定态散射

定态散射)()()()(22rrUrkrfeArkkrrkriie),(),(渐进解散射截面dd),(Nn2|),(|frrrVrrkGrrd)()(),()()(20||i22)(0e||1π2),(rrkrrRkGrrrVfrnkd)()(eπ2)π2(),()(i223李普曼-许温格方程)(22||π4ikfVkkkkrrkGrrkde12)π2(1),()(i222320iˆ1ˆ0)(0HEG自由格林算符),(|ˆ|2)(0)(0rrkGrGr)(i1|ˆ|)(0kkEEkGkiˆ1)(ˆ)(HEEGii全格林算符iipkVGki|ˆ||)()(全格林算符下的LS方程三、戴森方程DysonEquation散射波)()()(rrrs0)()(2222rk0)()]([2222rrUk)(2)(2rVrU)()(0)(|ˆ||VGkiˆ1ˆ0)(0HEG自由格林算符Lippmann-SchwingerEquation0)()(2222rk0)()]([2222rrUk)()()(rrrs)()()]([2222rVrrUks格林函数法)(),()]([22222rrrrkGrUk解记为),(2)(rrkGrrrVrrkGrsd)()(),()(2)()()()()()()(rrrs算符形式|ˆ||)()(VG坐标表象的全格林算符下的LS方程由于ik||),()(),(][22)(2)(222rrkVGrrrrkGk可用求解),(2)(0rrkGrrrkGrVrrkGrrkGrrkGd),()(),(),(),(2)(2)(02)(02)(rrrkGrVrrkGrrkGrrkGd),()(),(),(),(2)(2)(02)(02)(算符形式)(ˆ)(ˆ)(ˆ)(ˆ)()(0)(0)(EGVEGEGEG全格林算符的LS方程Dyson方程迭代法求解)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(0)(0)(0)(0)(0)(0)(EGVEGVEGEGVEGEGEG四、算符与算符TˆSˆrferkkrrkriii23e),()π2(),(rrkrVfirkfd),()(eπ2)π2(),()(i223)(22||π4ikfVk定义：)()(||ˆikiVkT)(22||π4),(ikfVkf)()(||ˆikiVkTifkTk|ˆ|π4)(22)(22π4fiTiffikTkT|ˆ|)()(跃迁矩阵元Tˆ跃迁算符2|),(|),(f2)(424||π)2(fiT)(|ˆ|)(|)()(0)(iiikVGkk)(|ˆ|)(|)()(0)(iiikVGVkVkViiikTGVkVkT|ˆˆ||ˆ)()(0)(iipkVGki|ˆ||)()()()(0)(ˆˆˆTGVVTVGVVT)()(ˆˆTˆ算符的LS方程)()(0)(ˆˆˆTGVVTVGVVT())(ˆˆ0ˆˆˆ)()()(0VGVTGV0ˆˆˆ)()()(0VGTG厄米共轭0ˆˆˆ)()(0)(GVGT相减Dyson方程)()(0)(0)(ˆˆˆˆGVGGG)(0)()(0)(0)(ˆˆˆˆˆGTGGG)()(||ˆikiVkT由定义ikkTVi|ˆ|)(1)(定义)(1)(ˆˆTVΩ摩勒（Moller）算符、波算符ikkΩi|ˆ|)()(iiiiikkkΩki||ˆ||)()(iikkΩi||ˆ)()(实际是积分iikkΩi||ˆ)()(ikiikΩ||ˆ)(†)(|||ˆˆ)()()(†)(jkjikikkΩΩji|||jjiijikk||iiikk1但jikijkijkkΩΩ|||ˆˆ)()(†)()(jikijkij||)()(ikkii||)()()(|ik只是哈密顿的散射（非束缚）本征态，不完备1||||)()(jjjikkffiiikkiiΩΩ||ˆˆ)()(†)()(Bˆ1jjjffB||ˆ向束缚态投影算符ikkΩi|ˆ|)()(ikkΩHHi|ˆˆ|ˆ)()()(|ikiEiikikΩEEi|ˆ|)()(iikEΩ|ˆ)(ikHΩ|ˆˆ0)(0)()(ˆˆˆˆHΩΩH†)(0†)(ˆˆˆˆΩHHΩfΩHfHΩ|ˆˆ|ˆˆ†)(0†)(fΩEfΩH|ˆ|ˆˆ†)(†)(00|ˆ†)(fΩ束缚本征态的湮灭算符Hˆ束缚态定义)(1)(ˆˆTVΩ)(†)(†ˆˆˆΩΩS)(†)()(†)(†ˆˆˆˆˆˆΩΩΩΩSS)(†)(ˆ)ˆ1(ˆΩBΩ)(†)()(†)(ˆˆˆˆˆΩBΩΩΩ0|ˆ†)(fΩ1同理)(†)()(†)(†ˆˆˆˆˆˆΩΩΩΩSS1)(†)(ˆˆˆΩΩS幺正算符散射算符)(†)(ˆˆˆΩΩS)()(||ˆikiVkT)(†)(ˆˆˆΩΩS)(1)(ˆˆTVΩiffikSkS|ˆ|ifkΩΩk|ˆˆ|)(†)()((|ifkk）ikkΩi|ˆ|)()()()(|ˆ||ikffGVkkiipkVGki|ˆ||)()()()(||i1|iikfifkfVkEEk)()(0)(|ˆ||iikikVGk)(0)(|iˆ|||iikififkfHEVkkkk)(||i1|ikffiifVkEEkk)()(||i1|iikffikffiVkEEkS)(||i1i1|ikffiififVkEEEEkk)(22||)(2iikfiffiVkEEififfikTkEE|ˆ|)(2i)(22)()(π2ifiiffiTEE)(πlim220xx矩阵与矩阵的关系TˆSˆ)(ˆ)(π2i1ˆTEESfiifkfkTkVkfi|ˆ|2π2||2π2),()(22)(22矩阵或矩阵的计算成了关键。TˆSˆ§3玻恩近似2|),(|),(f2)(424|ˆ|6π1ifkTk)()(0)(ˆˆˆTGVVT迭代法求解VGVGVGVVGVGVVGVVT)(0)(0)(0)(0)(0)(0)(ˆˆˆˆˆˆˆ算符的玻恩级数Tˆ收敛性不宜讨论0ˆHV收敛较快VT)(ˆ只取第一项2)(424)1(|ˆ|6π1),(ifkTk2424||6π1ifkVk2)(i422d)(eπ4rrVrkkfi玻恩一级近似VGVVT)(0)(ˆˆ保留两项，玻恩二级近似保留项数更多，得到高级近似高能散射逐级迭代，得态函数的各级近似)()(0)(|ˆ||VGk)()()0(rrrkii23e)π2(rrrVrrkGrrd)()(),()()(2)(0)1(rrrVrrkGrrd)()(),()()()1(2)(0)2(…………rrrVrrkGrrnnd)()(),()()()1(2)(0)(ifkTkf|ˆ|π4),()(22LS方程的近似解散射振幅的计算VGVGVGVVGVGVVGVVT)(0)(0)(0)(0)(0)(0)(ˆˆˆˆˆˆˆiiT)(VGVGVGVTn)(0)(0)(0)(ˆˆˆiif),()()(0ˆ1GnVn个，个ifkVkf||π4),(22)1(iiiffkTkf),(|ˆ|π4),()()(22rrVrkkfid)(eπ2)(i2ifkVVGkf||π4),()(022)2(ifkVkkGkkVkkk|||ˆ|||ddπ422)(0112122iifkVkEEkVkk||i1||dπ4111122ifinfnkVGVGVGVkkTkf|ˆˆˆ|π4|ˆ|π4),()(0)(0)(022)(22)(ifnkVGVGVGVkf|ˆˆˆ|π4),()(0)(0)(022)()(0ˆ1GnVn个，个ifnkVGVGVGVkf|ˆˆˆ|π4),()(0)(0)(022)(i1||ddddπ411132122ninfnEEkVkkkkkiinininnninnkVkEEkVkEEkVkEEkVkEEkVk||i1||i1||i1||i1||1112223332221),(f费曼图ikfkVik1kVfkV0ˆGikV1kV2kVfk0ˆG0ˆG………),()1(f),()2(f),()3(f),()(nf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dddsine1eπ22cos