Fundamental ConceptsYFIGURE 1.5.Orientations of the x'-and y'-axes.classical electrodynamics. Using Figure 1.5 we obtainEox'cos(kz-wt)=Excos(kz -wt)+cos(kzLV2Txcos(kz - wt)+Eoy'cos(kz-wt)= Eoycos( kz - wV2V2(1.1.8)In the triple-filter arrangement of Figure 1.4b the beam coming out of thefirst Polaroid is an x-polarized beam, which can be regarded as a linearcombination of an x'-polarized beam and a y'-polarized beam. The secondPolaroid selects the x'-polarized beam, which can in turn be regarded as alinear combination of an x-polarized and a y-polarized beam. And finally,the third Polaroid selects the y-polarized component.Applying correspondence (1.1.7) from the sequential Stern-Gerlachexperiment of Figure 1.3c, to the triple-filter experiment of Figure 1.4b suggests that we might be able to represent the spin state of a silver atom bysome kind of vector in a new kind of two-dimensional vector space, anabstract vector space not to be confused with the usual two-dimensional(xy) space. Just as x and y in (1.1.8) are the base vectors used to decomposethe polarization vector x' of the x'-polarized light, it is reasonable torepresent the S, + state by a vector, which we call a ket in the Diracnotation to be developed fully in the next section. We denote this vector by
91.1.The Stern-Gerlach Experiment[S; +> and write it as a linear combination of two base vectors, JS.; +)and JS,; -), which correspond to the S, + and the S, - states, respectively.So we may conjecture12(1.1.9a)IS.;S.:+>+/S.;V2V211(1.1.9b)SIS.:-IS.:?2V2in analogy with (1.1.8). Later we will show how to obtain these expressionsusing the general formalism of quantum mechanics.Thus the unblocked component coming out of the second (SGx)apparatus of Figure 1.3c is to be regarded as a superposition of S + andS, - in the sense of (1.1.9a). It is for this reason that two componentsemerge from the third (SG2) apparatus.The next question of immediate concern is, How are we going torepresent the S, + states? Symmetry arguments suggest that if we observean S, ± beam going in the x-direction and subject it to an SGy apparatus,the resulting situation will be very similar to the case where an S, ± beamgoing in the y-direction is subjected to an SGx apparatus. The kets for S, +should then be regarded as a linear combination of JS,; ±>, but it appearsfrom (1.1.9) that we have already used up the available possibilities inwriting ISx; ±). How can our vector space formalism distinguish S,±states from S,± states?An analogy with polarized light again rescues us here. This time weconsider a circularly polarized beam of light, which can be obtained byletting a linearly polarized light pass through a quarter-wave plate. When wepass such a circularly polarized light through an x-filter or a y-filter, weagain obtain either an x-polarized beam or a y-polarized beam of equalintensity. Yet everybody knows that the circularly polarized light is totallydifferent from the 45-linearly polarized (x'-polarized or y-polarized) light.Mathematically, how do we represent a circularly polarized light? Aright circularly polarized light is nothing more than a linear combination ofan x-polarized light and a y-polarized light, where the oscillation of theelectric field for the y-polarized component is 90° out of phase with that ofthe x-polarized component:*1(1.1.10)xcos(kz - wt)+yCE=Ez-wtV2It is more elegant to use complex notation by introducing e as follows:(1.1.11)Re(e) = E/Eo.*Unfortunately, there is no unanimity in the definition of right versus left circularlypolarized light in the literature
10Fundamental ConceptsFor a right circularly polarized light, we can then writeel(kz-wr)+Yel(kz-(1.1.12)V2where we have used i = elm/2.We can make the following analogy with the spin states of silveratoms:S, + atom right circularly polarized beam,(1.1.13)S, - atom left circularly polarized beamApplying this analogy to (1.1.12), we see that if we are allowed to make thecoefficients preceding base kets complex, there is no difficulty in accommo-dating the S, ± atoms in our vector space formalism:？11(1.1.14)[Sy; ±)[S,; +)±IS,;->,V2V2which are obviously different from (1.1.9). We thus see that the two-dimen-sional vector space needed to describe the spin states of silver atoms mustbe a complex vector space; an arbitrary vector in the vector space is writtenas a linear combination of the base vectorsS,; ±) with, in general, complexcoefficients. The fact that the necessity of complex numbers is alreadyapparent in such an elementary example is rather remarkable.The reader must have noted by this time that we have deliberatelyavoided talking about photons. In other words, we have completely ignoredthe quantum aspect of light; nowhere did we mention the polarization statesof individual photons. The analogy we worked out is between kets in anabstract vector space that describes the spin states of individual atoms withthe polarization vectors of the classical electromagnetic field. Actually wecould have made the analogy even more vivid by introducing the photonconcept and talking about the probability of finding a circularly polarizedphoton in a linearly polarized state, and so forth; however, that is notneeded here. Without doing so, we have already accomplished the main goalof this section: to introduce the idea that quantum-mechanical states are tobe represented by vectors in an abstract complex vector space.*1.2. KETS, BRAS, AND OPERATORSIn the preceding section we showed how analyses of the Stern-Gerlachexperiment lead us to consider a complex vector space. In this and the* The reader who is interested in grasping the basic concepts of quantum mechanics througha careful study of photon polarization may find Chapter 1 of Baym (1969) extremelyilluminating
111.2.Kets,Bras,andOperatorsfollowing section we formulate the basic mathematics of vector spaces asused in quantum mechanics. Our notation throughout this book is the braand ket notation developed by P. A. M. Dirac. The theory of linear vectorspaces had, of course, been known to mathematicians prior to the birth ofquantum mechanics, but Dirac's way of introducing vector spaces has manyadvantages, especially from the physicist's point of view.Ket SpaceWe consideracomplexvector spacewhosedimensionality is specifiedaccording to the nature of a physical system under consideration. InStern-Gerlach-type experiments where the only quantum-mechanical de-gree of freedom is the spin of an atom, the dimensionality is determined bythe number of alternative paths the atoms can follow when subjected to aSG apparatus; in the case of the silver atoms of the previous section, thedimensionality is just two, corresponding to the two possible values S, canassume.* Later, in Section 1.6, we consider the case of continuousspectrafor example, the position (coordinate) or momentum of a particlewhere the number of alternatives is nondenumerably infinite, in whichcase the vector space in question is known as a Hilbert space after D.Hilbert, who studied vector spaces in infinite dimensions.In quantum mechanics a physical state, for example, a silver atomwith a definite spin orientation, is represented by a state vector in a complexvector space. Following Dirac, we call such a vector a ket and denote it byJa).This state ket is postulated to contain complete information about thephysical state; everything we are allowed to ask about the state is containedin the ket. Two kets can be added:(1.2.1)[α)+/β)= /)The sum [> is just another ket. If we multiply lα) by a complex number c,the resulting product ca) is another ket. The number c can stand on theleft or on the right of a ket; it makes no difference:(1.2.2)c[α) = [α)c,In the particular case where c is zero, the resulting ket is said to be a nullket.One of the physics postulates is that la) and cla), with c+ O,represent the same physical state. In other words, only the "direction"' invector space is of significance. Mathematicians may prefer to say that we arehere dealing with rays rather than vectors.*For many physical systems the dimension of the state space is denumerably infinite. Whilewe will usually indicate a finite number of dimcnsions, N, of thc ket space, the rcsults also holdfordenumerablyinfinitedimensions
12Fundamental ConceptsAn observable, such as momentum and spin components, can berepresented by an operator, such as A, in the vector space in question. Quitegenerally, an operator acts on a ket from the left,(1.2.3)A.(α)) = A[α),which is yet another ket. There will be more on multiplication operationslater.In general, Ala) is not a constant times [α). However, there areparticular kets of importance, known as eigenkets of operator A, denoted by(1.2.4)[a'"),[a"),[a"),...with the property(1.2.5)A|a')=a'la"),A|a")=a"la"),..where a', a", ... are just numbers. Notice that applying A to an eigenketjust reproduces the same ket apart from a multiplicative number. The set ofnumbers (a',a",a",...), more compactly denoted by (a'), is called theset of eigenvalues of operator A. When it becomes necessary to ordereigenvalues in a specific manner, (a(1), a(2), a(3), ..) may be used in placeof fa',a",a",...The physical state corresponding to an eigenket is called an eigen-state. In the simplest case of spin systems, the eigenvalue-eigenket relation(1.2.5) is expressed ashh(1.2.6)S,Is,; +)IS;+>，2IS;-),SS,;->=2.where [S.; ±) are eigenkets of operator S, with eigenvalues + h/2. Here wecould have used just |h/2> for IS,; +>in conformity with the notation [a'),where an eigenket is labeled by its eigenvalue, but the notation IS,; +),already used in the previous section, is more convenient here because wealso consider eigenkets of S,:(1.2.7)SxIS;±)=±ISt;±>25We remarked earlier that the dimensionality of the vector space isdetermined by the number of alternatives in Stern-Gerlach-type experi-ments. More formally, we are concerned with an N-dimensional vectorspace spanned by the N eigenkets of observable A. Any arbitrary ket lα)can be written as[α)=Zcala'),(1.2.8)with a', a", .. up to a(N), where ca. is a complex coeficient. The questionof the uniqueness of such an expansion will be postponed until we prove theorthogonality of eigenkets