Feynman Lectures on Physics Volume 3 Chapter 20

Feynman Lectures on Physics Volume 3 Chapter 20 - 20...

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Unformatted text preview: 20 0perat0rs 20—1 Operations and operators All the things we have done so far in quantum mechanics could be handled w1th ordinary algebra, although we dld from time to time show you some special ways of writing quantum-mechanical quantities and equations. We would like now to talk some more about some interesting and useful mathematical ways of describing quantum-mechanical things. There are many ways of approaching the subject of quantum mechanics, and most books use a different approach from the one we have taken. As you go on to read other books you might not see right away the connections of what you will find in them to what we have been doing. Although we Will also be able to get a few useful results, the main purpose of this chapter is to tell you about some of the different ways of writing the same physics. Knowmg them you should be able to understand better what other people are saying. When people were first working out classical mechanics they always wrote all the equations in terms of x-, y-, and z-components. Then someone came along and pointed out that all of the writing could be made much simpler by inventing the vector notation. It’s true that when you come down to figuring something out you often have to convert the vectors back to their components. But it’s generally much easier to see what’s going on when you work with vectors and also easier to do many of the calculations. In quantum mechanics we were able to write many things in a simpler way by using the idea of the “state vector.” The state vector Ii/z) has, of course, nothing to do with geometric vectors in three dimensions but is an abstract symbol that stands for a physical state, identified by the “label,” or “name,” 1/2. The idea is useful because the laws of quantum mechanics can be written as algebraic equations in terms of these symbols. For instance, Our fundamental law that any state can be made up from a linear com- bination of base states is written as Iw=2om, mo where the C, are a set of ordinary (complex) numbers—the amplitudes CI = (i I up) -—while I l), I 2), I 3), and so on, stand for the base states in some base, or repre- sentation. If you take some physical state and do something to it—like rotating it, or like waiting for the time At—you get a different state. We say, “performing an operation on a state produces a new state." We can express the same idea by an equation: I ¢> = A I it). (20.2) An operation on a state produces another state. The operator xi stands for some particular operation. When this operation is performed on any state, say It), it produces some other state I qs). What does Eq. (20.2) mean? We define it this way. If you multiply the equation by (i I and expand W) according to Eq. (20.1), you get mw=2mimmw. we (The states I j) are from the same set as I 1).) This is now just an algebraic equation. The numbers (1’ I ¢) give the amount of each base state you will find in I¢), and it is given in terms of a linear superposition of the amplitudes (j I 1/1) that you find 20—1 20—1 Operations and operators 20—2 Average energies 20—3 The average energy of an atom 20—4 The position operator 20—5 The momentum operator 20—6 Angular momentum 20—7 The change of averages with time | ill) in each base state. The numbers (i l AA [ j) are just the coeflicients which tell how much of (j 1 III) goes into each sum. The operator 151‘ is described numerically by the set of numbers, or “matrix,” An E (i l A” U). (20.4) So Eq. (20.2) is a high-class way of writing Eq. (20.3). Actually it is a little more than that; something more is implied. In Eq. (20.2) we do not make any reference to a set of base states. Equation (20.3) is an image of Eq. (20.2) in terms of some set of base states. But, as you know, you may use any set you wish. And this idea is implied in Eq. (20.2). The operator way of writing avoids making any particular choice. Of course, when you want to get definite you have to choose some set. When you make your choice, you use Eq. (20.3). So the operator equation (20.2) is a more abstract way of writing the algebraic equation (20.3). It’s similar to the difl'erence between writing c = a X b instead of CI = aubz — azby, c1, = asz — 01b1,, C; = aIbJ, — aybz. The first way is much handier. When you want results, however, you will eventually have to give the components with respect to some set of axes. Similarly, if you want to be able to say what you really mean by A”, you will have to be ready to give the matrix AU in terms of some set of base states. So long as you have in mind some set A“, Eq. (20.2) means just the same as Eq. (20.3). (You should remember also that once you know a matrix for one particular set of base states you can always calculate the corresponding matrix that goes with any other base. You can transform the matrix from one “representation” to another.) The operator equation in (20.2) also allows a new way of thinking. If we imagine some operator AA, we can use it With any state I (b) to create a new state .«l | 50). Sometimes a “state” we get this way may be very peculiar—it may not represent any physical situation we are likely to encounter in nature. (For instance, we may get a state that is not normalized to represent one electron.) In other words, we may at times get f‘states” that are mathematically artificial. Such artificial “states" may still be useful, perhaps as the mid-point of some calculation. We have already shown you many examples of quantum-mechanical op- erators. We have had the rotation operator Ry(6) which takes a state |ip) and produces a new state, which is the old state as seen in a rotated coordinate system. We have had the parity (or inversion) operator P, which makes a new state by reversing all coordinates. We have had the operators 6",, fig, and 6', for spin one- half particles. The operator J; was defined in Chapter 17 in terms of the rotation operator for a small angle 6. R.(e) = 1 + i; e L. (20.5) This just means, of course, that Rz(e) I l) = M + g e J‘. i it). (20.6) In this example, jz ] zp) is li/ie times the state you get ifyou rotate I (L) by the small angle 6 and then subtract the original state. It represents a “state” which is the difi'erence of two states. One more example. We had an operator Iii—called the momentum operator (x-component) defined in an equation like (20.6). If DAL) is the operator which 20—2 displaces a state along x by the distance L, then [3, is defined by 1m) = 1 + g, 6a, (20.7) where 6 is a small displacement. Displacmg the state I 1%) along x by a small dis- tance 6 gives a new state l W). We are saying that this new state is the old state plus a small new piece ,1 air). The operators we are talking about work on a state vector like | 11/), which is an abstract description of a phySical situation. They are quite different from algebraic operators which work on mathematical functions. For instance, d/dx is an “operator” that works on f(x) by changing it to a new function f’(x) = df/dx. Another example is the algebraic operator V2. You can see why the same word is used in both cases, but you should keep in mind that the two kinds of operators are different. A quantum-mechanical operator xi does not work on an algebraic function, but on a state vector like lW- Both kinds of operators are used in quantum mechanics and often in Similar kinds of equations, as you Will see a little later. When you are first learning the subject it is well to keep the distinction always in mind. Later on, when you are more familiar With the subject, you will find that it is less important to keep any sharp distinction between the two kinds of operators. You will, indeed, find that most books generally use the same notation for both! We’ll go on now and look at some useful things you can do with operators. But first, one special remark. Suppose we have an operator xi whose matrix in some base is A” E (i | A“ lj). The amplitude that the state xi | 1/!) 1s also in some other state [ ¢>) is (¢> | xi l tp). Is there some meaning to the complex conjugate of this amplitude? You should be able to show that <¢l1l¢>* = <¢ l 2* I «a. (20.8) where AA (read “A dagger”) is an operator whose matrix elements are Al. = (A..)*. (20.9) To get the i, j element of A’r you go to the j, 1‘ element of a? (the indexes are reversed) and take its complex conjugate. The amplitude that the state A“r | ¢) is in l i/x) is the complex conjugate of the amplitude that xi [1/1) is in l 4)). The operator all is called the “Hermitian adjoint" of xi. Many important operators of quantum mechanics have the special property that when you take the Hermitian adjoint, you get the same operator back. If B is such an operator, then at: a, and it is called a “self-adjoint” or “Hermitian," operator. 20—2 Average energies So far we have reminded you mainly of what you already know. Now we would like to discuss a new question. How would you find the average energy of a system—say, an atom? If an atom is in a particular state of definite energy and you measure the energy, you will find a certain energy E. If you keep repeating the measurement on each one of a whole series of atoms which are all selected to be in the same state, all the measurements will give E, and the “average” of your measurements will, of course, be just E. Now, however, what happens if you make the measurement on some state N) which is not a stationary state? Since the system does not have a definite energy, one measurement would give one energy, the same measurement on another atom in the same state would give a different energy, and so on. What would you get for the average of a whole series of energy measurements? 2073 We can answer the question by proyecting the state I it) onto the set of states of definite energy. To remind you that this is a special base set, we’ll call the states I 1),). Each of the states I 17,) has a definite energy E,. In this representation, W) = E Q] m). (2010) When you make an energy measurement and get some number 15,, you have found that the system was in the state m- But you may get a different number for each measurement Sometimes you will get E1, sometimes E2, sometimes E3, and so on. The probability that you observe the energy E1 is Just the probability of finding the system in the state | 771), which is, of course, just the absolute square of the amplitude C1 = (771 | tb). The probability of finding each of the possible energies E, is P, = IC, [2. (20.11) How are these probabilities related to the mean value of a whole sequence of energy measurements? Let’s imagine that we get a series of measurements like this: E1, E7, E11, E9, E1, E10, E7, E2, E3, E9, E5, E4, and so on. We contlnue for, say, a thousand measurements. When we are finished we add all the energies and divide by one thousand. That’s what we mean by the average. There’s also a short-cut to adding all the numbers. You can count up how many times you get E1, say that is N1, and then count up the number of times you get E2, call that N 2, and so on. The sum of all the energies is certainly just NlEl + N2E2 + N353 + - -- = 2 ME. The average energy is this sum divided by the total number of measurements which iSJuSt the sum of all the N,’s, which we can call N; 21. NIE? E,V ~ ——N——- (20.12) We are almost there. What we mean by the probability of something happen- ing 18 just the number of times we expect it to happen divided by the total number of tries. The ratio N ,/ N should—for large N—be very near to P,, the probability of finding the state | 11,), although it Will not be exactly P, because of the statistical fluctuations. Let’s write the predicted (or “expected”) average energy as (E),,,.; then we can say that <E>av = Z PIE“ (20.13) The same arguments apply for any measurement. The average value of a measured quantity A should be equal to <A)av = 2 P114“ where A, are the various possible values of the observed quantity, and P, is the probability of getting that value. Let’s go back to our quantum-mechanical state | 1/1). lt’s average energy is (E),,v = Z [c.le, = Z CTC,E,. (20.14) Now watch this trickery! First, we write the sum as 1/ Next we treat the left-hand 0p l as a common “factor.“ We can take this factor out of the sum, and write it as 0| (2 l 170120.104. 1 20-4 This expression has the form (It I ¢>, where I ¢> is some “cooked-up” state defined by |¢> = Z | m>Et<m I it). (20.16) It is, in other words, the state you get if you take each base state I 1],) in the amount E1071 I 1”)- Now remember what we mean by the states I 17,). They are supposed to be the stationary states—by which we mean that for each one, HIM) 2 Elm)- Since E, is Just a number, the right-hand side is the same as | n,)E,, and the sum in Eq. (20.16) is the same as 2 RI 771><77zI ¢)- Now 1' appears only in the famous combination that contracts to unity, so 2 HI meI‘J’) = HZ I m><ml¢> = HM. Magic! Equation (20.16) is the same as l¢> = H | 1/)- (20.17) The average energy of the state I IL) can be written very prettily as <5)... = (w I 1W). (20.18) To get the average energy you operate on lip) with H, and then multiply by (‘P I. A simple result. Our new formula for the average energy is not only pretty. It is also useful, because now we don’t need to say anything about any particular set of base states. We don’t even have to know all of the possible energy levels. When we go to calculate, we’ll need to describe our state in terms of some set of base states, but if we know the Hamiltonian matrix 11,, for that set we can get the average energy. Equation (19.18) says that for any set of base states Ii), the average energy can be calculated from <E>nv = 2 (WI i><i I H I no I a. (2019) where the amplitudes (i I H I j) are just the elements of the matrix H”. Let's check this result for the special case that the states I i) are the definite energy states. For them, H Ij) = E, Ij), so (1 I H Ij) = E, 5,, and (15>... = Z (t I i>E.6.,<jI w) = 2 EM I WW), U which is right. Equation (20.19) can, incidentally, be extended to other physical measure- ments which you can express as an operator. For instance, 1:, is the operator of the z-component of the angular momentum L. The average of the z-component for the state I 4/) is (Lz>av = (1” I132 I1»- One way to prove it is to think of some situation in which the energy is proportional to the angular momentum. Then all the arguments go through in the same way. 20—5 In summary, if a physical observable A is related to a suitable quantum- mechanical operator ff, the average value of A for the state I ¢> is given by (Am = («I l A It). (20.20) By this we mean that Aav = (w l ¢), (20.21) with A 44>) = A |¢>- (20.22) 20-3 The average energy of an atom Suppose we want the average energy of an atom in a state described by a wave function Mr); How do we find it? Let’s first think of a one-dimensional situation with a state | (0) defined by the amplitude (x 1 up) = fix). We are asking for the special case of Eq. (20.19) applied to the coordinate representation. Follow- ing our usual procedure, we replace the states | i) and [j) by l x) and I x’), and change the sums to integrals. We get (5)“ = [/0 | x)(x 1 H] x’)(x’ 1 (0) dx dx’. (2023) This integral can, if we wish, be written in the following way: [GP l X><x I d>> dx, (20.24) with (x l 4,) = f (x 1 m x’)(x’ l 1P) dx’. (20.25) The integral over x’ in (20.25) is the same one we had in Chapter l6~see Eq. (16.50) and Eq. (l6.52)—and is equal to ’12 2 ~ m 2d)? (00 + V0020). We can therefore write (x l ¢> = {— g7; 3‘1};— + V(x)]¢(x). (20.26) Remember that (1; l x) = (x | l//>* = ¢*(x); usmg this equality, the average energy in Eq. (20.23) can be written as 2 2 <E>av = /¢*(X) {— 2h": 5; + V] 1/06) dx. (20.27) Given a wave function 11/(x), you can get the average energy by doing this integral. You can begin to see how we can go back and forth from the state-vector ideas to the wave-function ideas. The quantity in the braces of Eq. (20.27) is an algebraic operator.I We will write it as 50 h2 d2 3": 7539+" With this notation Eq. (20.23) becomes <E>nv = f¢*(x)3‘cip(x)dx. (20.28) The algebraic operator 3‘0 defined here is, of course, not identical to the quantum-mechanical operator 1-7. The new operator works on a function of position up(x) = (x I t/I) to give a new function of x, ¢(x) = (x 1 ¢); while H IThe “operator” V(x) means “multiply by V(x).” 20—6 operates on a state vector | ll!) to give another state vector i 4:), without implying the coordinate representation or any particular representation at all. Nor is 50 strictly the same as H even in the coordinate representation. If we choose to work in the coordinate representation, we would interpret H in terms of a matrix (xl 1-7 l x’) which depends somehow on the two "indices" x and x’; that is, we expect—according to Eq. (20.25)rthat (x | (to) is related to all the amplitudes (x I to) by an integration. On the other hand, we find that 5C is a differential op- erator. We have already worked out in Section 16—5 the connection between (xl [all x’) and the algebraic operator CR3. We should make one qualification on our results. We have been assuming that the amplitude $(x) = (x lip) is normalized. By this we mean that the scale has been chosen so that /I¢(X)l2dx=1; so the probability of finding the electron somewhere is unity. If you should choose to work with a fix) which is not normalized you should write (E)av = ———— (20.29) It‘s the same thing. Notice the Similarity in form between Eq. (20.28) and Eq. (20.18). These two ways of writing the same result appear often when you work with the x-repre- sentation. You can go from the first form to the second with any}? which is a local operator, where a local operator is one which in the integral [<x I A | x'><x' | to) we can be written as 5% ll/(X), where (i is a difierential algebraic operator. There are, however, operators for which this is not true. For them you must work with the basic equations in (20.21) and (20.22). You can easily extend the derivation to three dimensions. The result is thatI (E).w = fip(r)5c.p(r)dei, (20.30) with - 212 9 3c = — ‘27: v- + V(r), (20.31) and With the understanding that [I ‘0 ldeol = 1. (20.32) The same equations can be extended to systems with several electrons in a fairly obvious way, but we won’t bother to write down the results. With Eq. (20.30) we can calculate the average energy of an atomic state even without knowing its energy levels. All we need is the wave function. It‘s an important law. We’ll tell you about one interesting application. Suppose you want to know the ground-state energy of some system~—say the helium atom, but it’s too hard to solve Schrodinger’s equation for the wave function, because there are too many variables. Suppose, however, that you take a guess at the wave function—pick any function you like—~and calculate the average energy. That is, you use Eq. (20.29)—generalized to three dimensions—to find what the average energy would be if the atom were really in the state described by this wave function. This energy Will certainly be higher than the ground-state energy which is the lowest IWe write d Vol for the element of volume. It is. of course. Just dx dy dz. and the integral goes from — ac to + w in all three coordinates. 20—7 P(X) X Fig. 20—1. A curve of probability density representing a localized particle. possible energy the atom can have.1t Now pick another function and calculate its average energy. If it is lower than your first choice you are getting closer to the true ground-state energy. If you keep on trying all sorts of artificial states you will be able to get lower and lower energies, which come closer and closer to the ground-state energy. If you are clever, you will try some functions which have a few adjustable parameters. When you calculate the energy it will be expressed in terms of these...
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