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Quantum fourier transforms

  1. Jan 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Assume ψ(x,0)=e^(-λ*absvalue(x)) for x ± infinity, find Φ(k)

    2. Relevant equations
    Φ(k)=1/√(2π)* ∫e(-λ*absvalue(x))e(-i*k*x)dx,-inf, inf



    3. The attempt at a solution, my thought was Convert the absolute value to ± x depending on what of the number line was being integrated.

    U=i*k*x
    du/(i*k)=dx

    1/√(2π)*∫e-λ*√(u2/(i*k)2)*e(-u)du,-inf,inf

    Now fixing abs value

    1/((2π)*(i*k))*∫eλ/(i*k)*ue(-u),du,-inf,o

    the integrand for one half of the number line looks like:

    E(u*(λ/(ik)-1)

    For which i get: after limits are taken for that half of the integral

    (1/((λ/ik)-1))

    Then similar integral for other half

    Is this the right track or am i totally off?

     
  2. jcsd
  3. Jan 30, 2015 #2

    TSny

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    Homework Helper
    Gold Member

    Hello and welcome to PF!

    Your work looks OK. I don't think you need to make the substitution u = ikx. Looks like a typo in one place where you left out the square root for the ##2 \pi## factor.

    If you feel more comfortable with working with real functions, write ##e^{-ikx} = \cos (kx) - i \sin (kx)##. You can then check to see if the resulting integrands are even or odd functions over the interval ##-\infty < x < \infty##.
     
  4. Jan 30, 2015 #3
    yea i was thinking that route also but i forgot about about the even or odd shortcut

    Thanks!
     
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