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

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# Fractional Fourier Transform in a QM Oscillator

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