131.2. Kets, Bras, and OperatorsBra Space and Inner ProductsThe vector space we have been dealing with is a ket space. We nowintroduce the notion of a bra space, a vector space “dual to" the ket space.Wepostulate that corresponding to everyket [α> there exists a bra, denotedby (αl, in this dual, or bra, space. The bra space is spanned by eigenbras(<a') which correspond to the eigenkets (Ja")). There is a one-to-onecorrespondence between a ket space and a bra space:D[α)<aa.........(1.2.9)Iα)+ Iβ) (α|+(βIwhere DC stands for dual correspondence. Roughly speaking, we can regardthe bra space as some kind of mirror image of the ket spaceThe bra dual to clα) is postulated to be c*(al, not c(al, which is avery important point. More generally, we have(1.2.10)Cαlα) + cplβ) α c(α)+ c(βl.We now define the inner product of a bra and a ket.* The product iswritten as a bra standing on the left and a ket standing on the right, forexample,(1.2.11)<βα) =(<βI)-(Iα)) .bra (c)ketThis product is, in general, a complex number. Notice that in forming aninnerproduct we always take one vectorfrom thebra spaceand onevectorfromtheket space.We postulate two fundamental properties of inner products. First,(1.2.12)(β[α) =(α[β)*.In other words, <βlα) and <alβ) are complex conjugates of each other.Notice that even though the inner product is, in some sense, analogous tothe familiar scalar product a-b, <βa) must be clearly distinguished from(αβ); the analogous distinction is not needed in real vector space becausea-b is equal to b-a. Using (1.2.12) we can immcdiatcly dcducc that (αlα)must bc a real number.To prove this just let <β|-→<αl.*In the literature an inner product is often referred to as a scalar product because it isanalogous to a·b in Euclidean space; in this book, however, we reserve the term scalar for aquantity invariant under rotations in the usual three-dimensional space
14Fundamental ConceptsThesccond postulate on innerproducts is(1.2.13)<αα) ≥0,where the equality sign holds only if a) is a null ket. This is sometimesknown as the postulate of positive definite metric. From a physicist's pointof view, this postulate is essential for the probabilistic interpretation ofquantum mechanics, as will become apparent later.*Two kets [α) and |β) are said to be orthogonal if(1.2.14)(α|β) =0,even though in the definition of the inner product the bra (αl appears. Theorthogonality relation (1.2.14)also implies,via(1.2.12),(1.2.15)<β|α>= 0.Given a ket which is not a null ket, we can form a normalized ket[α),where1(1.2.16)[ααV<αlα)with the property(1.2.17)(αα) =1.Quite generally, y(ala) is known as the norm of la), analogous to themagnitude of vector Va-a = [al in Euclidean vector space. Because [α) andca) represent the same physical state, we might as well require that the ketswe use for physical states be normalized in the sense of (1.2.17).tOperatorsAs we remarked earlier, observables like momentum and spin com-ponents are to be represented by operators that can act on kets. We canconsider a more general class of operators that act on kets; they will bedenoted by X, Y, and so forth, while A, B, and so on will be used for arestrictiveclass of operators that correspond toobservables.An operator acts on a ket from the left side,(1.2.18)X-(Iα))= X[α),and the resulting product is another ket. Operators X and Y are said to beequal,X=Y.(1.2.19)*Attempts to abandon this postulate led to physical theories with “indefinite metric." Weshall not bc conccrncd with such thcories in this book.tFor eigenkets of observables with continuous spectra,different normalization conventionswill be used; see Section 1.6
151.2. Kets, Bras, and Operatorsif(1.2.20)X[α) = Y[α)for an arbitrary ket in the ket space in question. Operator X is said to bethe null operator if, for any arbitrary ket [α), we have(1.2.21)X[α) = 0.Operators can he added; addition operations are commutative and associa-tive:(1.2.21a)X+Y-Y+ X,X+(Y+ Z)= (X+Y)+Z(1.2.21b)With the single exception of the time-reversal operator to be considered inChapter 4, the operators that appear in this book are all linear, that is,(1.2.22)X(calα)+cplβ))=CαXα)+CgXIβ).An operator X always acts on a bra from the right side(1.2.23)(<α). X =(α|X,and the resulting product is another bra. The ket Xla) and the bra (aαlXare, in general, not dual to each other. We define the symbol Xt asXja) <αlXt.(1.2.24)The operator Xt is called the Hermitian adjoint, or simply the adjoint, of X.An operator X is said to be Hermitian ifX= Xt.(1.2.25)MultiplicationOperators X and Y can be multiplied. Multiplication operations are,in general, noncommutative, that is,(1.2.26)XY+ YX.Multiplication operations are, however, associative:(1.2.27)X(YZ) =(XY)Z= XYZWe also haveX(Y[α)) =(XY)Iα) = XY[α),(<βIX)Y=<βI(XY) =<βIXY.(1.2.28)Notice that(Xy)t=ytxt(1.2.29)
16Fundamental ConceptsbecauseXYla) = X(Ylα)) ((αlt)xt=(α|Ytxt.(1.2.30)So far, we have considered the following products: <βlαa),Xjα),<a|X, and XY. Are there other products we are allowed to form? Let usmultiply Iβ)and<αl, in that order.The resulting product(1.2.31)(Iβ))((αl) =Iβ)(αlis known as the outer product of β) and (α|. We will emphasize in amoment that β)(αis to be regarded as an operator; hence it is funda-mentally different from the inner product βα), which is just a number.There are also “"illegal products." We have already mentioned thatan operator must stand on the left of a ket or on the right of a bra. In otherwords, [α)X and X(αl are examples of illegal products. They are neitherkets, nor bras, nor operators; they are simply nonsensical. Products likeJα)Iβ) and <αKBl are also illegal when la) and Iβ) ((αl and <βD are ket(bra) vectors belonging to the same ket (bra) space.*The Associative AxiomAs is clear from (1.2.27), multiplication operations among operatorsare associative. Actually the associative property is postulated to hold quitegenerally as long as we are dealing with “"legal"' multiplications anong kets,bras, and operators. Dirac calls this important postulate the associativeaxiom of multiplicationTo illustrate the power of this axiom let us first consider an outerproduct acting on aket:(1.2.32)(Iβ)<αl)-)Because of the associative axiom, we can regard this equally well as(1.2.33)Iβ).((α/))where <aly) is just a number. So the outer product acting on a ket is justanother ket; in other words, Iβ)(a can be regarded as an operator. Because(1.2.32) and (1.2.33) are equal, we may as well omit the dots and let[β)(α/standfortheoperator/β)(αactingonl)or,equivalently,thenumber (αl) multiplying Iβ). (On the other hand, if (1.2.33) is written as((aly))-lβ), we cannot afford to omit the dot and brackets because the*Later in the book we will encounter products like a)Iβ),which are more appropriatelywritten as [α)iβ).but in such cases lα)and IB)always refer tokets from different vectorspaces. For instance, the first ket belongs to the vector space for electron spin, the second ket tothe vector space for electron orbital angular momentum; or the firstket lies in thevector spaceof particle 1, the second ket in the vector space of particle 2,and so forth
171.3. Base Kets and Matrix Representationsresulting expression would look illegal.) Notice that the operator Iβ)Kαlrotates l> into the direction of 1β). It is easy to see that if(1.2.34)X=Iβ>αl,thenXt= [α)<βl,(1.2.35)which is left as an exerciseIn a second important illustration of the associative axiom, we notethat(1.2.36)(<β) . (XJα)) =((βX) . ([α))ketketbrabraBecause the two sides are equal, we might as well use the more compactnotation(1.2.37)<β[X[α)to stand for either side of (1.2.36). Recall now that <a|Xt is the bra that isdual toXα),so<BIXIα)=<BI-(XIα))= ((<α|Xt)-Iβ)) *(1.2.38)=(α|Xβ)*,where, in addition to the associative axiom, we used the fundamentalproperty of the inner product (1.2.12). For a Hermitian X we have(1.2.39)<β[X|α) =(α|XIβ)*1.3. BASE KETS AND MATRIX REPRESENTATIONSEigenkets of an ObservableLet us consider the eigenkets and eigenvalues of a Hermitian oper-ator A. We use the symbol A, reserved earlier for an observable, because inquantum mechanics Hermitian operators of interest quite often turn out tobe the operators representing some physical observables.We begin by stating an important theorem:Theorem. The eigenvalues of a Hermitian operator A are real; theeigenkets of A corresponding to different eigenvalues are orthogonal.Proof. First, recall that(1.3.1)A|a')= a[a')