This video shows a series of simulations involving a small particle. We'll start using the assumptions of classical physics. First we'll describe the initial setup, and then play the simulation. For simplicity, we'll assume a one dimensional space, so the particle can only move horizontally along this line, either left or right. Here is our particle. It is starting at position zero, and has an initial velocity of 4 units per second. Now we'll play the simulation. Since our particle floats in empty space, it maintains a constant velocity. Note that time is slowed down by a factor of two. We'll play all simulations this way. Here's how quantum physics describes the same thing. The particle is represented by a wave function... such as this one. It contains information both about the particle's position and its velocity, but it will take us a while to explain how to read it all. First, we'll explain how to read information about position. The wave function is rotating around the main axis, and its distance from it varies. Here, for example, the distance is high. Here it is lower. And here even lower still. The particle is more likely to be found at a position where the distance is high. So it's more likely to be found here, ... Less likely to be found here, ... And even less so here. To better see the varying distances, let's draw a circle that touches the wave. Its radius shows us the distance at a given point. We'll slide it along the main axis, and wrap the function with the resulting surface. So we don't know exactly where the particle is, but we do know the probability distribution of its position. The wave function evolves with time according to an equation called the Schrödinger equation. This particular wave shape evolves by gliding to the right. In fact it's moving at 4 units per second, but we'll explain about velocity later. Let's watch it again. Note how the wave is gradually spreading. We'll explain this too later on. The particle is too small to be observed directly, but we have a measurement device. It can only detect the particle within this range. When we click this button, a 1-second timer is activated. When the timer is up, the device makes a measurement, and tells us if it detected the particle. It will say "yes" if the particle is anywhere within the measurement range. Now let's run a little experiment. The particle again starts at position zero, and has a velocity of 4 units per second. Our device has just been activated, and will make a measurement in one second. It's easy to see that by that time the particle will be inside the measurement range. So we predict the device will say "yes". This is the determinism of classical physics. If we know the initial state, we can predict the results of measurements. And now back to quantum physics. We paused just before the measurement. Can we now predict the result? The wave function doesn't tell us if the particle is inside the range or not. This is the famous indeterminism of quantum physics. But we can calculate probabilities. The volume of this slice is 35% of the total volume of this shape, And the remainder is outside. These are the probabilities of the two possible outcomes. This experiment ended with a "yes" result. If we run it again, we may end up with a "no". The measurement also changes the wave. If the result is "yes", the part outside the range gets zeroed out. Otherwise, the part inside the range is zeroed out. This is called wave function collapse. We'll now run the simulation in full, before and after the measurement. We'll repeat it twice, once for each possible outcome, and then explain The measurement made the wave function collapse. Afterwards, the function continues to evolve smoothly according to the Schrödinger equation, But since it is different now, it will evolve differently as well. Here are four wave functions. They all give us the same information about the particle's position. But they rotate differently. This one rotates clockwise. This one rotates clockwise faster. This one - counterclockwise. And the last one doesn't rotate at all. Let's see how they evolve with time. The rotations contains information about the particle's velocity, But it's not so easy to interpret. Here's how to make this information more readable. We'll use a mathematical operator called 'the Fourier Transform'. It decomposes the wave function into the frequencies that make it up,... creating a new wave function. This new function gives us information about the particle's velocity, In the same way the original wave function gave us information about the particle's position. To be more accurate, it gives us information about momentum, but we'll explain about momentum only near the end of the video. For now, it will work fine as the velocity wave function. The peak of the wave is at four units, Meaning the particle's most likely velocity is 4 units per second. But it's not the only velocity our particle can have. As before, we don't know its exact velocity, Only the probability distribution. This explains the spreading phenomenon we saw before. We'll demonstrate this using an example from classical physics. Here's a cluster of one hundred particles. We distributed them horizontally randomly according to the position wave function. We'll now assign them with random velocities according to the velocity wave function. So some of them are a little faster than 4 units per second, and some are a little slower. The center of the cluster moves at 4 units per second, and it is gradually spreading. But remember that our wave function represents just one particle, And that there's nothing quite like it in classical physic. The inverse operation of a Fourier transform is yet another Fourier transform, And we can in fact reconstruct the original wave function from the new one. So both functions contain all the information about the particle. We've already seen how adding curls to the position wave function changes velocity. It also works the other way around: Adding curls to the velocity wave function changes the particle's position. We'll now run our simulation again, this time showing both wave functions. When the position wave function collapses, the velocity wave function will snap as well, According to the Fourier transform. Given a wave function, we can calculate mean, And standard deviation. The standard deviation measures how far the wave is spread out from the mean. In Quantum physics, standard deviation is related to a concept called uncertainty. To see it better, let's draw four standard deviations side by side. When the standard deviation is low, we have pretty good knowledge where the particle is. So we say uncertainty is low. And when the standard deviation is high, the particle can be anywhere within a wide range. So we say uncertainty is high. Now here is curious property of the Fourier transform. If one uncertainty is low, the other one must be high, And vice versa. They can't both be low. So if we know approximately where the particle is, we'll know little about its velocity. And the other way around. We can formulate this rule more precisely. This rectangle represents the uncertainty we have about the particle. It can be proven mathematically that its area will always be at least 8 square units. The rule says the rectangle can't be any smaller, but it can be bigger. Running our simulation again, the position wave will start spreading... And here's some more insight into this spreading phenomenon. If initially the uncertainty in position is low, Then uncertainty in velocity must be high. So now the position wave will spread very fast. And vice versa. Now uncertainty in velocity is low, And the spreading phenomenon will hardly be noticed. As mentioned before, the velocity wave function is really the momentum wave function. This is explained in the final part of this video. Momentum is the product of mass and velocity. Here are two particles. The lower one is four times more massive. So if we give them the same velocity, then the lower one's momentum will be four times greater. And here's another way to look at it. Velocity is momentum divided by mass. Now we gave the two particles the same momentum, so the more massive particle is four times slower. To tie it all up, let's discuss units of measurement. The simulations so far didn't assume any particular units, so we can choose anything we want. For the position wave function, let's choose centimeters, so it will be close to the size actually shown on your screen. Applying the Fourier Transform, we get the momentum wave function, To put the display on a reasonable scale, we divided the momentum by a factor called the reduced Planck constant, or h-bar. Here is its value using our units of choice. For the mass of the particle, we chose the same value as h-bar, in kilograms. This is close to the mass of an electron. So the mass and h-bar cancelled each other out, And we ended up with plain velocity, in centimeters per second. Now let's increase the particle's mass. The two wave functions remained the same. The uncertainty principle is the same. But the same values of momentum now translate to much lower velocities. Let's shrink the upper part of the image, to bring velocity back to scale. Now let's reduce the uncertainty in position, ... And increase the velocity back to 4 centimeters per second. We'll repeat the same steps again: Increase the mass ... Shrink the image ... Reduce uncertainty in position ... And increase velocity. We are now ready for our final simulation. It's quite accurate to say the particle starts at position zero, And has an initial velocity of 4 centimeters per second. Spreading will be slow, because of the low uncertainty in velocity. So one second from now, the position wave will be almost entirely within the measurement range, And we can predict with near certainty the device will say "yes". For the same reason, the collapse will hardly be noticed.
An oldie but a goodie. I remember watching this several years back
I really like https://www.youtube.com/user/EugeneKhutoryansky . He's got an extensive collection of quantum theory, relativistic theory, electrical and even some mathematical visualizations.
The music can be a bit much (usually classical pieces, which for some reason includes Mendelssohn's Wedding March), but they do a pretty good job of explaining abstract concepts
I've always had it explained to me as imagine having a coin stuck down the side of a chair, and the very act of reaching for it causes it to slip down the crack further.
I wish I had found this gem before finishing the quantum mech series! It explains so much!
Nice
sweet, found this yesterday and its cool even though i am pretty new to advanced physics