 In Part A we developed the Schrodinger equation. In this video we want to visualize some of its solutions to try and figure out what it actually predicts. We let the computer numerically solve the general form in various two-dimensional situations. The basic idea is that if we know the wave function psi at some time, the equation gives us its slope, which we can follow to find the wave function a short time in the future. We repeat this process for as long as desired. Remember from Part A that the wave function, bizarrely, has both a real part and an imaginary part. Our animations will be of the sum of the squares of these. In a classical theory this would represent a wave intensity, like the intensity of light. In quantum mechanics it turns out to represent the probability of finding an electron at some point in space. We begin with one of the simplest situations, the so-called particle-in-a-box scenario. We have a square box and we place an electron at its center. The walls are impenetrable, so the wave function is zero outside the box. Inside the potential energy is zero, meaning no forces act on the electron. Classically we would expect to be able to place a particle at rest and have it stay put, but under wave particle duality we know the uncertainty principle will apply. If the electron's position is uncertain to delta x, its momentum will be uncertain to delta p, which is no smaller than Planck's constant over delta x. Therefore the electron cannot be perfectly at rest at a precise location. Let's look at the classical picture of an electron at the box's center. We'll represent momentum uncertainty by showing several copies of the electron moving with small velocity in different directions. The result is a cloud of particles that slowly expands. Now let's look at the prediction of the Schrodinger equation. We'll represent the wave function probability as the color-coded height of a surface, viewed from directly above. To show small details, we'll greatly amplify the heights and chop off the large parts, which then appear white in our top view. So the white area is where the probability is very large and the dark blue area has zero probability. We start with a broad wave function, meaning we don't constrain the electron location very tightly. This large uncertainty and position will produce a small uncertainty in momentum. So as the wave function evolves, according to the Schrodinger equation, it slowly expands. Eventually it starts interacting with the walls, modifying its initially circular shape. Now suppose we tightly constrain the initial location of the electron. This should produce a large momentum uncertainty, and classically we'd see a rapidly expanding group of particles that would quickly bounce off the walls. For the quantum mechanical case, we start with a narrow wave function. It rapidly spreads, reflects from the box, and then produces a kaleidoscopic sequence of interference patterns. Now the obvious question is what are we looking at here? What do these images physically represent? Schrodinger's proposal was that, quote, material points consist of, or nothing but, wave systems. Therefore the wave pattern we are seeing here is an electron. Period. Electrons, indeed all particles, are nothing but matter waves. Furthermore, he proposed that, quote, the wave function physically means and determines a continuous distribution of electricity in space, the fluctuations of which determine the radiation by the laws of ordinary electrodynamics. In the next video we'll see that Schrodinger had good reasons to think this. In fact, he hoped that his equation provided a classical deterministic basis for atomic theory, in which there would be no wave particle duality, and no quantum jumps occurring at random governed only by probabilities. At the same time, he realized that, quote, this extreme conception may be wrong. Indeed, it does not offer as yet the slightest explanation of why only such wave systems seem to be realized in nature as correspond to mass points of definite mass in charge. And this was the objection raised by many other physicists. If the electron is a wave, it should be possible for it to spread out, and for us to detect small pieces of it. But all we ever detect is in the Milliken experiment are whole electrons. It was Max Born who proposed what has become the standard interpretation of Schrodinger's wave function. The squared magnitude of the wave function represents the probability of finding the electron at some point in space at a given time. What Schrodinger's equation determines is not the behavior of the electron itself, but the evolution of the amplitude of this probability. When we experimentally detect it, the electron appears as a particle at a definite place, and the wave function will have collapsed. In a sense, the wave function represents not an electron, but our knowledge of the state of an electron. Let's mix things up by curving the bottom of our box, or equivalently, using a spring to attach the electron to the middle of the box. Everything is as before, except that the potential is now proportional to the square of the electron's distance from the center of the box. If it moves a small distance, it feels a small force pulling it back towards the center. If it's a large distance, the force is large. This is classically the nature of the force produced by a spring, or by gravity on a particle sliding on a curved surface. The force is always pulling the particle towards the center of the box. In the classical case, a particle released at rest simply oscillates back and forth across the box. If there's some initial range of momentum, the possible paths spread out, but then reconverge on the other side of the box. The greater the range of momentum, the greater the spreading, but the paths always reconverge. Now let's see what Schrodinger's equation predicts for the quantum mechanical case. We start with a broad wave function, and we see it behaves more or less like a classical particle, with a bit of spreading and contracting as we'd expect given the uncertainty principle. For each oscillation, we narrow the initial wave function, and we observe greater spreading. Eventually the wave function starts interacting with the sides of the box, and wave interference effects start showing up as distortion in its initially round shape. When wave effects become extreme, we've transcended the picture of a classically moving particle. We can see Bohr's correspondence principle manifested in these calculations. At relatively large scales, the behavior predicted by quantum mechanics corresponds to the classical prediction for a particle moving under a force. Even though the classical and quantum theories employ quite different physical concepts and mathematical expressions, at large enough scales they make similar predictions about the behavior of particles. It's only when we push down to very small scales that quantum theory strongly diverges in its predictions, and wave particle duality becomes very apparent. Of course it was behavior at these small scales that motivated the development of the theory in the first place. So quantum mechanics and classical mechanics are not two disjointed unrelated theories. Instead they represent two ends of a continuum. If the scale of a system being analyzed is really big, then quantum and classical mechanics give the same predictions. In this case, there's no need for the extra complexity of quantum mechanics, and we can think of this as a purely classical realm. On the other hand, if the scale is really small, then wave particle duality becomes important, and classical mechanics fails to accurately describe our observations. This is the purely quantum realm where Schrodinger's equation reigns supreme. The uncertainty principle is what delineates these extremes. If the product of the ranges of position and momentum is much bigger than Planck's constant, then quantum effects are negligible. This is true in the macroscopic world of our day-to-day experience. If, on the other hand, the product is on the order of Planck's constant, then quantum effects will be very important. Recall from video 4c how Bohr proposed the concept of stationary orbits to explain why atoms have discrete energy levels and don't suffer radiation collapse. De Broglie proposed that these were orbits in which fit a whole number of electron waves. The stationary state concept naturally arises in the solution of the Schrodinger equation without any additional assumptions. We can find certain solutions for which the wave function intensity, the probability distribution for finding the electron at a given place in space, doesn't change with time. As we discussed in part A, these are the solutions where the electron has a well-defined energy. Here are some of the stationary states for our curved bottom box. This is the lowest energy state. It's the quantum state that most nearly corresponds to the electron being at rest in the middle of the box. The next lowest energy state has two lobes. We can have three lobe and six lobe and so on stationary states where generally the more lobes the higher the energy. Let's look at the two lobe stationary state in a little more detail. This image tells us that if we measure the position of the electron it will most likely be in one of the two reddish regions. But we can't know which one. Likewise, although presumably the electron is oscillating back and forth, we can't know which direction it's traveling in at a given time. In this pure energy state the electron is equally likely to be on the left as on the right and equally likely to be traveling left to right as right to left. Following Schrodinger's interpretation of this figure as representing the distribution of electric charge in space, we conclude that there is no movement of charge hence no radiation is generated. But suppose that at some time we somehow determine that the electron cannot be on the left side of the box. So let's chop off that part of the wave function. Then Schrodinger's equation tells us that the electron which we now know is somewhere on the right is moving towards the left. We can see the oscillation. And if this does represent the motion of electric charge which produces radiation by the laws of classical electrodynamics, then we expect it to radiate away energy until it settles down into a stationary state of lower energy, the one lobe state. This is an example of the way in which Schrodinger thought his equation could bring classical determinism back into atomic physics. However, just as electrons only show up as discrete particles, so too does radiation only show up as discrete photons. Schrodinger's equation was a great triumph indeed, but his classical interpretation of the wave function failed to explain what is actually observed. Instead it was Born's interpretation of the wave function as a probability amplitude and the corresponding picture of wave particle duality that was found to be consistent with experimental results. The double slit experiment is the quintessential demonstration of photon wave particle duality. So let's see what Schrodinger's equation predicts for a double slit experiment performed with electrons. We'll simulate the experiment inside our flat bottom box. We're using a slightly different color scheme here where black represents zero amplitude up through blue, green, red and finally white, which represents large amplitude. We start off with a wave function that is wide in the horizontal dimension and narrow in the vertical dimension. It has an average momentum towards the top of the box. As it moves through the box, we see very little spreading horizontally, but significant spreading vertically, consistent with the demands of the uncertainty principle. Eventually the wave function hits the upper wall. Now we repeat with a single slit screen in the middle of the box. The wave function is reflected from the screen but passes through the slit and rapidly spreads fairly uniformly into the top half of the box. Moving the slit to the right we get a similar result with the transmitted part of the wave function simply shifted with the slit. Now with both slits open we see what looks basically like a composite of the two single slit cases until the two spreading lobes begin to overlap. At that point we see interference fringes forming. These fringes indicate that on average electrons should arrange themselves on the top screen in alternating bands of high and low probability, hence of high and low density. When the experiment is performed remarkably that is indeed what is observed. Schrodinger's equation provides us with a precise description of the physical world, but it's a description in terms of the probabilities of observing nondeterministic physical events. It seems that we must heed the words of Niels Bohr. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature and clearly there seem to be very fundamental limits to what it is we are able to say about nature.