LippmannSchwingerEquation

Lippmann-SchwingerEquationrrrVrrkGrrd)()(),()()(2)()(rrrVrrkGrrd)()(),()()()(2)(0)()(ˆ)(ˆ)(ˆ)(ˆ)()(0)(0)(EGVEGEGEGiipkVGki|ˆ||)()()()(0)(|ˆ||VGk)()(0)(ˆˆˆTGVVTVGVVT)()(ˆˆ)()(||ˆikiVkT)(1)(ˆˆTVΩiikkΩi||ˆ)()()(†)(ˆˆˆΩΩS)()(π2ifiiffifiTEES)(ˆ)(π2iIˆTEESfi玻恩近似VGVGVGVVGVGVVGVVT)(0)(0)(0)(0)(0)(0)(ˆˆˆˆˆˆˆmmT)(VGVGVGVTm)(0)(0)(0)(ˆˆˆ)(0ˆ1GmVm个，个mmff),(),()(ifmkVGVGVGVkf|ˆˆˆ|π4),()(0)(0)(022)(费曼图),()(nfikV1kV2kV3k0ˆG0ˆGVV2nkVVfk1nk0ˆG0ˆG0ˆGV：每作用一次，动量改变一次。mmkVk||1两次作用之间，粒子处于动量为的自由态（虚态）。mk贡献一个自由格林函数i1)(0miEEG传播子例：高速带电粒子被汤川势散射的散射截面areraVrV0)(解:汤川势（Yukawa）原子的屏蔽库仑场rrVfrkkfid)(eπ2),()(i2)1(rrVaarrqde1eπ2i02fikkqrrrVaarrqdddsine1eπ22cosi02rrVaarrqd)d(coseecosi02rrqsVqaard)ein(20202223)1(2Vqaa220d)sin(babxbxeax02223)1()1(2),(Vqaaffikkqikfkq2sin2kq202224622)1()1(4|),(|),(Vqaaf202222462)2sin41(4),(Vkaa总截面与无关，轴对称ddsin),(π0π20tdsin)(π2π0)41(π162242062kaVad)cos221(sinπ8π02222242062kakaVaπ02222222224222062)cos221()cos221d(2π8kakakakakaVa高能散射，k很大12sin2kq不是很小时2sin1614),(4442022kVaRutherford散射公式202222462)2sin41(4),(Vkaa库仑屏蔽散射areraVrV0)(aezzVπ4202sin161π44),(444222kezzze核电荷ez入射粒子电荷Ek2222sin161π44222Eezz一级玻恩近似与精确结果相同，与经典物理学结果也相同§4分波法ifkfkTkVkfi|ˆ|π4||π4),()(22)(22球对称势场的散射)(rVV表象，基矢}ˆ,ˆ,ˆ{2zLLHElm|),,(kekkk：方向的单位矢量kekiffikTkT|ˆ|)(,|||||kkekkkVGVVT)()(ˆˆiˆˆ)(HEVGi0)(,ˆrVL0ˆ,ˆ)(GL0ˆ,ˆ)(TLmmlllTlmTml|ˆ|)(),(Y||lmlmlme对角矩阵一、分波法的依据lmmlifklmlmTmlmlk||ˆ||)(lmmlififfiklmlmTmlmlkkTkT||ˆ|||ˆ|)()(lmmliiimmlllfffklmTmlk||||lmiilmlfflmifTkk)(Y)(Y|*lmmliilmmmlllffmlifTkk)(Y)(Y|*球函数的加法公式)(cosPπ412)(Y)(Y*lmiilmfflmlifeecos三维函数)()()()(|zzyyxxrrrr)()()(sin1||)(2rrrrrrr球坐标系下lllififiTlkkkT)(cosP4π12)(12)(1|2ifiifkkkkk)()()(ifififkkkkkk)(222ifikkkiiixfxxxf|)(|)()(如果的实数根均为单根：0)(xfix222222222ifikkk)(||1)(xaax)(2ifiEEklllififiTlkkkT)(cosP4π12)(12lliflifiTEElkT)()(cosP4π122)(ˆ)(π2i1ˆTEESfilfilTEES)(π2i1lllifiSlkT)(cos)P11)(2(π8i22fiifTkTkf22)(22π4|ˆ|π4),(llliSlk)(cos)P11)(2(2illliSlkf)(cos)P11)(2(2i),(lmmlllSSlmSml|ˆ|为对角矩阵Sˆ因为，故的模为1。1ˆˆ†SSlSllS2ie)ee(ee11iii2illlllSlliesini2lllllkf)(cosPesin1)2(1),(ikkkfi)(llf)(cosPsine12)(illllklfl不同，角动量不同l分波散射振幅等于不同分波上散射振幅的叠加分波法由此得名散射截面)(cosPsine12)(illllklf)(),(llff2|),(|),(f)()(*llllffllllllllklkl)(cosPsine12)(cosPsine12iillllllllkll)(cosP)(cos)Pcos(sinsin1)1)(2(22总截面)(ddsin)(π0π20tdsin)(π2π0lllllxxx122d)(P)(Pllllllllkl)cos(sinsin12π42lllk22sin)12(π4lllllk22sin)12(π4总截面取决于ll分波的相移)(cosPsine12)(illllklflllk22sin)12(π4)1(Psine12)0(illllklflllklsin)sini(cos12光学定理)()(llff)0(Imπ4llfkllt)0(Imπ4fkt光学定理：散射截面与向前()散射振幅的虚部成正比。0一个方向上的散射振幅包含了向所有方向散射总概率的信息！二、相移的由来)()()2(22rErVEVrLrrrr222222ˆ120)()1()(dddd12222rRrllrUkrRrrrEVrLrrrr222222ˆ12),()()(YrRr),(Y),(lmY)(2)(2rVrU222EklmlmllmrRcr),(Y)()()()(rRrRl轴对称llllPrRcr)(cos)(),(第l分波rrurRll)()(0)()1()(dd2222rurllrUkrull)sin()(lllkrArur)π21sin()(llllkrkrArR0)(dd222rukrulll分波的位相)(cos)π21sin(1),(0llllPlkrkrCrr)(coseei210)π21(i)π21(illlkrlkrlPkrCllrfrkrrkii23e),(e)π2()(又r0cosii)(cos)(i)12(llllkrkzPkrjlee而)(2π)(21krJkrkrjllrefPkrlrkrlllkrlkrli230)π21(i)π21(i23)()π2()(coseei2i)12()π2(),(l→l分波的相移)π21sin(1lkrkrr)π21sin()(llllkrkrArRrfkrkzii23e)(e)π2()π21(i)π21(ieei21lkrlkrkr)(cosi)12()π2()(i2)π2()(cosπ21i02323)π21(i0llllllllPelkfPeCl)(cosi)12()π2()(cosπ21i023)π21(i0llllllllPelPeClπ21i23)π21(ii)12()π2(lllleleCllelClli23i)12()π2()(cos)12()(i2)(cos)12(02i0llllPlkfPell)(cos)1()12(i21)(2i0llPelkfl)π21(i23)12()π2(llel)(coseei21)π21(i)π21(i0llkrlkrllPkrCllll)(ei2πirefPkrlrkrlllkrlkrli230)π21(i)π21(i23)()π2()(coseei2i)12()π2(),()(cos)()12(i21iii0llPeeelklll)(cossin)12(1i0lllPelkl)(cossin)12()(illlPeklfl0)(llf来自入射波中的l分波0l（入射）0l（排斥）0U0l（吸引）0U)(cos)()12(i21)(iii0llPeeelkflll)π21sin(1)(llllkrkrCrRr分波法：寻找各分波的相移来计算散射截面用分波法求散射截面是一个无穷级数的问题，实际上不可能也不必要一直计算到无穷大。一般l值越大的分波，对