231.4. Measurements, Observables, and the Uncertainty RelationsThe eigenket-eigenvalue relation(1.3.37)S,/±)= ±(h/2)/±)immediately follows from the orthonormality property of /±).It is also instructive to look at two other operators,S+=+><-I、S_=-><+ b(1.3.38)which are both scen to be non-Hermitian. The operator S+, acting on thespin-down ket [->, turns |-> into the spin-up ket I+ > multiplied by h. Onthe other hand, the spin-up ket /+), when acted upon by S+, becomes anull ket. So the physical interpretation of S+ is that it raises the spincomponent by one unit of h; if the spin component cannot be raised anyfurther, we automatically get a null state. Likewise, S_ can be interpreted asan operalor that lowers the spin coinponent by one unit of h. Later we willshow that S+ can be written as S ±isy.In constructing the matrix representations of the angular momentumoperators,itis customaryto label the column (row)indices in descendingorder of angular momentum components, that is, the first entry correspondsto the maximum angular momentum component, the second, the nexthighest, and so forth. In our particular case of spin systems, we have1-) =(I+)=(1.3.39a)0)h(1LS,=h(oS=hS,00)2(0-1(1.3.39b)We will come back to these explicit expressions when we discuss the Paulitwo-component formalism in Chapter 3.1.4.MEASUREMENTS,OBSERVABLES,ANDTHEUNCERTAINTYRELATIONSMeasurementsHaving developed the mathematics of ket spaces, we are now in aposition to discuss the quantum theory of measurement processes. This isnot a particularly easy subject for beginners, so we first turn to the words ofthe great master, P. A. M. Dirac, for guidance (Dirac 1958, 36): “Ameasurement always causes the system lo jump into an eigenstate of thedynamical variable that is being measured." What does all this mean? Weinterpret Dirac's words as follows: Before a measurement of observable A is
24Fundamcntal Conceptsmade, the system is assumed to be represented by some linear combination[α)=cala")=Ela'"(aα).(1.4.1)0.aWhen the measurement is performed, the system is " thrown into" one ofthe eigenstates, say la') of observable A. In other words,measuremen(1.4.2)[α][a').For example, a silver atom with an arbitrary spin orientation will changeinto either JS,; +) or [S,; -) when subjected to a SG apparatus of typeSG2. Thus a measurement usually changes the state. The only exception iswhen the state is already in one of the eigenstates of the observable beingmeasured, in which caseA measurement [a")(1.4.3)la'")with certainty, as will be discussed further. When the measurement causesJα) to change into a'), it is said that A is measured to be a'. It is in thissense that the result of a measurement yields one of the eigenvalues of theobservable being measured.Given (1.4.1), which is the state ket of a physical system before themeasurement, we do not know in advance into which of the various a")'sthe system will be thrown as the result of the measurement. We dopostulate, however, that the probability for jumping into some particularJa') is given byProbability for a'= Ka'α)/2,(1.4.4)provided that [α) is normalized.Although we have been talking about a single physical system, todetermine probability (1.4.4) empirically,we must consider a great numberof measurements performed on an ensemble-that is.a collection-ofidentically prepared physical systems, all characterized by the same ket [α).Such an ensemble is known as a pure ensemble.(We will say more aboutensembles in Chapter 3.) As an example, a beam of silver atoms whichsurvive the first SG2 apparatus of Figure 1.3 with the S, - componentblocked is an example of a pure ensemble because every member atom ofthe ensemble is characterized by IS,; +).The probabilistic interpretation (1.4.4) for the squared inner productKa'α>/2 is one of the fundamental postulates of quantum mechanics, so itcannot be proven. Let us note, however, that it makes good sense in cxtremecases. Suppose the state ket is a' itself even before a measurement ismade; then according to (1.4.4), the probability for getting a'-or, moreprecisely, for being thrown into [a')-as the result of the measurement ispredicted to be 1, which is just what we expect. By measuring A once again
251.4.Measurements,Observables,and theUncertaintyRelationswe, of course,get Ja") only; quite generally,repeated measurements of thesame observable in succession yield the same result.* If, on the other hand.weareinterested in the probabilityforthesvstem initiallycharacterized byla) to be thrown into some other eigenket la") with a"+ a, then (1.4.4)gives zero because of the orthogonality between [a') and [a").From thepoint of view of measurement theory, orthogonal kets correspond to mutu-ally exclusive alternatives; for example, if a spin system is in JS,; +),it isnot in [S,;->with certainty.Quite generally, the probability for anything must be nonnegative.Furthermore, the probabilities for the various alternative possibilities mustadd up to unity. Both of these expectations are met by our probabilitypostulate (1.4.4).We define the expectation value of A taken with respect to state [α)as(1.4.5)<A) =(α|A|α).To make sure that we are referring to state lα), the notation (A). issometimes used.Eguation(1.4.5)is a definition; however,it agrees with ourintuitive notion of average measured value because it can be written as(A)=ZE(αla")(a"Ala')(a'lα)a'a"(1.4.6)MKaa)/2a'+a'measured value a'probabilityforobtaininga"It is very important not to confuse eigenvalues with expectation values. Forexample, the expectation value of S, for spin systems can assume any realvalue between - h /2 and + h /2, say 0.273h; in contrast, the eigenvalue ofS, assumes only two values, h /2 and - h /2.To clarify further the meaning of measurements in quantum me-chanics we introduce the notion of a selective measurement, or filtration. InSection 1.1 we considered a Stern-Gerlach arrangement where we let onlyone of the spin components pass out of the apparatus while we completelyblocked the other component. More generally, we imagine a measurementprocess with a device that selects only one of the eigenkets of A, say [a'),and rejects all others; see Figure 1.6. This is what we mean by a selectivemeasuremcnt; it is also called filtration because only one of the A eigenketsfilters through the ordeal. Mathematically we can say that such a selective* Here successive measurements must be carried out immediately afterward. This point willbecome clear when we discuss the time evolution of a state ket in Chapter 2
26Fundamental Conceptsla'>lα>AMeasurementla">with a"+a'FIGURE1.6.Selectivemeasurement.measurement amounts to applying the projection operator A,- to [αa):(1.4.7)Alα)=[a"<aα).J. Schwinger has developed a formalism of quantum mechanicsbased on a thorough examination of selective measurements. He introducesa measurement symbol M(a') in the beginning, which is identical to Ag, orJa')(a'in our notation, and deduces a number of properties of M(a') (andalso of M(b', a') which amount to Jb')<u' by studying the outcome ofvarious Stern-Gerlach-type experiments.In this way he motivates the entiremathematics of kets, bras, and operators. In this book we do not followSchwinger's path; the interested reader may consult Gottfried's book.(Gottfried 1966, 192-9).Spin Systems, OnceAgainBefore proceeding with a general discussion of observables, we onceagain consider spin systems. This time we show that the results ofsequential Stern-Gerlach experiments, when combined with the postulatesof quantum mechanics discussed so far, are sufficient to determine not onlythe Sx.y eigenkets, ISx; ±> and IS,; ±), but also the operators S, and S,themselves.First, we recall that when the Sx + beam is subjected to an apparatusof type SG2, the beam splits into two components with equal intensities.This means that the probability for the S, + state to be thrown into [S,; ± ),simply denoted as[±), is cach; hence,1(1.4.8)K+IS; +>/=K-IS,; +>IV2We can therefore construct the S,+ ket as follows:1ei8i|-)(1.4.9)[Sx; +> =-/+)+V2V2with 8, real. In writing (1.4.9) we have used the fact that the overall phase(common to both I+ > and I-)) of a state ket is immaterial; the coeficient
271.4.Measurements, Observables, and the UncertaintyRelationsof I+) can be chosen to be real and positive by convention. The S,-ketmust be orthogonal to the S, + ket because the S, + alternative and S, -alternative are mutually exclusive. This orthogonality requirement leads to1101811(1.4.10)ISr; -+v2V2where we have, again, chosen the cocficicnt of I+ > to be real and positiveby convention. We can now construct the operator S, using (1.3.34) asfollows:h[(IS; +XSr; +D)-(ISr; -)Sr; -D)]S.h-181(I+X- D)+e81(1-X+D).(1.4.11)Notice that the S, we have constructed is Hermitian, just as it must be. Asimilar argument with S, replaced by S,leads to11ei82/-),(1.4.12)[Sw; ±>+>+V2V2?e-182(I+)- D)+ ei82(1-+ Dl.(1.4.13)S-Is there any way of determining S, and 8? Actually there is onepiece of information we have not yet used. Suppose we have a beam of spin atoms moving in the z-direction. We can consider a sequential Stern-Gerlachexperiment with SGx followed by SGy. The results of such an experimentare completely analogous to the earlier case leading to (1.4.8):1(1.4.14)KSy;±ISx; +>/=KS; ±/Sx; ->/=Vwhich is not surprising in view of the invariance of physical systems underrotations. Inserting (1.4.10) and (1.4.12) into (1.4.14), we obtain111 ± e (8, -82) =(1.4.15)which is satisfied only if(1.4.16)82-81=/2or—/2.We thus see that the matrix elements of S, and S, cannot all be real. If theS, matrix elements are real, the S, matrix elements must be purely imagin-ary (and vice versa). Just from this extremely simple example, the introduc-tion of complex numbers is seen to be an essential feature in quantummechanics. It is convenient to take the S, matrix elements to be real* and* This can always be done by adjusting arbitrary phase factors in the definition of I+) and-).This point will becomeclearer in Chapter 3,where the behavior of I±)underrotationswill be discussed