Lectures on Quantum Mechanics

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Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen. Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave. Later, however in the beginning of the twentieth century , it was found that light did indeed sometimes behave like a particle.

Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave.

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Lectures on Quantum Mechanics

So it really behaves like neither. Now we have given up. There is one lucky break, however—electrons behave just like light. They finally obtained a consistent description of the behavior of matter on a small scale. We take up the main features of that description in this chapter. Because atomic behavior is so unlike ordinary experience, it is very difficult to get used to, and it appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist.

Even the experts do not understand it the way they would like to, and it is perfectly reasonable that they should not, because all of direct, human experience and of human intuition applies to large objects. We know how large objects will act, but things on a small scale just do not act that way. So we have to learn about them in a sort of abstract or imaginative fashion and not by connection with our direct experience. In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.

In reality, it contains the only mystery. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics. To try to understand the quantum behavior of electrons, we shall compare and contrast their behavior, in a particular experimental setup, with the more familiar behavior of particles like bullets, and with the behavior of waves like water waves.

We consider first the behavior of bullets in the experimental setup shown diagrammatically in Fig.

We have a machine gun that shoots a stream of bullets. It is not a very good gun, in that it sprays the bullets randomly over a fairly large angular spread, as indicated in the figure. In front of the gun we have a wall made of armor plate that has in it two holes just about big enough to let a bullet through. It might be a box containing sand. Any bullet that enters the detector will be stopped and accumulated. When we wish, we can empty the box and count the number of bullets that have been caught.

A bullet which happens to hit one of the holes may bounce off the edges of the hole, and may end up anywhere at all. Or, if we assume that the gun always shoots at the same rate during the measurements, the probability we want is just proportional to the number that reach the detector in some standard time interval. For our present purposes we would like to imagine a somewhat idealized experiment in which the bullets are not real bullets, but are indestructible bullets—they cannot break in half.

In our experiment we find that bullets always arrive in lumps, and when we find something in the detector, it is always one whole bullet. If the rate at which the machine gun fires is made very low, we find that at any given moment either nothing arrives, or one and only one—exactly one—bullet arrives at the backstop. Also, the size of the lump certainly does not depend on the rate of firing of the gun.

The effect with both holes open is the sum of the effects with each hole open alone. So much for bullets. They come in lumps, and their probability of arrival shows no interference. Now we wish to consider an experiment with water waves. The apparatus is shown diagrammatically in Fig. We have a shallow trough of water. You can imagine a gadget which measures the height of the wave motion, but whose scale is calibrated in proportion to the square of the actual height, so that the reading is proportional to the intensity of the wave.

Our detector reads, then, in proportion to the energy being carried by the wave—or rather, the rate at which energy is carried to the detector. With our wave apparatus, the first thing to notice is that the intensity can have any size. If the source just moves a very small amount, then there is just a little bit of wave motion at the detector. When there is more motion at the source, there is more intensity at the detector.

6. Time Evolution and the Schrödinger Equation

The intensity of the wave can have any value at all. In this case we would observe that the original wave is diffracted at the holes, and new circular waves spread out from each hole.


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There will be such constructive interference wherever the distance from the detector to one hole is a whole number of wavelengths larger or shorter than the distance from the detector to the other hole. You will notice that the result is quite different from that obtained with bullets Eq. The intensity can have any value, and it shows interference.

Now we imagine a similar experiment with electrons. It is shown diagrammatically in Fig. We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole.

All the electrons which come out of the gun will have nearly the same energy. In front of the gun is again a wall just a thin metal plate with two holes in it. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker. We should say right away that you should not try to set up this experiment as you could have done with the two we have already described.

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This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in. We know the results that would be obtained because there are many experiments that have been done, in which the scale and the proportions have been chosen to show the effects we shall describe. Something like: click ….. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the two numbers are very nearly the same.

So we can speak of the average rate at which the clicks are heard so-and-so-many clicks per minute on the average. As we move the detector around, the rate at which the clicks appear is faster or slower, but the size loudness of each click is always the same.

PHYS - Lecture 21 - Quantum Mechanics III | Open Yale Courses

If we lower the temperature of the wire in the gun, the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once. Except that once in a while, if there were two clicks very close together in time, our ear might not sense the separation. That is the way electrons go. Now let us try to analyze the curve of Fig. Let us check this idea by experiment. The result seems quite reasonable. The result of this measurement is also drawn in the figure. How can such an interference come about?

Perhaps they go in a more complicated way. There are some points at which very few electrons arrive when both holes are open, but which receive many electrons if we close one hole, so closing one hole increased the number from the other. It is as though closing one hole decreased the number of electrons which come through the other hole.

It seems hard to explain both effects by proposing that the electrons travel in complicated paths. It is all quite mysterious. And the more you look at it the more mysterious it seems. None of them has succeeded. The mathematics is the same as that we had for the water waves! It is hard to see how one could get such a simple result from a complicated game of electrons going back and forth through the plate on some strange trajectory. We conclude the following: The electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave.

Incidentally, when we were dealing with classical waves we defined the intensity as the mean over time of the square of the wave amplitude, and we used complex numbers as a mathematical trick to simplify the analysis. But in quantum mechanics it turns out that the amplitudes must be represented by complex numbers. The real parts alone will not do. That is a technical point, for the moment, because the formulas look just the same. But there are a large number of subtleties involved in the fact that nature does work this way. We would like to illustrate some of these subtleties for you now.

But that conclusion can be tested by another experiment. We shall now try the following experiment. To our electron apparatus we add a very strong light source, placed behind the wall and between the two holes, as shown in Fig. We know that electric charges scatter light. So when an electron passes, however it does pass, on its way to the detector, it will scatter some light to our eye, and we can see where the electron goes. And we observe the same result no matter where we put the detector. From this observation we conclude that when we look at the electrons we find that the electrons go either through one hole or the other.

Experimentally, Proposition A is necessarily true. What, then, is wrong with our argument against Proposition A? Back to experiment!

Lectures on Quantum Mechanics

Let us keep track of the electrons and find out what they are doing. Well, that is not too surprising! So there is not any complicated business like going through both holes. When we watch them, the electrons come through just as we would expect them to come through. But wait! What do we have now for the total probability, the probability that an electron will arrive at the detector by any route?

We already have that information. We just pretend that we never looked at the light flashes, and we lump together the detector clicks which we have separated into the two columns. Quantum Mechanics. Lectures in this Course 1. Professor Susskind opens the course by describing the non-intuitive nature of quantum mechanics. With the discovery of quantum mechanics, the fundamental laws of physics moved into a realm that defies human intuition or visualization.

The Statistical Interpretations of Quantum Mechanics

He presents the fundamental logic of quantum mechanics in terms of preparing and measuring the direction of the spin. This fundamental He then introduces linear operators Professor Susskind opens the lecture by presenting the four fundamental principles of quantum mechanics that he touched on briefly in the last lecture. He then discusses the evolution in time of a quantum system, and describes how the classical Professor Susskind begins the lecture by introducing the Heisenberg uncertainty principle and explains how it relates to commutators.

He proves that two simultaneously measurable operators must commute.


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