# Fractional Fourier Transform in a QM Oscillator

1. Feb 1, 2005

### Hans de Vries

Last year I made a more modern version of a QM simulation
I did a long long time ago, It makes movies of time evolutions
of arbitrary wave functions in a QM harmonical oscillator.
(You can see the movies via the links below)

http://www.chip-architect.com/physics/gaussian.avi

http://www.chip-architect.com/physics/narrow_gaussian.avi

http://www.chip-architect.com/physics/square.avi

Interesting is the used method with a Fractional Fourier Transform.
The eigenfunctions of the QM oscillator are Gaussian Hermite
functions which are also eigenfunctions of the Fourier Transform.

They stay unchanged under the Fourier Transform up to a constant
value. If we decompose an arbitrary function with the Gaussian
Hermite functions as the orthogonal base then we get the
Fourier Transform by simply multiplying the components with ein
(were n is for the nth Gaussian Hermite function) and adding
them back together again.

Now the time evolution for the Harmonical Oscillator is eint
so after time '1' we get the Fourier Transform. At time is '2' we get
the original function back again but mirrored. At t=3 we get the mirrored
transform and finally at t=4 we're back where we started.

In the mean time we have Fractional Fourier Transforms. There are 3
movies. One of a Gaussian Pulse equal to the 0th eigen function but
displaced from the center so it oscillates back and forward. Two is
a narrow Gaussian pulse that spreads into a sine wave and back.
Third movie is a square wave that oscillates back and forward
between it's Fourier Transform which is sin(x)/x.

You may want to set your player in a repeat mode for continuous playing.
It's all based on the good old Schrodinger Equation. So OK, non-relativistic
and zero rest mass but still interesting. You can see the momentum
and energy if you look at the derivatives.

Regards, Hans

Last edited: Feb 1, 2005
2. Feb 2, 2005

### vanesch

Staff Emeritus
This is very similar to what happens in (classical) Fourier optics, when evolving from the "image" to the "focal plane", where you have the spatial fourier transform of the image amplitude !

cheers,
patrick.