CHAPTER 1. ANGULAR MOMENTUM
20
The + sign designates that the lines associated with j1 m1 , j2 m2 , and j3 m3 are oriented in such a way that a counter-clockwise rotation leads from j1 m1 to j2 m2 to j3 m3 . We use a − sign to designate that a clockwise rotation leads from j1 m1 to j2 m2 to j3 m3 . Thus, we can rewrite Eq.(1.106) as
j1 m1
j2 m2
j3 m3
j1 m1
=−
.
j2 m2
(1.107)
j3 m3
The symmetry relation of Eq.(1.78) is represented by the graphical identity: j3 m3
j2 m2 j2 m2
+
=
j1 m1
+
j1 m1
j1 m1
=
.
j3 m3
+
j3 m3
(1.108)
j2 m2
The symmetry relation (1.79) leads to the graphical relation: j3 m3
j3 m3 j2 m2
−
= (−1)j1 +j2 +j3 +
j1 m1
.
j2 m2
(1.109)
j1 m1
One can attach directed lines and three-j symbols to form combinations such as
j1 m1 +
✻
j3 m3
= (−1)j1 −m1
j2 m2
j1 −m1
j3 m3
.
(1.110)
j2 m2
Using this, the Wigner-Eckart theorem can be written j1 m1
j1 , m1 |Tqk |j2 , m2 = − ✻
j1 ||T k ||j2 .
kq
(1.111)
j2 m2
Furthermore, with this convention, we can write j1 m1
C(j1 , j2 , j3 ; m1 , m2 , m3 ) =
❄ 2j3 + 1 −
❄
j2 m2
j3 m3
.
(1.112)
√ Factors of 2j + 1 are represented by thickening part of the corresponding line segment. Thus, we have the following representation for a Clebsch-Gordan coefficient: j1 m1 ❄ j3 m3 . (1.113) C(j1 , j2 , j3 ; m1 , m2 , m3 ) = − ❄ j2 m2