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角动量算符的本征值方程(4)

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e?im??

1122(r1, r2)=ejm?im??m1m2Cm1m12,2jm???jj?j1m1, j2m2(r1, r2)

由于?jm,jm(r1, r2)线性无关,所以

这样

212Cjm1,m?Cj,m?m,m1?m2 1m21m2m?m1?m2.

(17)

?jj??jj? (18)

将其代人(10)式,得

Dm1?m1?R?Dm2?m2?R???j1??j2?j1?j2?j?j1?j21212Cj,mDRC??????mm?m,m?mj,m1212121m2?jj?(j)?jj? (19)

由§4.4节(10)式知:两不可约表示的矩阵元满足正交性

12D?,?,?D?,?,??W??,???d?d?d?? ????m2m2??m1m1*?j??j?G

?m1?m2??m1m2?j1j2?W(?,?,?)d?d?d? (20)

lj1

三个欧勒角的变化范围分别为: 0??(?)?2?, 0????. 权重因子W(?,?,?)?sin?,所以

?W(?,?,?)d?d?d??8??j1?2 (21)

lj1为表示的维数,对于表示D?R?,lj1?2j1?1.

*(j) 用Dm式两边并对欧勒角加权积分,利用正交??m?,m?m?R?乘(19)

1212关系(20)得

18?212D?,?,?D????,?,??Dm1??m2?,m1?m2??,?,??sin?d?d?d? ??mmm?112m2?j??j?*(j) ?

12j?112Cj, 1m2C??mj, m1m2 12?jj??jj? (22)

16

??为了确定C?j, jmjm,在上式中令m1??j1,m212125.4??j2,并由§

节(2)(3)

与(4)式知:

Dj1?j2,m1?m2(?,?,?) ?*(j)?(?1)kk(j?m1?m2)!(j?m1?m2)!(j?j1?j2)!(j?j1?j2)!k!(j?m1?m2?k)!(j?j1?j2?k)!(j1?j2?m1?m2?k)!12?

(cos12?)2j?m1?m2?j1?j2?2k(?sin?)j1?j2?m1?m2?2kei[(j1?j2)??(m1?m2)?] (22)D

(j1)j1m1(?,?,?)??2j1?!??j1?m1?!?j1?m1?!12?)m1?j1 (24)

j1?m1 (cos(?sin12?)e?i?j1??m1??D(j2)?j2m2(?,?,?)???1?j2?m2?2j2?!??j2?m2?!?j2?m2?!j2?m2 (25)

(cos12?)(?sin12?)j2?m2e?i??j2??m2??将(23)(24)(25)三式代人(22)式并注意??1?得:

2?j1?m1??1, ??1?2k?1,

11??2j1?!?2j2?!?j?m1?m2?!?j?m1?m2?!?j?j1?j2?!?j?j1?j2?!?2??2?8???j1?m1?!?j1?m1?!?j2?m2?!?j2?m2?!????1?Rk?j2?m21k!?j?m1?m2?k?!?j?j1?j2?k?!?j1?j2?m1?m2?k?!??(cos12?)2j?2j2+2m1?2k(sin12j?)2j1?2m1?2ksin?d?d?d?

?12j?1Cj, 1j12Cj, 1m2 ?j21m2?jj??j? (26)

利用积分

17

1 8?则

2?(cos12?)(sin2a12?)2bsin?d?d?d??a!b!?a?b?1?!

1??cos??2??8??2?12j?2j2?2m1?2k1??sin???2??2j1?2m1?2ksin?d?d?d? ??代入(26)式得:

2j?1?j?j2+m1?k?!?j1?m1?k?!

?j?j1?j2?1?!?j?j1?j2?1?!1

??2j1?!?2j2?!?j?m1?m2?!?j?m1?m2?!?j?j1?j2?!?j?j1?j2?!?2????j1?m1?!?j1?m1?!?j2?m2?!?j2?m2?!???k??1?k?j2?m2?j?j2?m1?k?!?j1?m1?k?!

k!?j?m1?m2?k?!?j?j1?j2?k?!?j1?j2?m1?m2?k?!j

?Cj, 1j12Cj, 1m2?j21m2?j??jj? (27)

在上式中再令m1?j1,m2??j2,得:

?

cj1j1?j2?j1j2??2??2j?1??j?j1?j2?!??j?j1?j2?1?!k ???1?k?j?j1?j2?!?j?j1?j2?k?!

k!?j?j1?j2?k?!?j?j1?j2?k?!再利用恒等式 ???1?kk?j?k!?j?j1?j2?!?j?j1?j2?k?!j1?j2?k?!?j?j1?j2?k?!??j??2j2?!?2j1?!j1?j2?!?j1?j2?j?!

这样

18

1 C(j1j2)j, j1?j2??2?2j?1??2j1?!?2j2?!???j?j?j?1!j?j?j!??12??12?? (28)

代入(27) 式得

1??j?j1?j2?!?j?j1?j2?!?j1?j2?j?!?j?m1?m2?!?j?m1?m2?!?2j?1??2Cj,m1m2=???j?j?j?1!j?m!j?m!j?m!j?m!???1121??11??22??22????j1j2? ???1?k?j?m22k?j?j2+m1?k?!?j1?m1?k?!

k!?j?m1?m2?k?!?j?j1?j2?k?!?j1?j2?m1?m2?k?!将其代入(18)式, 最后得C-G系数为:

1Cjm,m1m2?j1j2???j?j1?j2?!?j?j1?j2?!?j1?j2?j?!?j?m?!?j?m?!?2j?1??2?????j?j1?j2?1?!?j1?m1?!?j1?m1?!?j2?m2?!?j2?m2?!??k?j2?m2

???1?k?j?j2+m1?k?!?j1?m1?k?!. (29)

k!?j?m?k?!?j?j1?j2?k?!?j1?j2?m?k?!?1??j1??2?jm, m1m2表1与表2分别给出了C1?与C?jmj, 的C-G系数. mm112 表1

j 12C?1??j1??2?jm, m1m2系数

12m2?1 1m2??1 1j1?12 1?j?m??12??2j1?1?1?j?m??12???2j1?1?1?2?2?j?m?1??2?=???2j??????1 1?j?m??12??2j1?1?1?j?m??12??2j1?1?1?2?2?j?m?1??2?????2j??????11 1j1?12 1??j?m?1??2?=????2j?2??2?????2 1??j?m?1??2?=???2j?2??2?2????

19

表2

j m2?1 1Cjm1, m1m2?j1?系数

m2?0 1m2??1 1j1?1 ?(j1?m)(j1?m?1)?2???(2j?1)(2j?2)?11?? 1?(j1?m?1)(j1?m?1)?2????(2j1?1)(j1?1)?? ?(j1?m)(j1?m?1)?2???(2j?1)(2j?2)?11??1 j1 ?(j1?m)(j1?m?1)??????2j1(j1?1)??12m 1 [j1(j1?1)]2?(j1?m)(j1?m?1)?2?? ??2j1(j1?1)??11j1?1 ?(j1?m)(j1?m?1)?2?? ??2j1(2j1?1)???(j1?m)(j1?m)?2?????j1(2j1?1)?? ?(j1?m+1)(j1?m)?2??2j(2j?1)?11? 例如CC?1??j1??2?111j1?m1?,m1222?1??j1??2?jm, m1m2,

j?j1?12,

m2?12.

1??11????j1??j1??!?22???=????j1?1??11??11??11?!?j1??j1??!?j1??m1??!?j1??m1?22??22??22??2211???11??11??j1??j1??1?!?j1?m1?!?j1?m1?!???!???!22???22??22??j1?1?2?!2j?2???1??????????1?kk?111??j???m1?k?!?j1?m1?k?!?122??111111??????k!?j1??m1??k?!?j1??j1??k?!?j1??m1??k?!222222??????1??2j1?!1!0!?j1?m1?1?!?j1?m1?!?2j1?2??2????2j?2!j?m!j?m!1!0!??????11111????j1?m1?1?!?j1?m1?!?j1?m1?!?j1?m1?1?!? ????0!1!j?m?1!j?m?1!1!0!j?m!j?m!????????11111111??1

?j1?m1?1?2???????j1?m1???j1?m1?1????2j1?1?j1?m1?12j1?1j1?m??2j1?112

? 20

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