# Problem: properties of Fourier Transforms in Quantum Mechanics

1. May 4, 2009

### Sistine

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.
$$F[\psi(cx)]=\frac{1}{|c|}\overline{\psi}\left(\frac{p}{c}\right)$$

2. Relevant 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
$$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}px}\overline{\psi}(p)dp$$

3. 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.

2. May 4, 2009

### 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.

3. May 4, 2009

### Sistine

Thanks, but why is there a modulus involved?