13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 257
Both coefficients approach^12
√
3
2 for large values ofj.
The diagonal tensor operatorTμjj 1 ···μis essentially the hermitian tensor operator
PjJμ 1 ···Jμ,viz.
Tμjj 1 ···μ=Pj Jμ 1 ···Jμ
(
!
( 2 + 1 )!!
)− 1 / 2
( 2 j+ 1 )−^1 /^2 (j 0 j 1 ···j− 1 )−^1.
(13.72)
The quantitiesjkare analogousl to theSk, defined in (13.22), i.e.
jk^2 =j(j+ 1 )−
k
2
(
k
2
+ 1
)
.
The second rank tensor is
Tμνjj =
(
15
2
) 1 / 2
( 2 j+ 1 )−^1 /^2 (j 0 j 1 )−^1 PjJμJν. (13.73)
Apart from the factor( 2 j+ 1 )−^1 /^2 , the tensor operatorT
jj
μνis equal toPjφTμν,as
defined in (13.61), withc=1.