量子力学讲义田光善QM3第三章力学量的算符表示与表象变换.pdf

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1、?$3.1?Heisenberg?Schr odinger?p=hi?“?”?Landau?Lifschitz?“QuantumMechanics”?(Operation)?(Predictable)?“?”?n?En?(Operator)H?n?“?”?(Operation)?Hn=Enn.(1)1?Q?qn?n?n?=Xn=1Cnn.(2)?Q=QXn=1Cnn=Xn=1CnQn=Xn=1Cnqnn.(3)?“?”?(?)?Q?(3)?(3)?Q?n?$3.2?2?“+”?3.1:?(?=(0,a)?=(,)?L2()?f L2()?Z|f(r)|2dr (4)?L2()?L2()?f(r

2、)?g(r)?Z|f(r)|2dr ,Z|g(r)|2dr .(5)?G(r)=af(r)+bg(r)(6)?L2()?Z|G(r)|2dr=Z|af(r)+bg(r)|2dr=|a|2Z|f(r)|2dr+|b|2Z|g(r)|2dr+abZf(r)g(r)dr+abZf(r)g(r)dr|a|2Z|f(r)|2dr+|b|2Z|g(r)|2dr+2|ab|sZ|f(r)|2drsZ|g(r)|2dr.(7)?Cauchy-Schwartz?Zf(r)g(r)dr sZ|f(r)|2drsZ|g(r)|2dr.(8)?V?A:V V?A(c1v1+c2v2)=C1Av1+c2Av2,(9)3

3、?v1?v2?V?c1?c2?px=i h/x?L2(,)?(i)A=B?V?A=B?(ii)?I?I=?V?(iii)?A+B?=A+B?(iv)?AB?=A?B?AB 6=BA?L2()?x?px?x px px x=i hI(10)?Tr(x px px x)=Tr(x px)Tr(px x)=0.(11)?x?px?Lx,Ly?Lz?n?An=A AA.(12)?A1?A A1=A1A=I.(13)?Taylor?F(x)=Xn=01n!dndxnF(x)?x=0 xn,(14)4?F(A)=Xn=01n!dndxnF(x)?x=0An.(15)?“?”?u?v?(u,v)?V?(i)

4、?u V,?(u,u)0?(ii)(u,v)=(v,u)?(iii)(u,C1v1+C2v2)=C1(u,v1)+C2(u,v2)?(iv)(C1u1+C2u2,v)=C1(u1,v)+C2(u2,v)?(u,u)=0?u=0?V?ei?V?v?u?v?(u,v)=NXn=1unvn.(16)?L2()?(,)=Zdr(r)(r).(17)?A?A?V?u?v?u,Av?=?Bu,v?,(18)?B?A?A?A=A?A?3.2:?V?A?Aij=(i,Aj).(19)5?Aij=(i,Aj)=(Ai,j)=(j,Ai)=?A?ji.(20)?Aij=Aji.(21)?A?Aij=Aji=Aij

5、.(22)?,A?=?A,?(23)?(?)?3.3:?px=i h/x?L2(,)?(,px)=Z(x)i hx(x)!dx=(i h)(x)(x)?+i hZ x(x)!(x)dx=i hZ x(x)!(x)dx=Z i hx(x)!(x)dx=(px,).(24)?(,px)=?px,?(25)?px=px?L2(,)?px?(1)?A?B?C=A+B?6(2)?A?B?AB=BA?D=AB?3.1:?A?,A?,A?=?A,?=?A,?=?,A?.(26)?,A?,A?=?,A?=?A,?,(27)?1?2?1=1+2,2=1+i2.(28)?1,A1?=?1+2,A(1+2)?=?1

6、,A1?=?A(1+2),1+2?,?2,A2?=?1+i2,A(1+i2)?=?2,A2?=?A(1+i2),1+i2?.(29)?2,A1?+?1,A2?=?A2,1?+?A1,2?,i?2,A1?+i?1,A2?=i?A2,1?+i?A1,2?.(30)?i?2i?1,A2?=2i?A1,2?,(31)?1,A2?=?A1,2?.(32)?1?2?A?7?v?Av=v(33)?A?v?3.2:?,O?O?n?n,On?=(n,nn)=n(n,n).(34)?n=?n,On?(n,n)(35)?Dirac?3.3:?O?m6=n?m?n?m,On?=(m,nn)=n(m,n).(36)?m

7、,On?=?Om,n?=?Om,n?=(mm,n)=m(m,n)=m(m,n).(37)?(m n)(m,n)=0.(38)8?m6=n?(m,n)=0.(39)?O?n?n?n?(k?)?n?k?Schmidt?O?Hilbert?L2()?O?O?O?n?=XnCnn,(40)?On=nn?O?m?m?O?“?”?m?Cm?|Cm|2?m?O?m?hA,Bi=C 6=0.(41)?Heisenberg?A?B?A?B?9?hA,Bi=0?B?A?A?B?A?B?3.4:?hA,Bi=0?nm?Anm=nnm?Bnm=qmnm?140?4.3.3?3.4:?L=r p=r hi!,(42)?

8、hLx,Lyi=i hLz,hLy,Lzi=i hLx,hLz,Lxi=i hLy,(43)?hL2,Lzi=hL2x+L2y+L2z,Lzi=0.(44)?L2?Lz?Lz?Lz=x py y px=hi xy yx!.(45)?x=rsincos,y=rsinsin,z=r cos,(46)?r=qx2+y2+z2,ctg=zx2+y2,tg=yx,(47)?x=rrx+x+x(48)10?y=rry+y+y.(49)?Lz?Lz=hix rry+y+y!hiy rrx+x+x!=hi xy yx!=hix11+?yx?21x y11+?yx?2yx2=hi=i h.(50)?Lx=i h

9、 sin+ctgcos!,Ly=i h cos+ctgsin!.(51)?L2x+L2y=h2 sin+ctgcos!sin+ctgcos!h2 cos+ctgsin!cos+ctgsin!=h2 sin222 cosec2cossin+ctgcos2+2ctgsincos2+ctg2cos(sin)+ctg2cos2222ctgsincos2+cos222+cosec2cossin+ctgsin2+ctg2sincos+ctg2sin222!=h2 22+ctg+ctg222!(52)?L2z=h222.(53)11?L2=L2x+L2y+L2z=h222+cossin+?ctg2+1?22

10、#=h21sin sin!+1sin222#.(54)L2?h21sin sin!+1sin222#(,)=E(,).(55)?(?)?E=l(l+1)h2?Ylm(,)=(1)mvuut(l m)!(l+m)!s2l+14Pml(cos)eim,l m m.(56)?l=0,1,2,?Pml(x)?Pml(x)=12ll!?1 x2?m/2dl+mdxl+m?x2 1?l,m 0,Pml(x)=(1)m(l m)!(l+m)!Pml(x).(57)?Ylm(,)?Lz?LzYlm(,)=m hYlm(,).(58)?m?Legendre?Pml(cos)?Z0Pml(cos)Pml(cos)

11、sind=22l+1(l m)!(l+m)!ll,(59)?Ylm(,)?Z20Z0Ylm(cos)Ylm(cos)sind d=llmm.(60)12?516?$3.3?Hilbert?Dirac?Hilbert?L2()?r?(e1,e2)?(?)?r=xe1+ye2.(61)?x?y?x=(e1,r),y=(e2,r)(62)?L2()?1(r),2(r),n(r),.(63)?Zm(r)n(r)dr=mn.(64)?L2()?(r)?(r)=Xn=1ann(r),(65)13?an?an=Zn(r)(r)dr (n,)(66)?(e1,e2)?(e1,e2)?U?e1e2=U11U12

12、U21U22e1e2.(67)?U?UU=UU=I?r?r=xe1+ye2=xe1+ye1=x(U11e1+U12e2)+y(U21e1+U22e2),(68)?x=U11x+U21y,y=U12x+U22y,(69)?xy=U11U21U12U22xy.(70)?xy=U11U21U12U22xy=U11U12U21U22xy.(71)?Hilbert?L2()?m?(m,n)=mn.(72)?(r)?n?m?(r)=Xann(r)=Xamm(r).(73)14?12.n.=U11U12U21U22.Un1Un2.12.n.,(74)?U=U11U12U21U22=(1,1)(2,1)(1,

13、2)(2,2).(75)?am=XnUmnan.(76)$3.4?L2()?n?O?1?O?e1=O1?L2()?n?e1=O1=O111+O212+.(77)?e2=O2=O121+O222+.(78)?O=O11O12O21O22,Oij=?i,ej?=?i,Oj?.(79)?O?n?15?3.5:?x?L2(R)?Schr odinger?E(x)=h22md2(x)dx2+12m20 x2(x)(80)?n(x)=2nn!1/2e122x2Hn(x),=rm0 h.(81)?n(x)?x?en(x)?en(x)=xn(x)=xn(x)=x1n1(x)+x2n2(x)+=1rn2n1+s

14、n+12n+1(82)?(?85?(3.4.23)?xn1,n=1rn2,xn+1,n=1sn+12.(83)?n?x=101200120q2200q220q32.(84)?Schr odinger?Schr odinger?i ht(r,t)=H(r,t)(85)?H?Hilbert?L2()?L2()?n(?H?)?(r,t)?(r,t)=Xmam(t)m(r).(86)16?Schr odinger?i hXmdam(t)dtm(r)=Xmam(t)Hm(r).(87)?k(r)?r?i hXmdam(t)dtZk(r)m(r)dr=i hXmdam(t)dtkm=i hdak(t)dt

15、=Xmam(t)?k,Hm?=XmHkmam(t).(88)?i hta1(t)a2(t).=H11H12H21H22a1(t)a2(t).(89)?an(t)=anexp?i hEt?(90)?Ea1a2.=H11H12H21H22a1a2.(91)?E?DetH11 EH12H13H21H22 EH23=0(92)?En?a(n)1,a(n)2,?,|a(n)1|2+|a(n)2|2+=1.(93)17?n(r,t)=?a(n)11(r)+a(n)22(r)+a(n)mm(r)+?exp?iEn ht?.(94)?Hilbert?L2()?O?n?n?O?O=O11O12O21O22,(9

16、5)?O=O11O12O21O22.(96)?12.=U12.=U11U12U21U2212.(97)?O=O11O12O21O22=?1,O1?1,O2?2,O1?2,O2?=?(PmU1mm),O(PkU1kk)?(PmU1mm),O(PkU2kk)?(PkU2kk),O(PmU1mm)?(PmU2mm),O(PkU2kk)?18=PmPkU1m?m,Ok?U1kPmPkU1m?m,Ok?U2kPmPkU2k?k,Om?U1mPmPkU2k?k,Om?U2m=U11U12U21U22O11O12O21O22U11U12U21U22T=UOUT=(UT)TOUT=eUOeU.(98)?eU?

17、(?U?)?O?O?$3.5?Hilbert?Hilbert?L2()?L2()?S:L2()L2()?(i,j)=?Si,Sj?=(i,j)(99)?L2()?1?2?(i,j)=(i,j)=ij.(100)?S?Fourier?(p)=1(2 h)d/2Zdr(r)ei hpr.(101)?(1,2)=Zdp 1(p)2(p)19=Zdk 1(2)d/2Zdr11(r1)eikr1!1(2)d/2Zdr22(r2)eikr2!=Z Zdr1dr21(r1)(r2)1(2)dZdkeik(r1r2)#=Z Zdr1dr21(r1)(r2)d(r1 r2)=Zdr11(r1)2(r1)=(1,

18、2).(102)?Fourier?Dirac?(bra)?(ket)?|i?Hilbert?h|?|i?|i?(,)=h|i.(103)?h|i=h|i.(104)?Hilbert?L2()?|ri?r?|pi?hr|ri=(r r),hp|pi=(p p).(105)?Dirac?hr|i=(r)(106)?|i?hp|i=(p)(107)20?3.6:?hx|pi?|pi?p?hx|pi?hx|pi=12 hexp?i hpx?.(108)?pxexp?i hpx?=hixexp?i hpx?=pexp?i hpx?.(109)?hr|pi=1(2 h)3/2exp?i hp r?.(11

19、0)?I=Xn|nihn|.(111)?n?n?|i=Xmam|mi.(112)?Pn|nihn|?Xn|nihn|!|i=Xmam Xn|nihn|!|mi=Xmam Xn|nihn|mi!=Xmam Xn|ninm!=Xmam|mi=|i.(113)?(111)?(111)?(r r)=hr|ri=hr|I|ri=*r?Xn|nihn|!?r+=Xnhr|nihn|ri=Xnn(r)n(r).(114)21?O?h|O|i=h|O|i.(115)?h|O|i=Z Zdrdrh|rihr|O|rihr|i=Z Zdrdr(r)Or,r(r).(116)?h|O|i=Z Zdpdph|pihp

20、|O|pihp|i=Z Zdpdp(p)Op,p(p).(117)?O?O=12m20 r2?O=e2r?Or,r=hr|O|ri=Or(r r).(118)?O?O=p22m?Op,p=Op(p p).(119)?xp,p?px,x?xp,p?xp,p=hp|x|pi=Z Zdxdx hp|xihx|x|xihx|pi=Z Zdxdx 12 hei hpx!x(x x)12 hei hpx!=12 hZdx xei h(pp)x=12 hZdx hipei h(pp)x=hip?12 hZdx ei h(pp)x?=hip(p p).(120)22?12 hZdx ei h(pp)x=(p

21、p).(121)?508?4?px,x=hx|p|xi=hix(x x).(122)?S:L2()L2()(123)?O?S?L2()?O?L2()?O=SOS?L2()?e=S L2()?O?e?e=Oe?L2()?S=e(124)?=S1e=Se=SOe=SOS O.(125)?O=SOS,(126)?O=SOS.(127)?L2()?m?L2()?n?|1i?S?L2()?|i=S|1i?m?S|1i=|i=S11|1i+S21|2i+.(128)23?Si1=hi|S|1i?S|2i=|ei=S12|1i+S22|2i+.(129)?S=S11S12S21S22.(130)?Sij=h

22、i|S|ji?S?S?O?O?(O)h|O|i=h|?SS?O?SS?|i=XmXnh|S|mihm|SOS|nihn|S|i.(131)?h|S|mi=Sm,hn|S|i=?S?n=Sn,(132)?h|O|i=XmXnSm(O)mn?S?n.(133)?O=h1|O|1ih1|O|2ih2|O|1ih2|O|2i,O=h1|O|1ih1|O|2ih2|O|1ih2|O|2i(134)24?O=SOS.(135)?3.7:?Dirac?Schr odinger?L2?Dirac?Schr odinger?i ht|(t)i=H|(t)i.(136)?Dirac?*r?i ht?(t)+=hr

23、|H|(t)i(137)?*r?i ht?(t)+=i hthr|(t)i=i ht(r,t).(138)?hr|H|(t)i=*r?p22m+V(r)?(t)+=*r?p22m?(t)+hr|V(r)|(t)i.(139)?r?V(r)(r,t)?*r?p22m?(t)+=*r?p22mI?(t)+=Zdp*r?p22m?p+hp|(t)i=Zdpp22mhr|pihp|(t)i=Zdpp22m1(2 h)d/2ei hprhp|I(t)i=ZdpZdrp22m1(2 h)d/2ei hprhp|rihr|(t)i=Z Zdpdrp22m1(2 h)d/2ei hpr1(2 h)d/2ei

24、hpr(r,t)=1(2 h)dZ Zdpdrp22mei hp(rr)(r,t)=12m1(2 h)dZ Zdpdr hir!2ei hp(rr)(r,t)25=h22m2rZdr1(2 h)dZdpei hp(rr)#(r,t)=h22m2rZdr(r r)(r,t)=h22m2r(r,t).(140)?(137)?i ht(r,t)=h22m2r(r,t)+V(r)(r,t).(141)?Schr odinger?Dirac?Schr odinger?278?279?(1)?4.4?“?”?(2)?129?13?(4.1.52)?(3)?130?15?(4)?152?4.25?(5)?153?4.37?(6)?6.2?6.4?6.6?26