高等量子力学(学校讲义).pdf

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1、Chapter1.HilbertSpace§1.LinearVectorSpacesDefinealinearvectorspaceasfollows.Theoperation“+”onasetofelementsS={x,y,z,...}istohavethefollowing4properties:1){x∈S,y∈S}⇒z≡x+y∈S∀x,y∈S;2)Theoperation“+”iscommutativeandassociative;x+y=y+x,(x+y)+z=x+(y+z)=x+y+z.3)∃auniqueelement03x+0=x∀x∈S;4)∀x∈S∃−x3

2、x+(−x)=0.Thisis,alltheelementsofSformanabeliangroupwithrespectto“+”operation.Furthermore,ifthequantitiesα,β,γ,...aretheelementsofsomefieldF,thatwesimplycallscalars,thenwerequirethatmultiplicationofanelementofSbyascalarsatisfy(∀x,y∈S,∀α,β∈F):5)α(βx)=(αβ)x;6)(α+β)x=αx+βx;7)α(x+y)=αx+αy;8)e·x=x

3、(eistheunitelementinF).Definition:anyspaceSthatisclosedundertheoperationsofadditionandofmultiplicationbyscalarsiscalledalinearvectorspace.AllelementsinSarecalledvectors.Someexamplesoflinearvectorspaces:1.n-dimensionalEuclideanspaceEn,x=(ξ1,ξ2,...,ξn),y=(η1,η2,...,ηn),x+y=(ξ1+η1,ξ2+η2,...,ξn+

4、ηn),αx=(αξ1,αξ2,...,αξn).Thevectorspaceisrealorcomplexaccordingtowhetherthecomponentsofthevectorsarerealorcomplex.WehereaftertakeourfieldFofscalarstobethefieldofallrealorofallcomplexnumbers.Inquantummechanics,weshallusuallydealwithcomplexspaces,simplytreatrealspacesasspecialcasesofthese.Inthi

5、scase,theunitelementeofFisthenumber1.12.Acountableinfinite-dimensionalvectorspaceE∞whereavectorisrepresentedasx=(ξ1,ξ2,...,ξn...,)withacountableinfinityofcomponents,ξj,j=1,2,3,....3.Thesetofallcomplexnumbers.Thisisacomplexvectorspacewherex+yandαxareordinarycomplexadditionandmultiplication,res

6、pectively.4.Thesetofallcontinuousfunctions,f(x),x∈[0,1].Hereeachvectorhasanuncountableinfinityofcomponents.5.Thesetofallcomplex-valuedfunctions,f(x),x∈[a,b],suchthat

7、f(x)

8、2is(Lebesgue)square-integrableover[a,b].ThisspaceisusuallydenotedbyL2[a,b].§2.InnerProductandDual(Conjugate)SpaceNowwedefi

9、netheinnerproductoftwovectorsxandyinalinearvectorspaceanddenoteitby

10、y>.ForEnandE∞wesimplyextendthedefinitionoftheinner(orscalar)productfromE3.ForEnwewriteXn∗

11、y>≡ξjηj,(2.1)j=1whileforE∞X∞∗

12、y>≡ξjηj,(2.2)j=1providedthisinfiniteseriesconverges.Hereafterw