Problem: properties of Fourier Transforms in Quantum Mechanics

[Sistine]

Homework Statement

I am trying to derive a property of the Fourier Transform of the wave function.
F[\psi(cx)]=\frac{1}{|c|}\overline{\psi}\left(\frac{p}{c}\right)

Homework Equations

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

The Attempt at a Solution

I tried changing the integration variable
u=cx
Then
\psi(u)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}pu}\overline{\psi}(p)dp
But when I took the Fourier transform and substituted u=cx I did not manage to get the desired result.

 

[Cyosis]

You're using the wrong Fourier Transform.

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

Use the same substitution as you did previously and work it out from here.

 

[Sistine]

Thanks, but why is there a modulus involved?