### References

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 comments

#### Part 1

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.

#### Part 2

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 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.

#### Part 4

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.

#### Part 5

In this part we perform mathematical analysis of what we’ve seen so far, and reach additional conclusions.

First, we can define the mean and standard deviation of a given wave function. The formulas for these are the usual formulas known from statistics. The mean shows us what is the center of the wave in some sense, and the standard deviation measures how much the wave is concentrated or spread out around the center.

The Fourier transform has a property that when applied to a wave function with low standard deviation it produces a wave function with a high one, and vice versa. This is a purely mathematical property. In the context of quantum physics this property produces the principle of uncertainty.

The uncertainty principle applies to many aspects of quantum physics, not just position and velocity (momentum).

#### Part 6

In this part we discuss the scale of what we’ve seen so far, and explain why these phenomenon are not noticed on objects of ordinary size, e.g., a tennis ball.

The simulations shown in the video can be in fact interpreted in many scales, depending on the mass of the particle. To understand why, let’s explain again the uncertainty principle.

The usual formula that describes the uncertainty principle of position and momentum is:

Δx Δp >= h_bar/2,

where Δx is the uncertainty in position, and Δp the uncertainty in momentum.

h_bar is a physical constant having a value of 1.0545718e-34 m^2 kg /sec. In the video we use centimeters (cm) instead of meters (m), so its value becomes 10,000 times bigger. But here we’ll continue to use meters.

Substituting this number in the formula, we get:

Δx Δp >= 0.55e-34,

So the principle of uncertainty allows us precision of less than 1e-34 units, way beyond the pricision of any normal measurement device. So it has no implications when we deal with tennis balls.

However, if the mass of the object we measure is tiny too, then the uncertainty principle starts to play a significant role. We can replace Δp with ‘m Δv’, where m is the mass of the particle, and Δv is the uncertainty in velocity. So now we get:

Δx m Δv >= h_bar/2

or:

Δx Δv >= h_bar/2m.

If m is approximately the same size as h_bar, then this translates to:

Δx Δv >= 0.5.

Now this has severe implications. Say we know the object’s speed up to a few 0.01 meters per second of error, i.e., Δv=0.01. The uncertainty principle tells us Δx>=50. More than 50 meters!

### FAQ

#### How come the wave function is 3-dimensional? You said you assumed movement in 1 dimension.

The wave function contains information about the particle.

In particular there are two pieces of information we need:

X – A real number indicating the particle’s position in the 1-dimensional world.

V – A real number indicating the particle’s velocity in the 1-dimensional world.

Now imagine we don’t know X exactly but only probabilistically. To draw a probability distribution for X we need a 2-D diagram. But that’s just information about X. The world is still 1-dimensional and X is still just a real number.

The wave function is a complicated mathematical object that contains information about both X and V in a probabilistic way. To fully visualize this object we need a 3-D view, but again, this is just information about a particle that exists in a 1-dimensional world.