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Feynman Lectures on Physics Volume 3 Chapter 02

Feynman Lectures on Physics Volume 3 Chapter 02 - 2 The...

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Unformatted text preview: 2 The Relation of Wave and Particle Viewpoints 2—1 Probability wave amplitudes In this chapter we shall discuss the relationship of the wave and particle Viewpoints. We already know, from the last chapter, that neither the wave view- point nor the particle viewpoint is correct. We would always like to present things accurately, or at least precisely enough that they will not have to be changed when we learn more—it may be extended, but it will not be changed! But when we try to talk about the wave picture or the particle picture, both are approximate, and both will change. Therefore what we learn in this chapter will not be accurate in a certain sense; we will deal with some half-intuitive arguments which will be made more precise later. But certain things will be changed a little bit when we interpret them correctly in quantum mechanics. We are doing this so that you can have some qualitative feeling for some quantum phenomena before we get into the mathematical details of quantum mechanics. Furthermore, all our experiences are with waves and with particles, and so it is rather handy to use the wave and particle ideas to get some understanding of what happens in given circumstances before we know the complete mathematics of the quantum-mechanical amplitudes. We shall try to indicate the weakest places as we go along, but most of it is very nearly correct—~it is just a matter of interpretation. First of all, we know that the new way of representing the world in quantum mechanics—the new framework—43 to give an amplitude for every event that can occur, and if the event involves the reception of one particle, then we can give the amplitude to find that one particle at different places and at difierent times. The probability of finding the particle is then proportional to the absolute square of the amplitude. In general, the amplitude to find a particle in different places at dificrent times varies with position and time. In some special case it can be that the amplitude varies sinusoidally in space and time like ei("’tTk"), where r is the vector position from some origin. (Do not forget that these amplitudes are complex numbers, not real numbers.) Such an amplitude varies according to a definite frequency w and wave number k. Then it turns out that this corresponds to a classical limiting situation where we would have believed that we have a particle whose energy E was known and is related to the frequency by E = hw, (2.1) and whose momentum p is also known and is related to the wave number by p = hk. (2.2) (The symbol h represents the number k divided by 271'; h = h/21r.) This means that the idea of a particle is limited. The idea of a particleiits location, its momentum, etc—which we use so much, is in certain ways unsatis- factory. For instance, if an amplitude to find a particle at different places is given by e“‘”“'"""), whose absolute square is a constant. that would mean that the prob— ability of finding a particle is the same at all points. That means we do not know where it is—it can be anywhere—there is a great uncertainty in its location. On the other hand, if the position of a particle is more or less well known and we can predict it fairly accurately. then the probability of finding it in different places must be confined to a certain region, whose length we call Ax. Outside this region, the probability is zero. Now this probability is the absolute square of an amplitude, and if the absolute square is zero, the amplitude is also zero, so that 2—1 2—1 Probability wave amplitudes 2—2 Measurement of position and momentum 2—3 Crystal difiraction 2—4 The size of an atom 2—5 Energy levels 2—6 Philosophical implications Note: This chapter is almost exactly the same as Chapter 38 of Volume I. Fig. 2—1. A wave packet of length Ax. Fig. 2—2. Diffraction passing through a slit. of particles we have a wave train whose length is Ax (Fig. 2—1), and the wavelength (the distance between nodes of the waves in the train) of that wave train is what corre- sponds to the particle momentum. . Here we encounter a strange thing about waves; a very simple thing which has nothing to do with quantum mechanics strictly. It is something that anybody who works with waves, even if he knows no quantum mechanics, knows: namely, we cannot define a unique wavelength for a short wave train. Such a wave train does not have a definite wavelength; there is an indefiniteness in the wave number that is related to the finite length of the train, and thus there is an indefiniteness in the momentum. 2—2 Measurement of position and momentum Let us consider two examples of this idea—to see the reason that there is an uncertainty in the position and/or the momentum. if quantum mechanics is right. We have also seen before that if there were not such a thing—if it were possible to measure the position and the momentum of anything simultaneously—we would have a paradox; it is fortunate that we do not have such a paradox, and the fact that such an uncertainty comes naturally from the wave picture shows that every- thing is mutually consistent. Here is one example which shows the relationship between the position and the momentum in a circumstance that is easy to understand. Suppose we have a single slit, and particles are coming from very far away with a certain energy—so that they are all coming essentially horizontally (Fig. 2—2). We are going to concentrate on the vertical components of momentum. All of these particles have a certain horizontal momentum p0, say, in a classical sense. So, in the classical sense, the vertical momentum 1),, before the particle goes through the hole, is definitely known. The particle is moving neither up nor down, because it came from a source that is far away—and so the vertical momentum is of course zero. But now let us suppose that it goes through a hole whose width is B. Then after it has come out through the hole, we know the position vertically—the y-position—with considerable accuracy—namely j=B.’r That is, the uncertainty in position, Ay, is of order B. Now we might also want to say. since we known the momentum is absolutely horizontal, that A191, is zero; but that is wrong. We once knew the mo- mentum was horizontal. but we do not know it any more. Before the particles passed through the hole, we did not know their vertical positions. Now that we have found the vertical position by having the particle come through the hole, we have lost our information on the vertical momentum! Why? According to the wave theory, there is a spreading out, or diffraction, of the waves after they go through the slit, just as for light. Therefore there is a certain probability that particles coming out of the slit are not coming' exactly straight. The pattern is spread out by the ditl‘raction effect, and the angle of spread, which we can define as the angle of the first minimum, is a measure of the uncertainty in the final angle. How does the pattern become spread? To say it is spread means that there is some chance for the particle to be moving up or down, that is, to have a component of momentum up or down. We say chance and particle because we can detect this diffraction pattern with a particle counter, and when the counter receives the particle, say at C in Fig. 2—2, it receives the entire particle, so that, in a classical sense, the particle has a vertical momentum, in order to get from the slit up to C. To get a rough idea of the spread of the momentum, the vertical momentum 1),, has a spread which is equal to [70 A0, where p“ is the horizontal momentum. And how big is A6 in the spread-out pattern? We know that the first minimum occurs at an angle A0 such that the waves from one edge of the slit have to travel one wavelength farther than the waves from the other side—we worked that out before (Chapter 30 of Vol. 1). Therefore A0 is VB, and so Am, in this experiment is pox/B. Note that if we make B smaller and make a more accurate measurement T More precisely, the error in our knowledge of y is i[3/2. But we are now only in- terested in the general idea, so we won‘t worry about factors of 2. 2—2 of the position of the particle, the diffraction pattern gets wider. So the narrower we make the slit, the wider the pattern gets, and the more is the likelihood that we would find that the particle has sidewise momentum. Thus the uncertainty in the vertical momentum is inversely proportional to the uncertainty of y. In fact, we see that the product of the two is equal to pot. But >\ is the wavelength and [)0 is the momentum, and in accordance with quantum mechanics, the wavelength times the momentum is Planck’s constant 11. So we obtain the rule that the uncertainties in the vertical momentum and in the vertical position have a product of the order h: Ay Ap, x h. (2.3) We cannot prepare a system in which we know the vertical position of a particle and can predict how it will move vertically with greater certainty than given by (2.3). That is, the uncertainty in the vertical momentum must exceed h/Ay, where Ay is the uncertainty in our knowledge of the position. Sometimes people say quantum mechanics is all wrong. When the particle arrived from the left, its vertical momentum was zero. And now that it has gone through the slit, its position is known. Both position and momentum seem to be known with arbitrary accuracy. It is quite true that we can receive a particle, and on reception determine what its position is and what its momentum would have had to have been to have gotten there. That is true, but that is not what the uncertainty relation (2.3) refers to. Equation (2.3) refers to the predictability of a situation, not remarks about the past. It does no good to say “I knew what the momentum was before it went through the slit, and now I know the position," because now the momentum knowledge is lost. The fact that it went through the slit no longer permits us to predict the vertical momentum. We are talking about a predictive theory, not just measurements after the fact. So we must talk about what we can predict. Now let us take the thing the other way around. Let us take another example of the same phenomenon, a little more quantitatively. In the previous example we measured the momentum by a classical method. Namely, we considered the direction and the velocity and the angles, etc., so we got the momentum by classical analysis. But since momentum is related to wave number, there exists in nature still another way to measure the momentum of a particle—photon or otherwise— which has no classical analog, because it uses Eq. (2.2). We measure the wave- lengths of the waves. Let us try to measure momentum in this way. Suppose we have a grating with a large number of lines (Fig. 2—3), and send a beam of particles at the grating. We have often discussed this problem: if the particles have a definite momentum, then we get a very sharp pattern in a certain direction, because of the interference. And we have also talked about how accu- rately we can determine that momentum, that is to say, what the resolving power of such a grating is. Rather than derive it again, we refer to Chapter 30 of Volume I. where we found that the relative uncertainty in the wavelength that can be measured with a given grating is l/Nm, where Nis the number of lines on the grat- ing and m is the order of the diffraction pattern. That is, AVA = l/Nm. (2.4) Now formula (2.4) can be rewritten as AVA2 : l/Nmtt : l/L, (2.5) where L is the distance shown in Fig. 2—3. This distance is the difference between the total distance that the particle or wave or whatever it is has to travel if it is reflected from the bottom of the grating. and the distance that it has to travel if it is reflected from the top of the grating. That is, the waves which form the diffrac- tion pattern are waves which come from different parts of the grating. The first ones that arrive come from the bottom end of the grating, from the beginning of the wave train. and the rest of them come from later parts of the wave train, coming from different parts of the grating, until the last one finally arrives, and that involves a point in the wave train a distance L behind the first point. So in order that we 2—3 Fig. 2—3. Determination of momen- tum by using a diffraction grating. Fig. 2—4. crystal planes. Scattering of waves by shall have a sharp line in our spectrum corresponding to a definite momentum, with an uncertainty given by (2.4), we have to have a wave train of at least length L. If the wave train is too short, we are not using the entire grating. The waves which form the spectrum are being reflected from only a very short sector of the grating if the wave train is too short, and the grating will not work right—we will find a big angular spread. In order to get a narrower one, we need to use the whole grating, so that at least at some moment the whole wave train is scattering simul- taneously from all parts of the grating. Thus the wave train must be of length L in order to have an uncertainty in the wavelength less than that given by (2.5). Incidentally, A)\/)\2 = A(l/)\) = Ak/27r. (2.6) Therefore Ak = 21r/L, (2.7) where L is the length of the wave train. This means that if we have a wave train whose length is less than L, the un- certainty in the wave number must exceed 27r/L. Or the uncertainty in a wave number times the length of the wave train*we will call that for a moment Ax— exceeds 271'. We call it Ax because that is the uncertainty in the location of the particle. If the wave train exists only in a finite length, then that is where we could find the particle, within an uncertainty Ax. Now this property of waves, that the length of the wave train times the uncertainty of the wave number associated with it is at least 277', is a property that is known to everyone who studies them. It has nothing to do with quantum mechanics. It is simply that if we have a finite train, we cannot count the waves in it very precisely. Let us try another way to see the reason for that. Suppose that we have a finite train of length L; then because of the way it has to decrease at the ends, as in Fig. 2—1, the number of waves in the length L is uncertain by something like 3:1. But the number of waves in L is kL/27r. Thus k is uncertain, and we again get the result (2.7), a property merely of waves. The same thing works Whether the waves are in space and k is the number of radians per centimeter and L is the length of the train, or the waves are in time and w is the number of oscillations per second and T is the “length” in time that the wave train comes in. That is, if we have a wave train lasting only for a certain finite time T, then the uncertainty in the fre- quency is given by Aw = 27r/T. (2.8) We have tried to emphasize that these are properties of waves alone, and they are well known, for example, in the theory of sound. The point is that in quantum mechanics we interpret the wave number as being a measure of the momentum of a particle, with the rule that p = hk, so that relation (2.7) tells us that Ap z h/Ax. This, then, is a limitation of the classi- cal idea of momentum. (Naturally, it has to be limited in some ways if we are going to represent particles by waves!) It‘is nice that we have found a rule that gives us some idea of when there is a failure of classical ideas. 2—3 Crystal diffraction Next let us consider the reflection of particle waves from a crystal. A crystal is a thick thing which has a whole lot of similar atoms—we will include some com- plications later—in a nice array. The question is how to set the array so that we get a strong reflected maximum in a given direction for a given beam of, say, light (x-rays), electrons, neutrons, or anything else. In order to obtain a strong reflection, the scattering from all of the atoms must be in phase. There cannot be equal num- bers in phase and out of phase, or the waves will cancel out. The way to arrange things is to find the regions of constant phase, as we have already explained; they are planes which make equal angles with the initial and final directions (Fig. 2—4). If we consider two parallel planes, as in Fig. 2—4, the waves scattered from the two planes will be in phase, provided the difference in distance traveled by a wave 24 front is an integral number of wavelengths. This difference can be seen to be 2d sin 6, where d is the perpendicular distance between the planes. Thus the condition for coherent reflection is 2dsin 6 = m (n = 1,2,. . .). (2.9) If, for example, the crystal is such that the atoms happen to lie on planes obey- ing condition (2.9) with n = 1, then there will be a strong reflection. If, on the other hand, there are other atoms of the same nature (equal in density) halfway between, then the intermediate planes will also scatter equally strongly and will interfere with the others and produce no eflect. So (1 in (2.9) must refer to ad- jacent planes; we cannot take a plane five layers farther back and use this formula! As a matter of interest, actual crystals are not usually as simple as a single kind of atom repeated in a certain way. Instead, if we make a two-dimensional analog, they are much like wallpaper, in which there is some kind of figure which repeats all over the wallpaper. By “figure” we mean, in the case of atoms, some arrangement—calcium and a carbon and three oxygens, etc., for calcium carbonate, and so on—which may involve a relatively large number of atoms. But whatever it is, the figure is repeated in a pattern. This basic figure is called a unit cell. The basic pattern of repetition defines what we call the lattice type; the lattice type can be immediately determined by looking at the reflections and seeing what their symmetry is. In other words, where we find any reflections at all determines the lattice type, but in order to determine what is in each of the elements of the lattice one must take into account the intensity of the scattering at the various directions. Which directions scatter depends on the type of lattice, but how strongly each scatters is determined by what is inside each unit cell, and in that way the structure of crystals is worked out. Two photographs of x-ray diflraction patterns are shown in Figs. 2—5 and 2—6; they illustrate scattering from rock salt and myoglobin, respectively. Incidentally, an interesting thing happens if the spacings of the nearest planes are less than V2. In this case (2.9) has no solution for n. Thus if A is bigger than twice the distance between adjacent planes, then there is no side diffraction pattern, and the light—or whatever it is—will go right through the material with- out bouncing oif or getting lost. So in the case of light, where >\ is much bigger than the spacing, of course it does go through and there is no pattern of reflection from the planes of the crystal. This fact also has an interesting consequence in the case of piles which make neutrons (these are obviously particles, for anybody’s money!). If we take these neutrons and let them into a long block of graphite, the neutrons difluse and work their way along (Fig. 2—7). They diffuse because they are bounced by the atoms, but strictly, in the wave theory, they are bounced by the atoms because of diflraction from the crystal planes. It turns out that if we take a very long piece of graphite, the neutrons that come out the far end are all of long wavelength! In fact, if one plots the intensity as a function of wavelength, we get nothing except for wavelengths longer than a certain minimum (Fig. 2—8). In other words, we can get very slow neutrons that way. Only the slowest neutrons come through; they are not diffracted or scattered by the crystal planes of the graphite, but keep going right through like light through glass, and are not scattered out the sides. There are many other demonstrations of the reality of neutron waves and waves of other particles. 2-4 The size of an atom We now consider another applicatio...
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