with v=0, and v=1 by evaluating their normalization integrals and show that they correspond to N=[tex]\frac{1}{\pi^{.5} * 2^v * v!}[/tex]


    2. Relevant equations



    3. The attempt at a solution

    [tex]\int \psi^{2}d\tau[/tex]=1
    [tex]\psi[/tex]1=N*2y*exp(-y[tex]^{2}[/tex]/2)
    [tex]\psi^{2}[/tex]=N2*4y[tex]^{2}[/tex]exp(-y[tex]^{2}[/tex])
    [tex]\int \psi^{2}d\tau[/tex] = [tex]\int N^{2}4y^{2}e^{-y^{2}}dy[/tex]=1

    By a table of integrals, integral from 0->inf of y[tex]^{2n}[/tex]exp(-ay[tex]^{2}[/tex]) => ([tex]\frac{(2n!)\pi^{.5}}{2^{2n+1}n!a^{n+1/2}}[/tex])

    So what I end up with is N1=[tex]\sqrt{\frac{1}{\pi^{.5}}}[/tex]

    The given equation of N I evaluate to be [tex]\sqrt{\frac{1}{2\pi^{.5}}}[/tex]

    I've looked through my integration several times, but am not sure where I could be coming up with an error (and checked the formula given by table with various other tables), any help is appreciated.


    EDIT: Noticed that in cartesian, all space is -inf to inf, not 0 to inf as in spherical polar....
     
    Last edited: Oct 8, 2009
  2. jcsd
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