Fractional Fourier Transform in a QM Oscillator

[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 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

 

[vanesch]

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.

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.

 

[R0man]

.


Thank you for sharing your work on the Fractional Fourier Transform in a QM oscillator. It is fascinating to see the time evolution of arbitrary wave functions in a harmonic oscillator through your simulation. The use of Gaussian Hermite functions as eigenfunctions of both the QM oscillator and the Fourier Transform is a clever approach. It allows for a simple decomposition and reconstruction of an arbitrary function using the Fourier Transform, and the time evolution of the harmonic oscillator can be easily visualized by applying the Fractional Fourier Transform.

I am also intrigued by the three movies that you have shared. The first one, with a Gaussian pulse oscillating back and forth, demonstrates the periodicity of the Fourier Transform and the time evolution of the harmonic oscillator. The second one, with a narrow Gaussian pulse spreading into a sine wave and back, shows the effect of the Fourier Transform on different types of functions. And the third one, with a square wave oscillating between its Fourier Transform and back, highlights the connection between the QM oscillator and the Fourier Transform.

Your work is a great example of the power and versatility of the Fractional Fourier Transform in quantum mechanics. It is impressive to see how the Schrodinger Equation, although non-relativistic and with zero rest mass, can still provide such interesting insights into the behavior of quantum systems. Thank you for sharing your simulation and providing a deeper understanding of the Fractional Fourier Transform in a QM oscillator.