空间的无限小转动

空间的无限小转动绕轴转角nϕdrnRrnrr),(δϕδϕ=×+=′LnnDˆie)(ˆ⋅−=δϕδϕ若哈密顿量具有旋转对称性(空间各向同性)0]ˆ,ˆ[=⋅LnH系统在方向的角动量守恒n有限转动rQr=′}{Q群)3(O1det−=Q非正当转动正当转动群1det+=Q群的三维表示)3(SO所有正当转动都可由两类简单转动完成：)()()()()()(32323eQeQeQeQeQnQψϕθϕψθ−−=欧拉角)()()()(kQjQkQQγβααβγ=−+−+−−−=βγβγββαγαγβαγαγβαβαγαγβαγαγβαcossinsincossinsinsincoscossincossinsincoscoscossinsincoscossinsincoscossinsincoscoscos2)3()3(SSOO⊗=四、两个角动量的耦合1.角动量算符满足下面对易关系的矢量算符[]γαβγβαεJJJˆiˆ,ˆ=JJJˆiˆˆ=×0ˆ,ˆ2=zJJ共同本征态jm|+=jmjjjmJ|)1(|ˆ22=jmmjmJz||ˆ,2,23,1,21,0=jjjjjm,1,,0,,1,−+−−=±±−+=±1|)1()1(|ˆjmmmjjjmJ±+±=1|)1)((jmmjmjyxJJJˆiˆˆ±=±对给定的j值：jm|张成群的一个2j+1维不变子空间)3(SOyxJJˆˆ、2、两个角动量的耦合0ˆ,ˆ21=JJzJJJJˆˆˆˆ22221、、、mjjj,,,|21耦合表象zzJJJJ222121ˆˆˆˆ、、、=⊗22112211,,,|,|,|mjmjmjmj无耦合表象+=2211211221121,,,|)1(,,,|ˆmjmjjjmjmjJ+=2211222221122,,,|)1(,,,|ˆmjmjjjmjmjJ=2211122111,,,|,,,|ˆmjmjmmjmjJz=2211222112,,,|,,,|ˆmjmjmmjmjJz+=mjjjjjmjjjJ,,,|)1(,,,|ˆ212112121+=mjjjjjmjjjJ,,,|)1(,,,|ˆ212222122+=mjjjjjmjjjJ,,,|)1(,,,|ˆ212212=mjjjmmjjjJz,,,|,,,|ˆ2121：基矢2ˆJ22|mj：基矢1ˆJ11|mj21ˆˆˆJJJ+=Clebsch–Gordan系数对给定的21jj、1|,,,,,,|21,22112211=∑mmmjmjmjmj=∑mjjjmjmjmjmjmjjjmm,,,|,,,,,,|,,,|21,221122112121∑=212121,2211,,,|mmjjjmmmmjmjC=mjjjmjmjCjjjmmm,,,|,,,2122112121Clebsch–Gordan系数、Wigner系数21ˆˆˆJJJ+=zzzJJJ21ˆˆˆ+=21mmm+=CG系数的性质(1)2121212121,mmmjjjmmmjjjmmmCC+⋅=δ求和实际为单层21ˆˆˆJJJ+=||,,2,1,21212121jjjjjjjjj−−+−++=(2)1111,,1,jjjm+−−=2222,,1,jjjm+−−=2121||jjjjj+≤≤−对给定的21jj、，总维数为)12)(12(21++=jjN(3)幺正变换，为幺正矩阵{}2121jjjmmmC2121jjjmmmC习惯上，取为实数。正交矩阵=221121212211,,,|,,,,,,|,,,mjmjmjjjmjjjmjmj(4)正交性∑∑−=′′−=11121212222121jjmjjmjmmjjmjjjmmmCCmmjjjjmjjmmjjjmjmjmjjjmjmj′′−=−==′′=∑∑δδ111222,,,|,,,,,,|,,,212211212211∑∑+−=−=′′212121212121||jjjjjjjjmmmjjmjjjmmmCCmmjjjjmjjmjmmjjmjjjmmmCC′′−=′′−==∑∑δδ1112121222212122112121||212211212211,,,|,,,,,,|,,,mmmmjjjjjjjmmjjjmjmjmjjjmjmj′′+−=−==′′=∑∑δδ21122212211112211121212112122121211212)1(1212)1(1212)1()1()1(122jjmjmmmjjjmjmmmjjjmmjmmjjjmjmmjjjjjjmmmjjjjjjmmmCjjCjjCjjCCC−−+−−−−−−−−−+−+++−=++−=++−=−=−=(5)(6)其他记号jmmjmjC2211),(2121mmmjjjC3j符号−+−=−−mjjjmjmjjmmmjjjmjj,,,|,,,12)1(212211212121几个常用的CG系数mmjjjjmmδδmjjmjC1111,,0,|0,0,,11100==121)1(0,0,,|,,,00+−=−=−−jjjmjmjCmjjjmm−=mjjmmmjCjjmmm,,21,|,21,,122121121计算结果如下212=j计算结果如下12=j−=mjjmmmjCjjmmm,,1,|,1,,12211121JnnDQDˆie)(ˆ)(ˆ⋅−==ϕϕ)()()()()()()(12kQjQkQkQjQkQQγβααβγαβγ=={})(αβγQ)d(ˆϕnD同构)(ˆ)(ˆ)(ˆ)(ˆ12kDjDkDDαβγαβγ=JkJjJkˆiˆiˆieee12⋅−⋅−⋅−=αβγzyzJJJˆiˆiˆieee12αβγ−−−=yzyzJJJJ1112ˆiˆiˆiˆieeeeβγβγ−−−=zyzyJJJJˆiˆiˆiˆieeee1αβαβ−−−=zzzzJJJJˆiˆiˆiˆieeee1αγαγ−−−=zyzJJJDˆiˆiˆieee)(ˆ12αβγαβγ−−−=zyzJJJˆiˆiˆieeeγβα−−−=ααβyxz2x1y1z1x2y2z3x3y3zγ3.转动算符（群）的矩阵表示)3(SOjm|张成群的一个2j+1维不变子空间)3(SO转动算符的矩阵元zyzJJJDˆiˆiˆieee)(ˆγβααβγ−−−==′=jmQDjm|)(ˆ||ϕ+=jmjjjmJ|)1(|ˆ22=jmmjmJz||ˆ∑−==jjnnmjnc|′=′jmQDmjmj|)(ˆ||ϕ∑−=′=jjnnmjncmj||∑−=′=jjnnmjnmjc|∑−=′=jjnnmnmcδmmc′=mmjmmcjmQDmjQD′′=′=|)(ˆ|)(D矩阵、D函数、Wigner函数群的2j+1维不可约表示的矩阵元)3(SO不同！1)(ˆ)(ˆ)(ˆ)(ˆ††==QDQDQDQD′=′jmQDmjQDjmm|)(ˆ|)(转动矩阵()())()()(1*†−′′′==QDQDQDjmmjmmjmm)(ˆ)(ˆ)(ˆ2121QQDQDQD=)()()(2121QQDQDQDjmmnjnmjnm′′=∑取QQ=112−=QQ()∑′=njmnjnmQDQD*)()(mmnjnmjnmQDQD′−′=∑δ)()(1′=′jmQDmjQDjmm|)(ˆ|)(∑′′′=mjmQDmjmjjmQD|)(ˆ|||)(ˆ磁量子数的反转对称性()),,()1(),,(*γβαγβαjmmmmjmmDD−′−−′′−=∑′′′=mjmmQDmj)(|′==′′|jmDmjDQDjmmjmm)(ˆ|)()(αβγαβγ′=|jmkDjDkDmj)(ˆ)(ˆ)(ˆ|γβα′=−−−|jmmjzyzJJJˆiˆiˆieee|γβαγβαmJm|jmmjyiˆiiee|e−−′−′=′=−′|jmmjdyJjmmˆie|βkmmkmmjkkmkjkmmkkmjmjmjmjmj22202sin2cos)!(!)!()!()!()!()!()!()1(+−′−′−+=×′−−′+−−+′−′+−+−=∑ββ矩阵)12()12(+×+jj2j+1维不可约表示k的取值上限是使分母中不出现负数的阶乘下的最大值。jmmmmd′+′−=)i(eγα′=−′|jmmjdyJjmmˆie|β21=jyyJσˆ21ˆ=βσβσβsinˆcoseˆiyiy−=−1ˆ2=yσ−=)2cos()2sin()2sin()2cos()(21βββββd−=+−−−+−2cose2sine2sine2coseˆ)(2i)(2i)(2i)(2i21ββββγαγαγαγαD1=j+−−−−+=−−−+−−−−−−+−2cos1e2sine2cos1e2sinecos2sine2cos1e2sine2cos1eˆ)(ii)(iii)(ii)(i1βββββββββγααγαγγγααγαD4、D矩阵与球函数的关系球函数lmlmlmmllmlmxxxmlmlll)1(dd)1(e)!()!(π412!2)1(),(Y222i−−+−⋅+−=++ϕϕθ),(Y)1(),(Y*ϕθϕθlmmml−=−kmmkmmjkkmmjmmmkjkmmkkmjmjmjmjmjD2220)i(2sin2cos)!(!)!()!()!()!()!()!()1(e)(+−′−′−+=+′−′×′−−′+−−+′−′+−+−⋅=∑ββαβγγαθcos=x),(Y12π4)0(*0ϕθϕθlmlmlD+=ϕmmlmxmlmllie)(P)!()!(π412)1(+−⋅+−=,2,1,0=lllllm,1,,0,,1,−+−−==jmlm|),(Yθϕϕθ),(Y12π4)0(*0ϕθϕθlmlmlD+=zyzJJJDˆiˆiˆieee)(ˆγβααβγ−−−==θϕϕθ|00|)0(ˆD==⇒0,0|00|ϕθ∑′′′=mmlmlD00||)0(ˆϕθ∑′′′=mmlmlDlmlm00||)0(ˆ||ϕθθϕ∑′′′=mmlmlDlm00||)0(ˆ|ϕθ∑′′′=mmllmmlmD)0,0(Y)0(),(Y**ϕθϕθ0π412)0,0(Ymlmlδ+=)0(π412),(Y0*ϕθϕθlmlmDl+=)(e)()i(αβγαβγγαjmmmmjmmdD′+′−′=)()(0000ϕθψϕθψllDd=),(Y12π40ϕθll+=)(cosPθl=()21212122211112π8d)()(12*mmmmjjjmmjmmjΩDDδδδαβγαβγ′′′′+=⋅∫正交性),(Y12π4)0(*0ϕθϕθlmlmlD+=kkllkklkkllD22202002sin2cos!)!(!)1()(⋅−−=−=∑ββαβγ)(0ϕθψlmD=§4群)2(SU{})(αβγQ)d(ˆϕnD同构)3(SO)2(SU同态二维幺模幺正矩阵群幺模1det=U1=−bcad两个方程幺正1††==UUUU1**=+ccaa1**=+ddbb0**=+dcba三个方程U只有三个独立的实参数一、群)2(SU群元)2(SU=dcbaU=****†dbcaU−=**abbaUU只有三个独立的实参数1**=+bbaa习惯上βαiecos⋅