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Expectation Values Question

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data
    A wave function ψ is A(eix+e-ix) in the region -π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x=0 and x=π/8, and (b) between x=0 and x=π/4.

    2. Relevant equations



    3. The attempt at a solution

    So to normalize the function, I multiplied it by its complex conjugate (A(e-ix+eix) and got:
    ∫A2[(eix+e-ix)(e-ix+eix)dx=1 From -π to π
    ∫2A2dx=1
    2A2x(from -π to π)=1
    2A2π+2A2π=1
    4A2π=1

    A=sqrt(1/4π)

    Now that I have the function normalized, I can find the probability the question asks for. The problem I'm having is however do you take the integral of complex numbers the same way as a real number?

    The best attempt I can get is:
    ∫(sqrt(1/4π)(eix+e-ix)dx From 0 to π/8
    (sqrt(1/4π))∫(eix+e-ix)dx
    (sqrt(1/4π))(ieix-ie-ix)

    Would I now just plug in 0 and π/8 and leave my answers in terms of i?

    Thanks for taking the time to look at this.
    Aglo6509
     
  2. jcsd
  3. Jan 30, 2012 #2

    ehild

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    Homework Helper
    Gold Member

    Hi aglo,

    The probability density is ψψ* ("*" for complex conjugate). The wave function can be complex, but probability can not!

    ehild
     
    Last edited: Jan 31, 2012
  4. Jan 31, 2012 #3
    It might be helpful for you if you note that
    [itex]\frac{e^{ix}+e^{-ix}}{2} = Cos(x)[/itex]
    so your answers won't even involve any i since those trig functions are real!

    also, the density you're looking for is [itex]\int \psi ^* \psi dx[/itex]
     
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