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Normalize wave equation

  1. Oct 2, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex]\Psi(x) = \frac{C}{a^2 + x^2}[/tex]


    2. Relevant equations
    I know to do this I need to solve for:
    [tex]\int_{-\infty}^{\infty} \left|\Psi(x)\right|^2 = 1[/tex]


    3. The attempt at a solution
    I'm not sure how to do it for this function. I've tried various methods to solve
    [tex]C^2 \int_{-\infty}^{\infty} \frac{dx}{(a^2 + x^2)^2} = 1[/tex]
     
  2. jcsd
  3. Oct 2, 2009 #2
    By substiting
    [tex]x =a * tan(\theta)[/tex]
    I was able to solve for
    [tex]\frac{C^2}{a^3} = \frac{2}{\pi}[/tex]

    I now need to find the expectation values of [tex] x [/tex] and [tex] x^2 [/tex] I'm not sure how to get started (are they just zero? if I plot this function I get a curve peaked at x=0...)
     
  4. Oct 2, 2009 #3
    [tex] E[X]=\int\limits_{-\infty}^{+\infty} x \left|\Psi(x)\right|^2 {\rm d}x [/tex]

    [tex]f(x)=\Psi^2(x)[/tex] is even and [tex]g(x)=x[/tex] is odd [tex]\Rightarrow E[X]=\int\limits_{-\infty}^{+\infty} g(x)f(x) {\rm d}x= 0[/tex]

    [tex] E[X^2]=\int\limits_{-\infty}^{+\infty} x^2 \left|\Psi(x)\right|^2 {\rm d}x =?[/tex]
     
    Last edited: Oct 2, 2009
  5. Oct 2, 2009 #4
    Thanks, I got it. Am I supposed to post a full solution?
     
  6. Oct 2, 2009 #5
    I don't know but it would be nice.
     
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