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Problem: properties of Fourier Transforms in Quantum Mechanics

  1. May 4, 2009 #1
    1. The problem statement, all variables and given/known data
    I am trying to derive a property of the Fourier Transform of the wave function.
    [tex]F[\psi(cx)]=\frac{1}{|c|}\overline{\psi}\left(\frac{p}{c}\right)[/tex]

    2. Relevant equations
    The Fourier transform of [tex]\overline{\psi}(p)[/tex] is defined as
    [tex] \overline{\psi}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{-\frac{i}{\hbar}px}\psi(x)dx[/tex]
    and the inverse fourier transform
    [tex]\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}px}\overline{\psi}(p)dp[/tex]

    3. The attempt at a solution
    I tried changing the integration variable
    [tex]u=cx[/tex]
    Then
    [tex]\psi(u)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}pu}\overline{\psi}(p)dp[/tex]
    But when I took the fourier transform and substituted [tex]u=cx[/tex] I did not manage to get the desired result.
     
  2. jcsd
  3. May 4, 2009 #2

    Cyosis

    User Avatar
    Homework Helper

    You're using the wrong Fourier Transform.

    [tex]F[\psi(c x)]=\frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^\infty e^{-\frac{i}{h} p x} \psi(c x) dx[/tex]

    Use the same substitution as you did previously and work it out from here.
     
  4. May 4, 2009 #3
    Thanks, but why is there a modulus involved?
     
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