Tensors for Physics

256 13 Spin Operators

∑

m 1

∑

m 2

|j 1 m 1 〉|j 2 m 2 〉(j 1 m 1 ,j 2 m 2 |jm)=|jm〉.

The nondiagonal elements of the spherical harmonicY(m)(u)are related to the spher-

ical tensor operators by

[

Y(m)(u)

]jj′

=PjY(m)(u)Pj

′

=

√

2 j+ 1

4 π

(

j 0 , 0 |j′ 0

)

T

jj′

m. (13.67)

By analogy to (13.67), the operator form of the Cartesian tensoruμ 1 ···uμ is

(uμ 1 ···uμ)jj

′

=

√

!

( 2 + 1 )!!

√

2 j+ 1 (j 0 , 0 |j′ 0 )T

jj′

μ 1 ···μ. (13.68)

The tensor operators have the properties:

(i) The hermitian adjoint ofTjj

′

μ 1 ···μis

(Tjj

′

μ 1 ···μ)†=(−^1 )j−j

′

Tj

′j

μ 1 ···μ. (13.69)

(ii) Orthogonality and normalization

tr{Tjj

′

μ 1 ···μ(T

jj′

ν 1 ···ν′)

†}=δ

′Δ

()

μ 1 ···μ,ν 1 ···ν. (13.70)

The theoretical description of therotational Raman scatteringinvolves the elements

oftheanisotropicmolecularpolarizabilitytensorwhicharenon-diagonalwithrespect

to the rotational quantum number, cf. (13.53). The relevant operators are

PjuμuνPj

′

=(uμuν)jj

′

,

withj′=j±2. In particular, one has

(uμuν)jj±^2 =

√

2

15

√

2 j+ 1 (j 0 , 20 |j± 20 ) Tμνjj±^2. (13.71)

The Clesch-Gordan coefficients are

(j 0 , 20 |j+ 20 )=

√

3

2

√

(j+ 1 )(j+ 2 )

( 2 j+ 1 )( 2 j+ 3 )

,

(j 0 , 20 |j− 20 )=

√

3

2

√

j(j− 1 )

( 2 j+ 1 )( 2 j− 1 )

.