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?