Last year I made a more modern version of a QM simulation(adsbygoogle = window.adsbygoogle || []).push({});

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 e^{in}

(were n is for the nth Gaussian Hermite function) and adding

them back together again.

Now the time evolution for the Harmonical Oscillator is e^{int}

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fractional Fourier Transform in a QM Oscillator

**Physics Forums | Science Articles, Homework Help, Discussion**