对称性原理及其应用

2397.17.1.117-12400122233223()1()()2()33ababababaabbabaababb+=+=++=+++=+++290907.1-12417-2a7-2bab1807.1-1abbab342yx=y22()yxx=−=x12(1)yx=−x12(1)yx=−y22(1)(1)yxx=−−=+2427.1.21PPrr=−KK(7.1.1a)Pxx=−s()PnKnKs()Pnrrr⊥=−&KKKK(7.1.1b)()rrnn=××&KKKKs()rrnn⊥=⋅KKKKs()PxPixx==−KaK243()TaK()Tarra=+KKKK(7.1.2)T3x()()TaTai=KT11(),{()|}TaxxaTTaaR=+=∈(7.1.3)rxiyjzk=++KKKK''''rxiyjzk=++KKKKzα()Cα'cossin0()'sincos0'001xxxCyyyzzzααααα−==(7.1.4)2p(2)Cαπ+()Cαz()Rα1{()|[0,2)}RCααπ=∈2()TaK()TbK()()()()()()()TbTarTbrarabrabTabr=+=++=++=+KKKKKKKKKKKKKKK(7.1.5a)()()()TbTaTab=+KKKK(7.1.5b)()()()(0)TaTarTaarTrr−=−==KKKKKKKK(7.1.6)()Ta−K()TaK(0)TI=244()[()()]()[()()]()TaTbTcTabcTaTbTc=++=KKKKKKKKK(7.1.7)I(7.1.7)T3T3(7.1.5b)()()()()()()TaTbTbaTabTbTa=+=+=KKKKKKKK(7.1.8)xT1T3T3T1SO33RSO21()RnKnK1()RnK3RzR111()RRk=K3R(2)()CCαπα+=()Cα(2)Cπα−(),2()()(2),2CCCCαβαβπαβαβπαβπ++=+−+≥(7.1.9)SO3SO2PPI⋅=(7.1.10){PI}{P}()()PnPnI⋅=KK(7.1.11){(),}PnIK245{()}PnK3()rψKQ()rψKrKQ'rQr=KK'rK(')()()rQrrψψψ==KKK(7.1.12a)1()()rQrψψ−=KK(7.1.12b)ˆQQˆ()()Qrrψψ=KK(7.1.13)(7.1.12)(7.1.13)1ˆ()()QrQrψψ−=KK(7.1.14)ˆQ1ˆ()()()()PrPrPrrψψψψ−===−KKKK(7.1.15)xˆ()()[()]()TaxTaxxaψψψ=−=−(7.1.16)ˆ()0011ˆ()()()()()()!!nnnaDnnxaxaaDxexnnψψψψ∞∞−==−=−=−=∑∑(7.1.17)xˆˆ()aDTae−=(7.1.18)246ˆ()aTae−⋅∇=KK(7.1.19)zˆ()()[()]()CCaαψϕψαϕψϕ=−=−(7.1.20)ˆ()()zLeαψϕαψϕ−−=(7.1.21)ˆzLϕ∂∂=zˆˆ()zLReαα−=(7.1.22a)ˆˆˆ,zyxLxykLLrϕ∂∂∂∂∂∂==−=⋅=×∇KKk(7.1.22a)ˆˆˆ(),kLLReekααααα−⋅−⋅===KKKK(7.1.22b)nKaˆˆˆ(),,LReLrnαααα−⋅==×∇=KKKKK(7.1.23)()rψK()()()()1()()()(11),()()()()2eeeooorrrrrrrrrrrψψψψψψψψψψψ+−=+==−−KKKKKKKKKKK(7.1.24)()()()10ˆˆ()(11)(11)(11)()()()01eeeooorrrPrPrrrψψψψψψψ+===−−KKKKKKK(7.1.25)24710ˆ01P=−(7.1.26)4()/Qaxxa=(7.1.27)1ˆ()()()[()]()QaxxQaxaxψψψψ−===(7.1.28)ˆlnˆ()axDQae−⋅=(7.1.29)()'Qtttττ==+(7.1.30)()'/Qattta==(7.1.31)'()'xxxvtQvttt−==(7.1.32)ˆ()(,)[()(,)](,)QvfxtfQvxtfxvtt=−=+(,)fxt001(,)1(,)()()(,)(,)!!xnvtnnnnnfxtfxvttvtvtfxtefxtnxnx∞∞∂==∂∂+===∂∂∑∑248222'1()'/1/xxxvtQvtttvxcvc−==−−(7.1.33)222sinh,cosharctanh(/)xssctxctsxctθθθ==−==2222222222221(')''[(/)()]1/'tanhtanhtanh'tanh()'/11tanhtanhxvctcxvctcsctxctvxcxvtctxsvcxxvtctctvxcθβθθβθβ=−=−−−=−=−−−−=====−−−⋅−⋅tanh/vcβ='()'sssQvθθθβ==−ˆ()(,)[()(,)](,)QvfsfQvsfsθθθβ=−=+7.1.31Q()rψK1ˆ()()()QrQrrψψψ−==KKK(7.1.34)()rψK()rψKG(7.1.34)11ˆ()()()()QrQrQQrrψψψψ−−===KKKK(7.1.35)G()rψK()rψK2497.1-1()(,)rfxyψ=Kzz1{()|}GTzkzR=∈Kxy2()GPk=KG12GGG=×7.1-2()(,)rfzψρ=KjzSO2z2(),GPnnk=⊥KKK7.1-3()()rfrψ=KSO3SO3O37.1-4()()rfψρ=Kzzz2()(,)sin,rfrmmZψθϕ=∈Kjsin(2/)sinmmmϕπϕ+=ˆˆ()()(,)()sin(,)sin()RrfrRmfrmαψθαϕθϕα==−K(7.1.36)2/mαπ=ˆ()()()Rrrαψψ=KK(7.1.37)ˆ()Rα250ˆ()()(),0Rnrrnmαψψ=≤KK(7.1.38)ˆˆˆ()[()],()nRnRRmIααα==ˆ{()|2/,0}Rnmnmααπ=≤(7.1.39)mmCmCmR1{ˆ()Rα}1122ˆ()()(,)sin()(,)sin()()Rrfrmfrmrαψθϕαθϕπψ−=−−=−−=KK12ˆ()Rα−()(,)sinrfrmψθϕ=KmC()(,)cosrfrmψθϕ=K()(,)cos,rfyzkxkRψ=∈Kx2/akπ=ˆˆ()()(,)()cos(,)cos()(,)cos()TarfyzTakxfyzkxafyzkxrψψ==−==KK(7.1.40)ˆ()Taˆ()()(),TnarrnZψψ=∈KK(7.1.41)||ˆ[()],0ˆ()ˆ[()],0nnTanTnaTan≥=−ˆ{()|2/,}TnaaknZπ=∈(7.1.42){ˆ()Ta}32512(,)ntuaufrt∂=∆+K(7.1.43)Q2ˆˆˆ(,)ntQuaQuQfrt∂=∆+K(7.1.44)21ˆˆˆˆˆ(,)ntQuaQQQuQfrt−∂=∆+K(7.1.45)ˆQu1ˆˆˆ,(,)(,)QQQfrtfrt−∆=∆=KK(7.1.46)1221ˆˆˆ()aaQTaeeQ−⋅∇⋅∇−∆=∆−=∇=∇=∆KKK(7.1.47)1ˆˆQQ−∆=∆ˆˆ1221ˆˆˆ()LLQReeQααα−⋅⋅−∆=∆−=∇=∇=∆KKK(7.1.48)1ˆˆQQ−∆=∆(7.1.43)(,)frtK(7.1.43)2(,)tuaufrt∂=∆+(,)frt(,)frtO(3)O(3)7.27211252(PierreCurie1859—1906)27.2-17.2-27-32537-330,1raurau=∆==()uur=32220,()1uurarrua∂∂+=∂∂=ˆLuf=(7.2.1)254ˆLfu12{,,,}nGggg=igu1iivgun=∑(7.2.2)G11ˆˆˆˆjjikikgvgguguvnn=⋅==∑∑(7.2.3)ˆ,jjjgvvCλλ=∈(7.2.4)2''()()0(0),'(0)0yxxyxyay+===()()()Pyxyxyx=−=72211918•(EmmyNoether,1882-1935)2552(,,)Hqptααqαpα,dpdqHHdtqdtpαααα∂∂=−=∂∂(7.2.5)(,,,,,)xyzHxyzpppx(,,,,,)(,,,,,)xyzxyzHxayzpppHxyzppp+=a(,,,,,)(,,,,,)(,,,,,)0xyzxyzxHxayzpppHxyzpppHxyzxyza+−=∂⋅=(,,,,)xyzHHyzppp=x0xdHpdtx∂==∂constxp=(7.2.6)z(,,,,,)zHzpppρϕρϕ(,,,,,)zHzpppρϕραϕ+(,,,,,)zHzpppρϕρϕ=256(,,,,)zHzpppρϕρj0dHpdtϕϕ∂==∂2constpmϕρϕ==(7.2.7)3iHtψψ∂=∂=(7.2.8)ˆHψ=xˆ1ˆˆˆˆˆ()(),()aDTaHTaHTae−−⋅⋅==(7.2.9)aˆˆˆˆ(1)(1)aDHaDH−⋅⋅+⋅=ˆˆˆˆHDDH=(7.2.10)ψxˆ*xpDdiψψτ=∫∫∫=*ˆˆˆˆˆˆ[*][**]0xdpDDdHDDHddtittψψψψτψψψψτ∂∂=+=−=∂∂∫∫∫∫∫∫=(7.2.11)x72312572019562V(x)7-4a241124(),0Vxaxxa=+(7.2.12)3'()0Vxaxx=+=(7.2.13)x0(7.2.12)2584114(),0Uxbxxb=−≠7-4b21112()()()VxVxUxbxax=+=+(7.2.14)'()0Vxbax=+=xb/a''()0Vxa=-2-1122468-2-1.5-1-0.50.510.511.52-1-0.50.51-0.05-0.0250.0250.050.0757-4a7-4b7-4c3a7-4c259(7.2.13)x0xa=±−''(0)0,''()320VaVaaaa=±−=−=−x0xa=±−127.37311''()()()()yxVxyxEyx+=(7.3.1)Exx↔−()()()Pyxyxyxλ=−=(7.3.2)22()()()PyxPyxyxλλ==(7.3.3)2602PI=21λ=l≤1()()()()eeooyxyxyxyx−=+−=−(7.3.4)2''()()()()yxVxyxEyx−+⋅=(7.3.5)()Vxaxxa→+()()()()Tayxyxayxλ=+=(7.3.6)()()nyxnayxλ±±=(7.3.7)|l|∫1n|l|1liKaeλ=(7.3.8)KKaππ−≤//aKaππ−≤(7.3.9)()()iKxyxeux=(7.3.6)()()iKaiKxiKaiKxeeuxaeeux+=()()uxaux+=(7.3.10)732126120,0()trbtuaurbuufr===∆==(7.3.11)(,)uurt=(7.3.12)22cos()1[(,)]ikrikrrukuUruuAeferθθϕ⋅→∞∆+=→+(7.3.13)