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Visualization of Quantum Physics
The following video gives a visual demonstration of some basic principles of Quantum Physics.
The mathematics needed for the simulations is well explained in this book: Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena.
Additional commentsMORE COMMENTS WILL BE ADDED IN THE NEXT FEW DAYS
The mathematical formula of the initial wave function is phi(x)=exp(-x^2/4+4ix). This is a complex function. For example, phi(1) =~ -0.51-0.59i, which is a complex number. In the video, the y axis (up/down) is used for the real part and the z axis (towards/away from camera) is used for the imaginary part.
This part shows how we can use the wave function to calculate probabilities. It also shows how the wave function collapses during measurement.
Part 3 - The Observer Effect
The wave function represents that information we have about the particle. When we make a measurement we gain new information, so the wave function must change. For example, when the device says "yes", we know for certain the particle is not outside the measurement range. Therefore, the part of the wave function outside this range collapses to zero.
When we don't make measurements, the wave function evolves smoothly and predictably according to the Schrödinger equation. How it evolves depends on its initial shape. Different wave shapes evolve differently. The example shown repeatedly throughout this video (e.g., at 2:23), shows an initial wave shape that evolves by gliding to the right and spreading a little.
But here's what happens in part 3's simulations 08 and 09 (at 4:56): We start with the same initial wave shape, and it starts to evolve the same. A second later, the device makes a measurement. This causes the wave function to collapse and change, as explained above. After the measurement, the wave function continues to evolve according to the Schrödinger equation, but it has a new initial shape now. Applying the Schrödinger equation to this new shape reveals it now evolves in a different way. It's still basically moving to the right, but it spreads faster, and develops a bumpy, wavy shape.
So this is quantum physic's observer effect: When we perform measurements, we gain new information about the objects we measure. This changes their wave function, which causes them to evolve differently from now on. So it's impossible to make measurements without changing the objects we measure.
Fourier transform is also used to process sound waves. A pure tone is a sine wave. The frequency of the wave determines the pitch of the tone. More complicated waves can be broken down to different sines, or different tones. This is basically what's done in an equalizer display. The sound wave is broken down to sine waves, and the display shows you how much of each frequency range the original wave contains.
This is also what we do here. Since we deal with complex waves (see comment in part 1), what's here considered a "pure tone" is a perfect helix. A helix that performs 4 complete oscillations over a range of 2 pi units, represents a velocity of 4 units per second. The FT applied on our position wave reveals the composition of helices that make up our wave. It shows the 4-helix has the largest contribution, but helices near 4 also contribute. Hence our particle has a distribution of velocities around 4.