Tensors for Physics

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.