We're going to start our explanation with probably the most famous experiments in the area of quantum mechanics, the double-slit experiment. I already slightly discussed it on the previous week, now it's time to go a bit deeper. I'm now going to follow the description provided by Richard Feynman that you can read in his famous lectures on physics volume 3. First, imagine that you have a wall with two narrow slits and a sandbox after it. Right in front of the wall, we have a machine gun. Since we have a machine gun and not something else, you might expect that we are going to shoot. Yes, indeed we are going to shoot in the direction of this wall having two slits. Now let's state that this gun is far enough from the wall and the slits are very close to each other. The bullets that pass through the slits have equal chance to pass through the upper or the lower slit. Many bullets will not pass through the slits because they will miss them, so they will be stopped by the wall. Those bullets which pass through the slits are going to reach the sandbox and gets stuck there. After the experiment, we can retrieve the bullets from the sandbox and count them. When counting the bullets, we can draw this graph. The x axis here corresponds to the position of the bullet in the box, it's how far it went from the line of symmetry of the whole set. The y axis will correspond to the number of bullets found in this position. It is convenient for us to rotate the set like this. The bullets now go in the direction of the earth, so there's no distinction for them between two paths. Now if you perform this experiment with only one slit opened, doesn't matter left or right, we are going to obtain a very straightforward result. Most of the bullets go to the position right in front of the open slit while fewer of them stray away from the straight line because they hit the edge of the slit and bounce off it. You can see on the slide the resulting graphs for the experiment with the left slit opened on the left and the right slit opened on the right, and some supposed trajectories are the bullets that formed these graphs. Now the question is, having these two graphs, can they predict the resulting graph for the experiment with both slits opened? Let's reason for a while. When we look at the graph on the left, we see this single bulge in front of the open slit, but no bulge in front of another closed slit. Why? Because the right slit was closed and no bullets were allowed to go through it. On the other hand, when the right slit was opened, we got this bulge on the right format of those bullets which did not reach the box in the left experiment. Now, if you open both slits, can't we expect that the bullets from the right will add up to the bullets from the left? Yes, we can. It is exactly what happens. The resulting graph for the experiment with the two slits open, you just assume all the graphs for the left and the right slits opened. Our reason worked well for this case and we were able to give the correct prediction. Let's go to the second part of the experiment. In the previous experiment without counting bullets, now let's count something completely different. The screen with two slits is now going to be placed in the water, like this shown on the slide. Instead of a machine gun, we are going to have some source of water waves and oscillating bobber. We will make this source bobber to oscillate with some frequency. It will produce the waves of the same frequency and these waves are going to reach the screen being the slit. The screen itself is a very high-tech thing. The point is it doesn't reflect the waves. Any wave which hits the screen is absorbed by it. We don't have the reflected waves in the area between the source bobber and the screen. But the slits in the screen don't absorbs the waves since they are just empty space. They are going to act as sources of waves in the area outside the screen. To detect those secondary waves, we are going to place many smaller bobbers along the line parallel to the screen at some distance. These bobbers will swing due to the secondary waves and produce some work we are going to measure. Since you probably remember from school that waves don't transmit matter, but they do transmit energy. Again, we will start the experiment with only one slit open. The graph above shows the power produced by each bobber. The bobbers which are closer to the slit, produce more power, while those far away produce less power. It is absolutely normal since the energy of the wave is distributed along the whole wave front. The bobbers which are close to the wave source meet the waves whose front is small circle, while those far away meet the waves whose front is a bigger circle, thus the same energy is distributed along the longer front. The interesting part here is that every bobber in this line oscillates and produces work for us. Let's explore what happens in more detail. First, we measure the energy of each bobber movement per unit of time. The energy per unit of time is power. Second, each bobber represents for us a small share of the wave front. Which means we measure not just the power, but the power per unit surface of the wave. This thing is called the intensity of the wave. Third, from physics we know that intensity of the wave is proportional to its squared amplitude. The graph on the slide shows the intensity of the wave measured by each bobber while the bobbers themselves are captured in different positions at some point of time. For example, these two different bobbers oscillate in antiphase. Which means when the bobber 1 goes up, the bobber 2 goes down. Because the bobber 1 is on the wave peak, the bobber 2 finds itself exactly between the peaks in its lowest position. But the power produced by both bobbers is positive. Now, when we have the graphs of intensities with the left and the right slits being opened separately, can we predict the resulting graph for the two slits opened simultaneously? We can, of course. But since we understand that now we can build energy out of waves per unit area and not the number of bobbers, we understand that this prediction is going to be a bit more tricky than the previous one. When they open two slits, the thing we are going to deal with will be the sum of two waves, with the waves superpositioned. On this slide, you can see the intensities graphs for both slits open separately, and the positions of the bobbers at some point of time, again, for both slits open separately. When we have two waves interacting with the bobbers, when both slits are open, the resultant wave will be the sum of these two waves. So it is not the intensities we add up, but the waves themselves. The resultant wave will look like this. At some points where the waves are in phase, we have bigger amplitudes. While at points where waves arrive in antiphase, we can have much smaller amplitudes and some points even zero amplitudes. What about the intensity graph? The intensity is proportional to the square of the amplitude. So we calculate it and obtain something like this. The point here is, the resulting graph of the wave intensity with two slits open is not the sum of the graphs with the left and the right slits opened separately. That's what we have learned from waves. Let's continue.