文档介绍：两个角动量的耦合,4、两个角动量的耦合在讨论了粒子的轨道角动量和自旋角动量后,一个系统总角动量,等于有关的各个独立的角动量的矢量和。在量子力学中,即使是单个粒子的系统,也会出现自旋角动量和轨道角动量的相加或耦合。在讨论有关具体问题前,先要学会量子力学中角动量算符的加法。一、对易关系,,JJ和设有两个独立的角动量算符,它们分别可以是轨道角动量算符,或自12旋角动量算符,或由它们相加而成的一些角动量算符。,,JJ和由角动量算符的一般定义知,都是厄米算符。且满足如下一些对易关12系:,,,JJiJ,,111xyz,,,,,,,,,JJiJJJiJ,,,,,,111111yzx,,,JJiJ,,,,111zxy,,,,,,,,,2222JJJJJJ,,0,,,,,,,,111111xyz,,,,,,,,,JJiJ,,222,,,,,,,,2JJJJ,0,,,0,,,,,,2212,,,,,,,,,现定义,JJJ,,12,显然:也是一个角动量算符J,,,,,,,,22易证:,,JJJJ,,0,,,,12,,,,,,,,22,,,,,,,,,22而由:JJJJJJJ,,,,,2,,121212,,,,,,,,,,,,22可推得:,,JJJJ,,0,,,,12,,,,,,,,,,,,,,,,22另外,易证:JJJJ,,,,0,,,,12zz,,,,,,,,,,,,,,,,22JJ,,,JJ,0,,,,1zz2zz,,,,,,,,,,,,,,,,,,,,,,,,JJJJJJJJ,,,,0,,,,,,,,,,,,1212xxxxyyyy,,,,,,,,,,,,,,,,22注意:JJJJ,0,,0,,,,,,12,,,,,,,,,,,,,JJiJ,,例、证明:xyz,,,,证明:,,,,,,,,,,,,,,JJJJJJJJJJ,,,,,,,1212xxyyxyxyyx,,,,,,,,,,,,,,,,,,JJJJJJ,,,,,,112212xyyxyy,,,,,,,,,,,,,,,,,,,,,,,,JJJJJJJJ,,,,,,,,11122122xyxyxyxy,,,,,,,,,,,,,,,,,,iJiJ00,,,,12zz,iJ,z二(无耦合表象(直乘表象)2,,,,2JJ和J有上述对易关系知,,,这四个算符两两相互对易J12z2z1222,,,,,,,,,,,,,,,,,,,,2,,,,JJJJJJJJJJ,,0,,0,0,,0,,,,,,112,,,,,,zzzz1221212(),,,,,,,,,,,,,,,,由算符对易的性质知,这四个算符有共同本征态,可以建立以这些共同本征态为基的表象,这种表象叫无耦合表象或直乘表象。为简单,采用狄拉克算符。,,,,22JJJ|,jm,J根据角动量算符的性质,设和的共同本征态为,而与的12111z2z|,jm,共同本征态为,它们的本征值方程分别为:222,2Jjmjjjm|,1|,,,,,1,,111111,Jjmmjm|,|,,,111111z2,2Jjmjjjm|1|,,,,2,,222222,Jjmmjm||,,,222222z,这里和分别属于两个独立的空间,只对|||jmjmJmj,,,1122111,作用,而只对作用Jmj|,222,,JJjmjm,分别只对和起作用||,,121122?,,,若定义,|||jmjmjm