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Normalization Constant for Gaussian

  1. May 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the normalization constant N for the Gaussian wave packet
    [tex]\psi (x) = N e^{-(x-x_{0})^{2}/2 K^{2}}[/tex]
    2. Relevant equations
    [tex]1 = \int |\psi (x)|^{2} dx[/tex]
    3. The attempt at a solution
    [tex]1 = \int |\psi (x)|^{2} dx = N^{2} \int e^{-(x-x_{0})^{2}/K^{2}} dx[/tex]
    Substitute [itex]y=(x-x_{0})[/itex]
    [tex]N^{2} \int e^{-y^{2}/K^{2}} dy[/tex]
    Substitute again [itex]z = y/|K|[/itex]
    [tex]N^{2} \int e^{-z^{2}} dz = N^{2} x_{0} K \sqrt{\pi}[/tex]
    [tex]N= ( \frac{1}{K x_{0} \sqrt{\pi}})^{1/2}[/tex]
    Where my question lies is with the [itex]x_{0}[/itex] in N. Should that be there?
     
    Last edited: May 16, 2012
  2. jcsd
  3. May 16, 2012 #2
    Actually... that first substitution may be flawed.
     
  4. May 16, 2012 #3
    Ok, I think I see where I went wrong. The [itex]x_{0}[/itex] doesn't belong in the final answer.
     
  5. May 16, 2012 #4

    BruceW

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    Homework Helper

    yes, you've figured it out.

    P.S. on this line, on the left hand side, there should be K since you have changed dy for dz:
    [tex]N^{2} \int e^{-z^{2}} dz = N^{2} x_{0} K \sqrt{\pi}[/tex]
    But after that you've remembered the K, so I guess you just forgot to type the K here, but you understand the right answer.
     
  6. May 16, 2012 #5
    Yeah, I forgot about the K, so what I should end up with is:
    [tex]N^{2} K \int e^{-z^{2}} dz = N^{2} K \sqrt{\pi}[/tex]
     
  7. May 16, 2012 #6

    BruceW

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    Homework Helper

    yep, looks right to me :)
     
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