高量6-01多個(gè)角動(dòng)量耦合ppt課件

1、年月年月第六章第六章 多個(gè)角動(dòng)量耦合多個(gè)角動(dòng)量耦合引言引言構(gòu)造原子和原子核等全同多粒子體系的波構(gòu)造原子和原子核等全同多粒子體系的波函數(shù)，要求：函數(shù)，要求：、具有一定的交換對稱性、具有一定的交換對稱性、是角動(dòng)量算符的本征態(tài)、是角動(dòng)量算符的本征態(tài)涉及不同耦合方式之間的關(guān)系如涉及不同耦合方式之間的關(guān)系如 L-S耦耦合、合、j-j耦合），將碰到四個(gè)角動(dòng)量的耦合耦合），將碰到四個(gè)角動(dòng)量的耦合解決此類問題涉及多個(gè)角動(dòng)量的耦合，其解決此類問題涉及多個(gè)角動(dòng)量的耦合，其 基礎(chǔ)是兩個(gè)和三個(gè)角動(dòng)量的耦合基礎(chǔ)是兩個(gè)和三個(gè)角動(dòng)量的耦合定位：應(yīng)用上十分重要的技術(shù)性問題定位：應(yīng)用上十分重要的技術(shù)性問題6.1三個(gè)角動(dòng)量耦合三

2、個(gè)角動(dòng)量耦合Racah系數(shù)，系數(shù)，6-j符號(hào)符號(hào)Racah Coefficient; 6-j Symbol一、一、Racah系數(shù)；與系數(shù)；與C-G系數(shù)之關(guān)系系數(shù)之關(guān)系三角動(dòng)量耦合有次序問題，三角動(dòng)量耦合有次序問題，Racah系數(shù)由系數(shù)由此而來此而來考慮彼此對易且獨(dú)立的三個(gè)角動(dòng)量考慮彼此對易且獨(dú)立的三個(gè)角動(dòng)量知知3 , 2 , 1)() 1()(22imjjJJimjiiiimjiziiiii三角動(dòng)量相加三角動(dòng)量相加321JJJJ有三種方式有三種方式)1 (3121221aJJJJJJ；)1 (2312332bJJJJJJ；)1 (1321331cJJJJJJ；一、一、Racah系數(shù)；與系數(shù)；與

3、C-G系數(shù)之關(guān)系系數(shù)之關(guān)系(1a)對應(yīng)對應(yīng)zJJJJJJ2232122221的共同本征函數(shù)的共同本征函數(shù))3()12();(33312121233121231221mjmmmjjmmjmjCjmjjjj)2()1 ()12(2231211121222111212mjmmmjmjmjmjmjC(1b)對應(yīng)對應(yīng)zJJJJJJ2223232221的共同本征函數(shù)的共同本征函數(shù))23() 1 ();(23233121123231123321mjmmmjjmmjmjCjmjjjj323322232333222323)3()2()23(mmmjmjmjmjmjmjC一、一、Racah系數(shù)；與系數(shù)；與C-G系

4、數(shù)之關(guān)系系數(shù)之關(guān)系兩個(gè)表象通過一個(gè)幺正變換相聯(lián)系兩個(gè)表象通過一個(gè)幺正變換相聯(lián)系);();();(2332123123213122123jmjjjjjjjjjjUjmjjjjj 綜合以上各式可得綜合以上各式可得jmjjjjjmjjjjjjjjjjU;|;);(3122123321231232123123212323332223231133121212122211mmmmmmjmjmjjmmjmjjmmjmjmjmjmjCCCCU為為C-G系數(shù)乘積組合而成，因而為實(shí)數(shù)系數(shù)乘積組合而成，因而為實(shí)數(shù)稱稱U為重耦合為重耦合(recoupling)系數(shù)系數(shù)一、一、Racah系數(shù)；與系數(shù)；與C-G系數(shù)之關(guān)系

5、系數(shù)之關(guān)系U與所有磁量子數(shù)無關(guān)與所有磁量子數(shù)無關(guān)1、2312321,mmmmm已求和，已求和，U與之無關(guān)與之無關(guān)2、證明與、證明與m無關(guān)。以升降算符作用于變換式無關(guān)。以升降算符作用于變換式j(luò)mjjjjJjjjjjjUjmjjjjJj;|);(;|23321231232131221231;| );(1;|2332123123213122123jmjjjjjjjjjjUjmjjjjj m 不同對應(yīng)的變換系數(shù)相同不同對應(yīng)的變換系數(shù)相同一、一、Racah系數(shù)；與系數(shù)；與C-G系數(shù)之關(guān)系系數(shù)之關(guān)系變換系數(shù)的幺正性變換系數(shù)的幺正性由由31221312211212;|;jjjmjjjjjmjjjj23321

6、233212323; |;jjjmjjjjjmjjjj23123212312321121223);();(jjjjjjjjjUjjjjjjU23123212312321232312) ;();(jjjjjjjjjUjjjjjjU逆變換逆變換);();();(3122123123212332112jmjjjjjjjjjjUjmjjjjj一、一、Racah系數(shù)；與系數(shù)；與C-G系數(shù)之關(guān)系系數(shù)之關(guān)系定義定義Racah系數(shù)系數(shù)2/1231223123212312321)12)(12/();();(jjjjjjjjUjjjjjjW解析表達(dá)式解析表達(dá)式2/1231222312313212312332312

7、1221233223131212212312321)!()!()!()!()!()!()!( )!1()()()()()();(321jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjWvjjjj二、二、Racah系數(shù)的解析表達(dá)式系數(shù)的解析表達(dá)式三角關(guān)系三角關(guān)系2/112211122211212211221)!1()!()!()!()(jjjjjjjjjjjjjjjRacah系數(shù)包含四個(gè)系數(shù)包含四個(gè)C-G系數(shù)系數(shù)四個(gè)三角關(guān)系四個(gè)三角關(guān)系三、三、Racah系數(shù)的對稱性系數(shù)的對稱性由由C-G系數(shù)對稱性而來系數(shù)對稱性而來在量子數(shù)交換時(shí)，必須保持四個(gè)三角關(guān)系在量子數(shù)交

8、換時(shí)，必須保持四個(gè)三角關(guān)系三角關(guān)系圖形三角關(guān)系圖形一個(gè)圖形對應(yīng)一個(gè)一個(gè)圖形對應(yīng)一個(gè)Racah系數(shù)系數(shù)規(guī)定圖中彎形箭頭的次序?yàn)橐?guī)定圖中彎形箭頭的次序?yàn)镽acah系數(shù)中系數(shù)中角量子數(shù)的排列次序，共角量子數(shù)的排列次序，共24個(gè)圖形個(gè)圖形三、三、Racah系數(shù)的對稱性系數(shù)的對稱性兩種交換對稱性兩種交換對稱性分號(hào)兩邊量子數(shù)間無交換的分號(hào)兩邊量子數(shù)間無交換的W都相等都相等分號(hào)兩邊量子數(shù)有交換的分號(hào)兩邊量子數(shù)有交換的W相差一相因子相差一相因子);();();();();();(.122312312233212312213231212323123122312321jjjjjjWjjjjjjWjjjjjjWjj

9、jjjjWjjjjjjWjjjjjjWge);()();()();(.23231213123212231232122312312312jjjjjjWjjjjjjWjjjjjjWgejjjjjjjj三、三、Racah系數(shù)的對稱性系數(shù)的對稱性完全版完全版2312321jjjjjjabcdef );(),;(),;(),;();(),;(),;();(fedbcaWfebdacWfecadbWfeacbdWefdcbaWefcdabWefbadcWefabcdW);(),;(),;(),;();(),;(),;(),;()()(cbdefaWcbedafWcbfadeWcbafedWbcdfeaWb

10、cfdaeWbceadfWbcaefdWabcdefWfecb);(),;(),;(),;();(),;(),;(),;()()(dafbceWdabfecWdacefbWdaecbfWadfcbeWadcfebWadbefcWadebcfWabcdefWfeda四、四、6-j 符號(hào)符號(hào) Wigner引進(jìn)引進(jìn));()()12)(12();()(23123212/1231223123212331221321321jjjjjjWjjjjjjjjUjjjjjjjjjjjjjj對稱性更高，在下列變換下值不變對稱性更高，在下列變換下值不變?nèi)我鈨闪薪粨Q；任意兩列上下兩行交換任意兩列交換；任意兩列上下兩行交

11、換事實(shí)上，對應(yīng)事實(shí)上，對應(yīng)24 個(gè)個(gè)W的的6-j 符號(hào)全相等符號(hào)全相等五、五、Racah 系數(shù)求和公式系數(shù)求和公式 C-G系數(shù)與系數(shù)與Racah系數(shù)間求和公式系數(shù)間求和公式) 1 ();(2323112233211221122211223232233222312321jmjjjmjjjjjjjjCjjjjjjUCCCjmjjjjjjjjjjmjjCCjjjjjjUCC2233211221122211122323223322232311);()2(2312321五、五、Racah 系數(shù)求和公式系數(shù)求和公式 證明證明 把展開式代入變換關(guān)系把展開式代入變換關(guān)系);();();(23321231232

12、13122123jmjjjjjjjjjjUjmjjjjj3322112312321332211|);(|2332123233322232311231232133121212122211mjmjmjCCjjjjjjUmjmjmjCCmmmmmjmjmjjmmjmjjmmmmjmmjmjmjmjmj五、五、Racah 系數(shù)求和公式系數(shù)求和公式以左矢以左矢|332211jjj作用于兩邊作用于兩邊232332332223231123322333223223112333211221122211);();(23123212312321jjjjmjjjjjjjmjjjjmjjjjjCCjjjjjjUCCjj

13、jjjjUCC以以223233322jjjC作用上式兩邊得作用上式兩邊得 (1) ;(122312321jjjjjjjU以以作用得作用得 (2)五、五、Racah 系數(shù)求和公式系數(shù)求和公式 Racah系數(shù)間求和公式系數(shù)間求和公式（1）);();()();(31231322312321123121323312312321jjjjjjUjjjjjjUjjjjjjUjjjjjjjj 用用Racah系數(shù)表示系數(shù)表示);();() 12()();(3123132231232123123121323312312321jjjjjjWjjjjjjWjjjjjjjWjjjjjjjj 五、五、Racah 系數(shù)求和

14、公式系數(shù)求和公式 Racah系數(shù)間求和公式系數(shù)間求和公式（2）);();();();();(124231423141241241212412341232412341232312312321124jjjjjjUjjjjjjUjjjjjjUjjjjjjUjjjjjjUj);();();() 12();();(124231423141241241212412341231242412341232312312321124jjjjjjWjjjjjjWjjjjjjWjjjjjjjWjjjjjjWj用用Racah系數(shù)表示系數(shù)表示五、五、Racah 系數(shù)求和公式系數(shù)求和公式證明兩個(gè)求和公式要用到的關(guān)系式證明兩個(gè)

15、求和公式要用到的關(guān)系式j(luò)mjjjjjmjjjjjjjjjjU;|;);(31221233212312321jmjjjjjmjjjj;|;2332131221jmjjjjmjmjmjCCmjmjmjCCjmjjjjjjjmmmmjmmjmjmjmjmjjjjmmmmjmmjmjmjmjmj;|)(|)(|;|31212332211332211312211221123213312121212112212211232133121212122211以及以及jmjjjjjmjjjjjjj;|)(;|1221331221123五、五、Racah 系數(shù)求和公式系數(shù)求和公式 （1式的證明式的證明插入封閉關(guān)系j

16、mjjjjjmjjjj;|;3113231221jmjjjjjmjjjjjmjjjjjmjjjjj;|;|;311322332123321312212323;|;);(31132233212312321jjmjjjjjmjjjjjjjjjjUjmjjjjjmjjjj;|;3113231221而jjjjjjjmjjjjjmjjjj231312)(;|;)(2311312213);()(12312132311232jjjjjjUjjjjj五、五、Racah 系數(shù)求和公式系數(shù)求和公式 （1式的證明續(xù)）式的證明續(xù)）jmjjjjjmjjjj;|;3113223321又jmjjjjjmjjjjjjj;|;

17、)(3113212332231);()(3123132231jjjjjjUjjj將后兩式代入第一式，注意到將后兩式代入第一式，注意到，即得，即得1）SSjjjjjS)()(2311232整數(shù)jjjjjjjjjjjjjjjevenjjjjjjjjjjjS312312321312312321312312321231)()()()()(443六、六、Racah 系數(shù)基本應(yīng)用系數(shù)基本應(yīng)用 用于計(jì)算不可約張量算符矩陣元用于計(jì)算不可約張量算符矩陣元jmjjTTjmjjLL2121| )2() 1 (|2211212211) 12)(12(|)2(| |) 1 (| )(21jjjjLjjjjTjjTjLL

18、jjj六、六、Racah 系數(shù)基本應(yīng)用系數(shù)基本應(yīng)用 用于計(jì)算不可約張量算符矩陣元用于計(jì)算不可約張量算符矩陣元jmjjTmjjjLM2121| ) 1 (| 211111); () 12)(12(|) 1 (| )(1222mjjmLMLjjLjjjCLjjjjjWjjjTj六、六、Racah 系數(shù)基本應(yīng)用系數(shù)基本應(yīng)用 具體例子具體例子計(jì)算電子角動(dòng)量平均值計(jì)算電子角動(dòng)量平均值jmlLjmljmlLjmlz210212121|jmjmjlClljjWljlLl10211)1 ;() 12)(12()(|21利用利用) 1(|lllLl ) 1(/10jjmCjmjm) 12)(1() 12)(1(

19、2) 1() 1()()1:(432121llljjjlljjlljjWlj) 1() 1() 1(2|432121lljjjjmjmlLjmlz六、六、Racah 系數(shù)基本應(yīng)用系數(shù)基本應(yīng)用 計(jì)算電子角動(dòng)量平均值續(xù)）計(jì)算電子角動(dòng)量平均值續(xù)）jmLJjmjjjmJjmjmlLjmlz|) 1(|202121亦可利用一階張量投影定理計(jì)算亦可利用一階張量投影定理計(jì)算六、六、Racah 系數(shù)基本應(yīng)用系數(shù)基本應(yīng)用具體例子具體例子計(jì)算氦核電四極矩計(jì)算氦核電四極矩設(shè)氦核處于態(tài)設(shè)氦核處于態(tài)13P1111|13P電四極算符電四極算符)(202251620YreQ1111|11111111|11112022516

20、20YreQ11112011212022516)21;1111(33)(1|1CWYre因因451024510102021|1CY10/1111120C6/1)21;1111(W25120reQ6.2矢量球函數(shù)及梯度公式矢量球函數(shù)及梯度公式Vector Spherical Function一、矢量球函數(shù)一、矢量球函數(shù) 矢量函數(shù)描述矢量函數(shù)描述 S=1 的粒子的粒子010100000 iSx12122zSS01211i10010012111i一、矢量球函數(shù)一、矢量球函數(shù) 總角動(dòng)量本征函數(shù)總角動(dòng)量本征函數(shù)mLMmJMmLMzJMLLLYCsT11, 0, 111)()(稱為矢量球函數(shù)稱為矢量球函數(shù)

21、一、矢量球函數(shù)一、矢量球函數(shù) 梯度公式梯度公式lmllllmllllmTlrrrrTlrrrrYr111211121) 1()()()()()(6.3四個(gè)角動(dòng)量耦合四個(gè)角動(dòng)量耦合9-j符號(hào)符號(hào)9-j Symbol一、考慮四個(gè)角動(dòng)量的耦合一、考慮四個(gè)角動(dòng)量的耦合 考慮兩種耦合次序考慮兩種耦合次序?qū)?yīng)兩個(gè)表象對應(yīng)兩個(gè)表象力學(xué)量完全集力學(xué)量完全集共同本征函數(shù)展開式共同本征函數(shù)展開式一、考慮四個(gè)角動(dòng)量的耦合一、考慮四個(gè)角動(dòng)量的耦合 變換系數(shù)與變換系數(shù)與9-j 符號(hào)符號(hào)與與C-G系數(shù)的關(guān)系系數(shù)的關(guān)系一、考慮四個(gè)角動(dòng)量的耦合一、考慮四個(gè)角動(dòng)量的耦合 9-j 符號(hào)與符號(hào)與6-j 符號(hào)的關(guān)符號(hào)的關(guān)系系1)243434232243344322) 12)(12()(0342432jjjjjjjjjjjjjjjjjjjj2) 31241324234433412212241334431221) 12()(jjjjjjjjjjjjjjjjjjjjjjjjjjjjjj一、一、9-j 符號(hào)的對稱性符號(hào)的對稱性 9-j 符號(hào)與符號(hào)與