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Problem: properties of Fourier Transforms in Quantum Mechanics

  • Thread starter Sistine
  • Start date
  • #1
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Homework Statement


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]

Homework 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]

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.
 

Answers and Replies

  • #2
Cyosis
Homework Helper
1,495
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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.
 
  • #3
21
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Thanks, but why is there a modulus involved?
 

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