关于数学物理方法课程的若干问答-续2-吴崇试

307Vol．30No．720117COLLEGEPHYSICSJuly2011櫍櫍櫍櫍櫍櫍櫍櫍櫍櫍櫍櫍殻殻殻殻210087126、xy．x2－y2．．．．．．27x=1ddx1－x2dyd[]x+νν+1y=0x=1ρ1=ρ2=0x=1——Pνxx=1．Pν1=1ν．x=0x=1．x=1．—Pνx．28ddx1－x2dyd[]x+λ－μ1－x[]2y=0y±1．λ=ll+1l=012…．λμ．μΦ″+μΦ=0Φ0=Φ2πΦ'0=Φ'2πμ=m2m=012…．μλλ．μ=m2mλ=ll+1l=mm+1m+2…mlmlμ=m2m=012…λ=ll+1l=012…．λ=ll+1l=mm+1m+2…m≠0l=01…m－1“”Φ≡0．29ddx1－z2dwd[]z+νν+1－m21－z[]2w=0m=123…|z－1|＜2Pmνz=z2－1m/2dmPνzdzmQmνz=z2－1m/2dmQνzdzm230ν=ν＜mPmνz≡0．Qmνzm=0．PmνzQmνz．WPmνzQmνz=－m1－z2Γν+m+1Γν－m+1ν－m=ν－m+1WPmνzQmνz=0PmνzQmνz．Pmνz≡0．Γν－m+1Γν+m+1Pmνz=limμ→mΓν－μ+1Γν+μ+1Pμνzν－m=QmνzQmνz．P－mνz．ddz1－z2dwd[]z+νν+1－μ21－z[]2w=0．ν→－ν+1μ→－μz→－zPμνz=1Γ1－μz+1z()－1μ/2F－νν+11－μ1－z()2、．20005．16Pμν－zPμ－ν－1±zP－μν±zP－μ－ν－1±z．Qμνz=eiπμ槡π2ν+1Γν+μ+1Γν+3/2z－ν－μ－1z2－1μ/2·Fν+μ2+1ν+μ+12ν+32z()－2、．20005．17Qμν－zQμ－ν－1±zQ－μν±zQ－μ－ν－1±z．1614．．30rlPmlcosθcosmrlPmlcoθsinmr－l－1Pmlcosθcosmr－l－1Pmlcosθsinml=012…m=012…l1r2rr2u()r+1r2sinθsinθu()θ+1r2sin2θ2u2=00＜r＜a0＜θ＜π0＜＜2πu|θ=0u|θ=π0≤r≤a0≤≤2πu|=0=u|=2πu=0=u=2π0≤r≤a0≤θ≤πu|r=0u|r=a=fθ0≤θ≤π0≤≤2πr＜arlPmlcosθcosmrlPmlcosθsinmr＜ar－l－1·Pmlcosθcosmr－l－1Pmlcosθsinm．25r=0．31urθ=Σ∞l=0Alrl·Plcosθzu．urθ=Σ∞l=0AlrlPlcosθ1Plcosθ0＜θ＜πΘ0Θπ．21sinθddθsinθdΘd[]θ+λΘ=0723Θ0Θπ/2=0—Θ'π/2=0αΘ'π/2+βΘπ/2=0{．321r2rr2u()r+1r2sinθsinθu()θ+1r2sin2θ2u2=0a＜r＜b0＜θ＜π0＜＜2πu|θ=0u|θ=π0≤r≤a0≤≤2πu|=0=u|=2πu=0=u=2π0≤r≤a0≤θ≤πu|r=0u|r=a=fθ0≤θ≤π0≤≤2πurθ=Σ∞l=0Σlm=0Alrl+Blr－l－1·PmlcosθCmsinm+Dmcosm4rlPmlcosθcosmrlPmlcosθsinmr－l－1Pmlcosθcosmr－l－1Pmlcosθsinm4urθ=Σ∞l=0Σlm=0rlPmlcosθAlmsinm+Blmcosm+Σ∞l=0Σlm=0r－l－1PmlcosθClmsinm+Dlmcosm4．rlr－l－1lsinmcosmmurθ=Σ∞l=0Σlm=0Alrl+Blr－l－1·PmlcosθCmsinm+Dmcosm=Σ∞l=0Σlm=0AlCmrl+BlAlr－l()－1Pmlcosθ·sinm+DmCmcosm=Σ∞l=0Σlm=0A'lmrl+B'lr－l－1Pmlcosθsinm+D'mcosm3AlmAlmB'lAlmB'lD'm．glθ=Σlm=0PmlcosθCmsinm+Dmcosmurθ=Σ∞l=0Alrl+Blr－l－1glθurθa＜r＜brlr－l－1θ．urθ=Σ∞l=0Σlm=－lAlmrl+Blmr－l－1P|m|lcosθeim．33ut－κ1rrru()r+1r22u[]2=00＜r＜a0＜＜2πt＞0u|=0=u|=2πu=0=u=2π0≤r≤at＞0u|r=0u|r=a=ft0≤≤2πt＞0u|t=0=grθ0＜r＜a0＜＜2π．ft=Acos2Aurt=vrt+wrt．1ww|=0=w|=2πw=0=w=2πw|r=0w|r=a=Acos22wwt－κ1rrrw()r+1r22w[]2=03t430wtw=wrwr．4rmcosmr－mcosmw=Ar/a2cos2．w=Aa/r2cos2w|r=0．5w=Acos2．vrt．6w=Ar/acos2vrtvrt．．34JνzNνzJνzNνzz=z0WJνzNνzz=z0=0WJνzNνz≡0JνzNνz．z=0JνzJ'νz．z=0JνzJ'νz0NνzWJνzNνz0．Reν＞1Jν0=J'ν0=0．．．35^A≡iddx^A^AD^A=D^AD^A=^A^Au=^Auu∈D^A．．．L201．^A≡iddxL201^AL201^Au≡idudxD^Auxu'x∈L201ux．vxux=∫x0vtdt+u0ux．0．∫10v*idud[]xdx=－i∫10dv*d[]xudx+iv*xux10=∫10idvd[]x*udx+iv*1u1－v*0v0〈v^Au〉=〈^Avu〉+iv*1u1－v*0v0^A≡iddx^Av=idvdxD^A+vxv'x∈L201v0=v1=0．^A．^A^Au≡idudxD^Auxu'x∈L201u0=0^Av=idvdxD^Avxv'x∈L201v1=0^A．^Au≡idudxD^Auxu'x∈L201u0=u1=0222230ThekinematicsequationsofafreeparticleintheSchwarzschildspace－timeNIUZhen-fengLIUWen-biao1．DepartentofPhysicsCollegeofScienceHebeiNorthUniversityZhangjiakouHebei07500China2．DepartmentofPhysicsandInstituteofTheoreticalPhysicsBeijingNormalUniversityBeijing100875ChinaAbstractTheusualformkinematicsequationsofafreeparticleareobtainedviadirectlysolvingthegeodesicsequationsintheSchwarzschildspacetime．Comparingwiththemethodinsometextbookswherethekinematicse-quationsaregivenbythesphericalsymmetryofspace－timeandnormalizedfour－velocitytheresultsarecom-pletelysame．Howeverthegeodesicsmethodismorephysicalanditisusefulandmeaningfultounderstandgeo-desicsequationsinacurvedspace－time．KeywordsSchwarzschildspace－timegeodesicsequationskinematicsequationsinacurvedspace－timemercuryprecessiondeflectionoflight4^Av=idvdxD^Avxv'x∈L201^A．^AD^A≠D^A．^A^A^A≡iddx．^A^AD^AD^AD^Auxu'x∈L201u0=u1=0D^Avxv'x∈L201v0=v1=0^A．^AL201^Au≡idudxD^Auxu'x∈L201u0=±u1^A^Av=idvdxD^Avxv'x∈L201v0=±v1．36〈v^Au〉=〈^Avu〉uv∈D^A^A．1．2．^Au≡idudxD^Auxu'x∈L201u0=u1=0．3．4nUnUn．Questionsandanswersinthecourseof“methodsofmathematicalphysics”WUChong-shiSchoolofPhysicsPekingUniversityBejing100871ChinaAbstractAseriesofquestionswhicharefrequentlyemergedintheteachingofcourseof“methodsofmathemati-calphysics”butwererarelydiscussedinthetextbooksarelisted．Theanswersorexplanationsaregiveninsomedetail．Keywordsmethodsofmathematicalphysicsfunctionofonecomplexvariableequationsofmathematicalphysics