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Feynman Lectures on Physics Volume 3 Chapter 07

Feynman Lectures on Physics Volume 3 Chapter 07 - The...

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Unformatted text preview: The Dependence of Amplitudes on Time 7—1 Atoms at rest; stationary states We want now to talk a little bit about the behavior of probability amplitudes in time. We say a “little bit,” because the actual behavior in time necessarily involves the behavior in space as well. Thus, we get immediately into the most complicated possible situation if we are to do it correctly and in detail. We are always in the difficulty that we can either treat something in a logically rigorous but quite abstract way, or we can do something which is not at all rigorous but which gives us some idea of a real situationipostponing until later a more careful treatment. With regard to energy dependence, we are going to take the second course. We will make a number of statements. We will not try to be rigorous—but will just be telling you things that have been found out, to give you some feeling for the behavior of amplitudes as a function of time. As we go along, the precision of the description will increase, so don’t get nervous that we seem to be picking things out of the air. It is, of course, all out of the air—the air of experiment and of the imagination of people. But it would take us too long to go over the historical development, so we have to plunge in somewhere. We could plunge into the ab- stract and deduce everything—which you would not understand—or we could go through a large number of experiments to justify each statement. We choose to do something in between. An electron alone in empty space can, under certain circumstances, have a certain definite energy. For example, if it is standing still (so it has no translational motion, no momentum, or kinetic energy), it has its rest energy. A more compli- cated object like an atom can also have a definite energy when standing still, but it could also be internally excited to another energy level. (We will describe later the machinery of this.) We can often think of an atom in an excited state as having a definite energy, but this is really only approximately true. An atom doesn’t stay excited forever because it manages to discharge its energy by its interaction with the electromagnetic field. So there is some amplitude that a new state is generated—with the atom in a lower state, and the electromagnetic field in a higher state, of excitation. The total energy of the system is the same before and after, but the energy of the atom is reduced. So it is not precise to say an excited atom has a definite energy; but it will often be convenient and not too wrong to say that it does. [Incidentally, why does it go one way instead of the other way? Why does an atom radiate light? The answer has to do with entropy. When the energy is in the electromagnetic field, there are so many different ways it can be—so many difierent places where it can wander—that if we look for the equilibrium condition, we find that in the most probable situation the field is excited with a photon, and the atom is de-excited. It takes a very long time for the photon to come back and find that it can knock the atom back up again. It’s quite analogous to the classical problem: Why does an accelerating charge radiate? It isn’t that it “wants” to lose energy, because, in fact, when it radiates, the energy of the world is the same as it was before. Radiation or absorption goes in the direction of increasing entropy.] Nuclei can also exist in different energy levels, and in an approximation which disregards the electromagnetic effects, we can say that a nucleus in an excited state stays there. Although we know that it doesn’t stay there forever, it is often useful to start out with an approximation which is somewhat idealized and easier to think about. Also it is often a legitimate approximation under certain circum- stances. (When we first introduced the classical laws of a falling body, we did not include friction, but there is almost never a case in which there isn’t some friction.) 7—1 7—1 Atoms at rest; stationary states 7—2 Uniform motion 7—3 Potential energy; energy conservation 7-4 Forces; the classical limit 7—5 The “precession” of a spin one-half particle Review: Chapter 17, Vol. I, Space-Time Chapter 48, Vol. 1, Beats Then there are the subnuclear “strange particles,” which have various masses. But the heavier ones disintegrate into other light particles, so again it is not correct to say that they have a precisely definite energy. That would be true only if they lasted forever. So when we make the approximation that they have a definite energy, we are forgetting the fact that they must blow up. For the moment, then, we will intentionally forget about such processes and learn later how to take them into account. Suppose we have an atom—~or an electron, or any particle—which at rest would have a definite energy E 0. By the energy E 0 we mean the mass of the whole thing times c2. This mass includes any internal energy; so an excited atom has a mass which is difierent from the mass of the same atom in the ground state. (The ground state means the state of lowest energy.) We will call E 0 the “energy at rest.” For an atom at rest, the quantum mechanical amplitude to find an atom at a place is the same everywhere; it does not depend on position. This means, of course, that the probability of finding the atom anywhere is the same. But it means even more. The probability could be independent of position, and still the phase of the amplitude could vary from point to point. But for a particle at rest, the complete amplitude is identical everywhere. It does, however, depend on the time. For a particle in a state of definite energy E o, the amplitude to find the particle at (x, y, 2) at the time t is ae—1(E0/fi)t , (7.1) where a is some constant. The amplitude to be at any point in space is the same for all points, but depends on time according to (7.1). We shall simply assume this rule to be true. Of course, we could also write (7.1) as tie—1"”, (7.2) with hw = E0 = Mcz, where M is the rest mass of the atomic state, or particle. There are three different ways of specifying the energy: by the frequency of an amplitude, by the energy in the classical sense, or by the inertia. They are all equivalent; they are just different ways of saying the same thing. You may be thinking that it is strange to think of a “particle” which has equal amplitudes to be found throughout all space. After all, we usually imagine a “particle” as a small object located “somewhere.” But don’t forget the uncer- tainty principle. If a particle has a definite energy, it has also a definite momentum. If the uncertainty in momentum is zero, the uncertainty relation, Ap Ax = it, tells us that the uncertainty in the position must be infinite, and that is just what we are saying when we say that there is the same amplitude to find the particle at all points in space. If the internal parts of an atom are in a different state with a different total energy, then the variation of the amplitude with time is difierent. If you don’t know in which state it is, there will be a certain amplitude to be in one state and a certain amplitude to be in another—and each of these amplitudes will have a dif- ferent frequency. There will be an interference between these difierent components —like a beat-note—which can show up as a varying probability. Something will be “going on” inside of the atom-even though it is “at rest” in the sense that its center of mass is not drifting. However, if the atom has one definite energy, the amplitude is given by (7.1), and the absolute square of this amplitude does not depend on time. You see, then, that if a thing has a definite energy and if you ask any probability question about it, the answer is independent of time. Although the amplitudes vary with time, if the energy is definite they vary as an imaginary exponential, and the absolute value doesn’t change. That’s why we often say that an atom in a definite energy level is in a stationary state. If you make any measurements of the things inside, you’ll find that nothing (in probability) will change in time. In order to have the probabilities change in 7—2 time, we have to have the interference of two amplitudes at two different frequencies, and that means that we cannot know what the energy is. The object will have one amplitude to be in a state of one energy and another amplitude to be in a state of another energy. That’s the quantum mechanical description of something when its behavior depends on time. If we have a “condition” which is a mixture of two different states with differ- ent energies, then the amplitude for each of the two states varies with time according to Eq. (7.2), for instance, as —i(E1/fi)t e—i(E2/if)t e and (7.3) And if we have some combination of the two, we will have an interference. But notice that if we added a constant to both energies, it wouldn’t make any difference. If somebody else were to use a different scale of energy in which all the energies were increased (or decreased) by a constant amount—say, by the amount A—then the amplitudes in the two states would, from his point of view, be Ai(E1+A)I/fi e—i(E2+A)t/fi e and (7-4) All of his amplitudes would be multiplied by the same factor e"“’“””, and all linear combinations, or interferences, would have the same factor. When we take the absolute squares to find the probabilities, all the answers would be the same. The choice of an origin for our energy scale makes no difference; we can measure energy from any zero we want. For relativistic purposes it is nice to measure the energy so that the rest mass is included, but for many purposes that aren’t rela- tivistic it is often nice to subtract some standard amount from all energies that appear. For instance, in the case of an atom, it is usually convenient to subtract the energy Mscz, where M, is the mass of all the separate pieces—the nucleus and the electrons—which is, of course, different from the mass of the atom. For other problems it may be useful to subtract from all energies the amount Macz, where M9 is the mass of the whole atom in the ground state; then the energy that appears is just the excitation energy of the atom. So, sometimes we may shift our zero of energy by some very large constant, but it doesn’t make any difference, provided we shift all the energies in a particular calculation by the same constant. So much for a particle standing still. 7-2 Uniform motion If we suppose that the relativity theory is right, a particle at rest in one inertial system can be in uniform motion in another inertial system. In the rest frame of the particle, the probability amplitude is the same for all x, y, and 2 but varies with t. The magnitude of the amplitude is the same for all 1, but the phase depends on t. We can get a kind of a picture of the behavior of the amplitude if we plot lines of equal phase~say, lines of zero phase—as a function of x and I. For a particle at rest, these equal-phase lines are parallel to the x-axis and are equally spaced in the t—coordinate, as shown by the dashed lines in Fig. 7—1. In a different frame—x’, y’, z’, t’—that is moving with respect to the particle in, say, the x-direction, the x’ and t’ coordinates of any particular point in space are related to ac and t by the Lorentz transformation. This transformation can be represented graphically by drawing x’ and t’ axes, as is done in Fig. 741. (See Chapter 17, Vol. I, Fig. 17—2.) You can see that in the x’-t’ system, points of equal phaseT have a different spacing along the t’-axis, so the frequency of the time variation is different. Also there is a variation of the phase with x’, so the prob- ability amplitude must be a function of x’. 1‘ We are assuming that the phase should have the same value at corresponding points in the two systems. This is a subtle point, however, since the phase of a quantum me- chanical amplitude is, to a large extent, arbitrary. A complete justification of this assump- tion requires a more detailed discussion involving interferences of two or more amplitudes. 7—3 Fig. 7-]. of the amplitude of a particle at rest in the x-f systems. Relativistic transformation Under a Lorentz transformation for the velocity 1.», say along the negative x-direction, the time t is related to the time I’ by t = t’ — x’v/c2 V1 —— v2/c2 ’ so our amplitude now varies as e—u/mb'ot _ e—(i/VLMEOt’/\/1—1)2/‘oil—Eo‘uac'tc2 ‘/l—712/(22) In the prime system it varies in space as well as in time. If we write the amplitude as —(i’fi)(E' t’—p'r’) e ‘ P , we see that E,’, = EU/x/l —— 112/02 is the energy computed classically for a particle of rest energy E0 travelling at the velocity Z), and p’ = E,§,v/c2 is the corresponding particle momentum. You know that x, = (t, x, y, z) and 1),, = (E, pr, p1), p,) are four-vectors, and that pflxfl = Et — p - x is a scalar invariant. In the rest frame of the particle, pm, is just El; so if we transform to another frame, E1 will be replaced by Elt! __ pl . x1. Thus, the probability amplitude of a particle which has the momentum p will be proportional to e~(i/h)(Ept—p-x) , (7.5) where E, is the energy of the particle whose momentum is p, that is, E12 2 \/(pc)2 + E5, (7.6) where E 0 is, as before, the rest energy. For nonrelativistic problems, we can write E, = M,c'~’ + W,,, (7.7) where W1, is the energy over and above the rest energy M,c2 of the parts of the atom. In general, W1, would include both the kinetic energy of the atom as well as its binding or excitation energy, which we can call the “internal” energy. We would write 2 p»; 2M Wp = Wint —l_ and the amplitudes would be [WW—W). (7.9) Because we will generally be doing nonrelativistic calculations, we will use this form for the probability amplitudes. Note that our relativistic transformation has given us the variation of the amplitude of an atom which moves in space without any additional assumptions. The wave number of the space variations is, from (7.9), _ 12. k _ h, (7.10) so the wavelength is =2I=£ A k p (7.11) This is the same wavelength we have used before for particles with the momentum p. This formula was first arrived at by de Broglie in just this way. For a moving particle, the frequency of the amplitude variations is still given by hw = W... (7.12) 74 The absolute square of (7.9) is just 1, so for a particle in motion with a definite energy, the probability of finding it is the same everywhere and does not change with time. (It is important to notice that the amplitude is a complex wave. If we used a real sine wave, the square would vary from point to point, which would not be right.) We know, of course, that there are situations in which particles move from place to place so that the probability depends on position and changes with time. How do we describe such situations? We can do that by considering amplitudes which are a superposition of two or more amplitudes for states of definite energy. We have already discussed this situation in Chapter 48 of Vol. I——even for prob- ability amplitudes! We found that the sum of two amplitudes with different wave numbers k (that is, momenta) and frequencies w (that is, energies) gives inter- ference humps, or beats, so that the square of the amplitude varies with space and time. We also found that these beats move with the so-called “group velocity” given by where Ak and Aw are the differences between the wave numbers and frequencies for the two waves. For more complicated waves—made up of the sum of many amplitudes all near the same frequency—the group velocity is dw v” = a}. (7.13) Taking w = Ep/h and k = p/f’i, we see that dB 1),, = (7.14) Using Eq. (7.6), we have dEp _ 2 i. 71; — c Ep (7.15) But Ep : Mc2, so dEp P Up— A—l (7.16) which is just the classical velocity of the particle. Alternatively, if we use the non- relativistic expressions, we have W w=7p and k=%, and dw_dW_ d p2 _ p at — :1;— a; (in) — 17! (7'17) which is again the classical velocity. Our result, then, is that if we have several amplitudes for pure energy states of nearly the same energy, their interference gives “lumps” in the probability that move through space with a velocity equal to the velocity of a classical particle of that energy. We should remark, however, that when we say we can add two amplitudes of ditTerent wave number together to get a beat-note that will corre- spond to a moving particle, we have introduced something new—something that we cannot deduce from the theory of relativity. We said what the amplitude did for a particle standing still and then deduced what it would do if the particle were moving. But we cannot deduce from these arguments what would happen when there are two waves moving with different speeds. If we stop one, we cannot stop the other. So we have added tacitly the extra hypothesis that not only is (7.9) a possible solution, but that there can also be solutions with all kinds of p’s for the same system, and that the different terms will interfere. 7—5 ; + Fig. 7—2. A particle of mass M and momentum p in a region of constant potential. Re (Am p) > DlST a x. e were [FOR ¢2‘ 4") Fig. 7—3. The amplitude for a par- ticle in transit from one potential to another. 7—3 Potential energy; energy conservation Now we would like to discuss what happens when the energy of a particle can change. We begin by thinking of a particle which moves in a force field de- scribed by a potential. We discuss first the efl‘ect of a constant potential. Suppose that we have a large metal can which we have raised to some electrostatic potential (1). as in Fig. 7—2. If there are charged objects inside the can, their potential energy will be qu, which we will call V, and will be absolutely independent of position. Then there can be no change in the physics inside, because the constant potential doesn’t make any difference so far as anything going on inside the can is concerned. Now there is no way we can deduce/what the answer should be, so we must make a guess. The guess which works is more or less what you might expect: For the energy, we must use the sum of the potential energy V and the energy Ep—which is itself the sum of the internal and kinetic energies. The amplitude is proportional to e»ri/fi>t<Ep+V) t—P-xr (7.18) The general principle is that the coefficient of t, which we may call co, is always given by the total energy of the system: internal (or “mass”) energy, plus kinetic energy, plus potential energy: hw=Ep+ V. Or, for nonrelativistic situations, (7.19) 0 m = Wint + + V. (7.20) Now what about physical phenomena inside the box? If there are several difiercnt energy states, what will we get? The amplitude for each state has the same additional factor ear/mm over what it would have with V = 0. That is just like a change in the zero of our energy scale. It produces an equal phase change in all amplitudes, but as we have seen before, this doesn’t change any of the probabilities. All the physical phenomena are the same. (We have assumed that we are talking about different states of the same charged object, so that qd; is the same for all. If an object could change its charge in going from one state to another, we would have quite another result, but conservation of charge prevents this.) So far, our assumption agrees with what we would expect for a change of energy reference level. But if it is really right, it should hold for a potential energy that is not just a constant. In general, V could vary in any arbitrary way with both time and space, and the complete result for the amplitude must be given in terms of a differential equation. We don’t want to get concerned with the general case right now, but only want to get some idea about how some things happen, so we will think only of a potential that is constant in time and varies very slowly in space. Then we can make a comparison between the classical and quantum ideas. Suppose we think of the situation in Fig. 7—3, which has two boxes held at the constant potentials ¢1 and ¢2 and a region in between where we will assume tha...
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