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Normalizing Wave Functions

  1. Nov 6, 2011 #1
    Pretty basic question here, but I was wondering if someone could explain to me how to normalize a wave function. Specifically:

    ψn(x) = A * √x * sin (n∏x2/L2), n = 1, 2, 3, ...

    Normalized in the range 0 to L.

    Thanks for the help, a little bit of a walk through would be much appreciated.
     
  2. jcsd
  3. Nov 6, 2011 #2

    jtbell

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    Staff: Mentor

    What have you tried so far, and where are you stuck?
     
  4. Nov 6, 2011 #3
    Well, I'm working it now, its been shed under some new light, so what I am doing is just integrating the square of psi and setting it equal to one, then solving for A. Which is what I tried initially, but I just got confused with multiplying the original by the complex conjugate.
     
  5. Nov 6, 2011 #4
    Also having trouble integrating the statement cos ((2*pi*x^2)/L^2) dx.
     
  6. Nov 8, 2011 #5
    Hint: you need to substitute x for some variable y so that you end up with cos(y^2) in the integrand. Then you can try messing around with trig, for example cos(y^2) = cos(y*y) = ?

    You realize there is no easy way to expand that. In fact, wolframalpha gives:

    http://www.wolframalpha.com/input/?i=cos(x^2)

    which uses the "Fresnel C Integral" which I haven't even come across until this example.

    So the integral is not representable by standard elementary functions...


    Edit: However integrating the square of your original wavefunction is quite straightforward once you calculate the integral of x(sin^2)(x^2).

    Use the fact that (sin^2)(y) = (1/2)(1-cos(y))
     
    Last edited: Nov 8, 2011
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