第七章 量子散射理论

1127.1()ABAB+→+*ABAB+→+(*AA)ABCD+→++ABABCD:Born,,Green:()7.21131zθ,θϕJJ,θϕdΩdndnJdndΩdNJdΩ(,)dnJdσθϕ=Ω(,)dnJdσθϕ=Ω(,)σθϕdΩ200(,)sintddππσσθϕθθϕ=∫∫2112()Vrr−1142222121212()22totalVrrEmm−∇−∇Ψ+−Ψ=Ψ22()2VrEµ−∇Ψ+Ψ=ΨµrEE2r→∞r→∞z1exp()AikzΨ=2(,)exp()/BikrrθϕΨ=kkB,θϕr→∞(,)ikrikzreAeBrθϕ→∞Ψ→+(,)(,)BAfθϕθϕ=(,)ikrikzreAefrψθϕ→∞→+(,)fθϕ()Ur(,)fθϕ3(,)fθϕ()()/ikzikrrAeferψθ→∞→+r(,)θϕ22(,)/Afrθϕ(,)θϕ2dSrd=Ω115222(,)(,)uAfdndSuAfdrθϕθϕ==ΩdΩuJ2JAu=2(,)(,)dnfJdσθϕθϕ==Ω(,)fθϕ(,)fθϕ4→(,)θϕ→(,)σθϕ→2(,)(,)fσθϕθϕ=(,)fθϕϕ(,)()ffθϕθ→11i(2)(2)0014(21)()2rkzlikrlikrlillelieeYikrππψπθ∞→∞−−−==→+⋅−∑2()(,)llmlRkrYψθϕ=∑22221dd10ddlllrkUrRrrrr++−−=22()()UrVrµ=3()fθ116()()llffθθ=∑()lfθl04(21)()()()ikrllllelijkrYfrπθθ+→l→l21++)(2)()12(4~)(krhakrjilkrRlllllπ[]ikreeailkrRlkrilkrillrl2)1()12(4)(2()2(πππ−−−∞→−++→la0la=1exp(2)2exp()sinllllaiiiδδδ=−=()4(21)sin(2)sin(2)likrlllllARkrliekrlkrlkrδππδπδ→∞→+−+=−+llR2(,)sin(2)(cos)lrlllArkrlPkrψθπδθ→∞→−+∑(,)(,)ikrikzrerAefrψθθϕ→∞→+31{}(cos)lPθi011(21)sin(cos)2rkzlllelikrlPkrπθ∞→∞=→+⋅−∑i2()fθ117()01sin(2)(cos)(21)sin/2(cos)()ikrlllllllAekrlPlikrlPfkrkrrπδθπθθ∞=−+=+⋅−+∑∑i3lδ+=+=−+=∑∑∑∞=∞=∞=lltllillillkYelkPelikfllδπσθδπθσθθδδ20220020sin)12(4)(sin124)()(cos)1)(12(21)(41)1(=lP∑∞=+=02sin)12(1)0(Imlllkfδlltlkδπσ202sin)12(4∑∞=+=)0(Im4fktπσ=)0(f5Lippman-Schwinger1dingeroSchr()()()()rrVrkΨ=Ψ+∇2222µ()rΨ()()referikrrkirϕθ,+→Ψ•∞→Green()rrG′,118()()()rrrrGk′−=′+∇δ,22()()()()rrVrrGrdr′Ψ′′′=Ψ∫,232µdingeroSchr()rψ()()()()()()rrVrrGrdrr′Ψ′′′+Ψ=Ψ∫,2320µdingeroSchr()()r0Ψ()()()0022=Ψ+∇rk2Lippman-Schwinger()rkiier•=Ψk()0ΨiΨ()()()()322,ikrredrGrrVrrµψψ•′′′′=+∫()()rrsciΨ+Ψ=Lippman-SchwingerGreen()()()()()refrrVrrGrdrikrrscϕθµ,,232→′Ψ′′′=Ψ∞→∫3Green()()()rrrrGk′−=′+∇δ,22()rrG′,()rrG′−Fourier()()()qGqedrrGrrqi~3′−•∫=′−()()()rrrrGk′−=′+∇δ,22()()()223121~kqqG−−=π()()()rrqiekqqdrrG′−•−−=′−∫2233121π119rrR′−=()ikReRRGπ41−=()rrerrGrrik′−−=′−′−π4()()()322ikrrikreredrVrrrrµψψπ′−•′′′=−′−∫Born1()VrLippman-Schwinger()r′Ψrkie′•()()rkirrikrkierVrrerder•′−•′′−′−=Ψ∫322πµBorn2∞→r()Vr∞→r()()1222221/rrrrrrrrrr′′′′−=−⋅+≈−⋅rr′−∞→rrr′()rkiikrrrikrrrikfeee′•−′•−′−=≈1ffkrkrk,=kkkf==′()()()rVerdrerrkkiikrrscf′′−→Ψ∫′⋅−−∞→322πµ120()rΨ()()rVerdfrqi′′−=′•−∫322,πµϕθkkqf−=qθqz3()()rdrqrVrqf′′′′−=∫∞sin2,02µϕθ2sin(/2)qkθ=()()()202422sin4rdrqrVrqf′′′′==∫∞µθθσq()θσk()θσBorn,Bornl7.3110IH→t→∞∓2321abc1212SoutinSΨ=ΨinipiΨ→=outfpfΨ→=fSi=δ30iHHH=+00//,()()()ttiiiiHtiiHtIIIHtHteHteeHeεεε−−−→==0ε0ε→εab,()()()iIIIitHtttεεε∂Ψ=Ψ∂00//,()tiiHtiiHtIHteeHeεε−−=,0()exp()tiIIiUtTHdεττ=−∫40H()inεΨ=Ψ−∞()outεΨ=Ψ+∞()()SεεΨ+∞=Ψ−∞S00limlim(,)(,)ttSUttU→+∞→−∞=≡+∞−∞aUUb0iHHH=+00()()iiinoutHHHH+=+122c0inputHiH0H0inputH3SSSS4Mllerφ±ΩS1(0,)U+Ω≡−∞(0,)U−Ω≡+∞2(,0)U++Ω=−∞(,0)U+−Ω=+∞±Ω3(,)(,)(,)SUUtUt=+∞−∞=+∞−∞(,0)(0,)SUU+−+=+∞−∞=ΩΩS11,1(,)()iIiSUIdHεττ+∞−∞−=+∞−∞=+∫212,1,2()()iiIIiddHHεεττττ+∞+∞−∞−∞−+∫∫3123,1,2,3()()()iiiIIIidddHHHεεεττττττ+∞+∞+∞−∞−∞−∞−++∫∫∫20iHiEi=0fHfEf=0mHmEm=(1)S(0)fifSiδ=(2)S(1)2()ififSiiEEfHiπδ=−−(3)S(2)112()iifimimfSiiEEfHmmHiEEiπδε=−−−+∑(4)S123(3)122()()()iiifimniminfHmmHnnHifSiiEEEEiEEiπδεε=−−−+−+∑1iffiSfSi=0t=0t=(0,)iUiiψ++=−∞=Ω0t=(0,)fUffψ−−=+∞=Ω(,0)(0,)fifiSfSifUUifiψψ+−+−+==+∞−∞=ΩΩ=2T1S()ifEEδ−S2TTSI=−2()fifiTfSIiiEEfTiπδ=−=−−2iπ3()222()limiffifiPfSiEETπδ∞→Τ→∞==−ΤΤ22()fifiifdPEETdtπδ→=−fEE=fEE=dE,θϕ124dΩ()ffdNEdEdρ=Ωf/22/22()()ffEEfifiEEifdPEETEdEdtπδρ+∆−∆→=−∫22()ifGroupffiEEEifdPETdtπρ==→=3(,)σθϕ,θϕ0I0(,)(,)dIdIσθϕσθϕ==Ω(,)GroupifdPIdtθϕ→=20()2(,)ifffiEEEETIρπσθϕ===422tmiimITkσ=−tσLippman-Schwinger0iHHH=+1Lippman-SchwingeriiiHEψψ=1250iH→iiψ→ifψ→iiiHEψψ=()()000iiiiiEHHEHiψψ−=−=Green()1000()limiiGEEHiεε+−±→=−±0ε0iiiiGHψψ±±±=+Lippman-Schwinger2Lippman-Schwinger{}iψ±{}iHiiiHEψψ±±=1{}iψ+{}iψ+{}iiHGreen0()iGE+H2{}iψ−{}iψ−iHGreen0()iGE−{}iH3iE3Green10H20HHilbertGreen0()iGE±0H0()iGE±()()011mimimmEEiEHiεε=−±−±∑3Green1[]E−1264TST2()fifiTfSIiiEEfTiπδ=−=−−ST12()()()iiiiiimmnimiminfHmmHifHmmHnnHifTifHiEEiEEiEEiεεε=+++−+−+−+∑∑()10iEHiε−−+()()011mimimmEEiEHiεε=−+−+∑01()()iiiimimifHmmHifHHiEEiEHiεε=−+−+∑01()iiiifTifHifHHiEHiε=+−+001211()()iiiiifHHHiEHiEHiεε++−+−+iE00012111()()()iiiiiiiiiTHHHHHHEHiEHiEHiεεε=+++−+−+−+0()iiTHHGET±=+5Green0iiiiGHψψ+++=+iH0()iiTHHGET±=+i()010iiiiHIHTiEHiψε+−−=−+iiHTiψ+=127TiiHiψ+iiHTiψ+=Hiψ+0iiGTiψ++=+0iiiiGHψψ+++=+iψ+0iiiiGHψψ+++=+()0iEHiε−+()iiEHiiiεψε+−+=Green()10()limGEEHiεε+−±→=−±0εGreeniψ+1()iiiiiGEiEHiψεε++==−+1()iiiiiGEiEHiψεε−−==−−iiiHEψψ=()iiiiGEHiψ±±=+7.41281Lippman-Schwinger0iiiiGHψψ±±±=+iiiHEψψ=P7022Green3sp4-0E→s012()0VraVrraσσ≤=i00V1σ2σPauli