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Determining Normalization Constant

  1. Aug 29, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine the normalization constant c in the wave function given by
    [itex]\psi[/itex](x) = c cos(kx) exp[(-1/2)(x/L)2 ]

    2. Relevant equations

    1=[itex]\int[/itex] |[itex]\psi[/itex](x)|2 dx

    limits of integration being -inf to inf.

    3. The attempt at a solution

    I'm very much sure that my math is wrong, I'm very rusty with improper integrals.

    1= [itex]\int[/itex] |c cos(kx) exp[(-1/2)(x/L)2|2 dx

    = [itex]\int[/itex] |c2 cos2(kx) exp[-(x2/L2)| dx

    it's at this point I start getting into trouble

    = c2 [itex]\int[/itex] |cos2(kx) exp[-(x2/L2)| dx

    = c2 [itex]\int[/itex] cos2(kx)dx [itex]\int[/itex] exp[-(x2/L2)dx

    = c2 (lim((2kx+sin(2kx))/4k)) ([itex]\pi[/itex])1/2/(1/L2)1/2

    I think I'm pretty solidly wrong by this point... where did I go wrong?
     
  2. jcsd
  3. Aug 29, 2011 #2
    You can't break up an integral like that. Think about it.

    [itex]\int x^2 dx = \int x * x dx = \int x dx \int x dx = x^4/4[/itex] ??

    Also this identity might make it less painful for you:

    [itex] cos(kx) = \frac{e^{ikx} + e^{-ikx}}{2}[/itex] (Euler's formula)
     
  4. Aug 29, 2011 #3
    Ah, right... well at least I made it three steps in before totally going off the deep end. Still working on it though.
     
  5. Aug 29, 2011 #4
    So my sticking point mathematically really seems to be the

    e(-1/2)(x/L)2

    This almost certainly simplifies down to something reasonably basic after being squared and/or integrated shouldn't it?
     
    Last edited: Aug 29, 2011
  6. Aug 30, 2011 #5
    do you know what a gaussian integral is?
     
  7. Aug 30, 2011 #6
    Yep, Arfken is a life-saver!
     
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