6.5 Matrix Elements and the Wigner–Eckart Theorem
127
Note that the general form of the operator |O Λ
λ | refers to a component of an irreducible set. Such a set of operators is usually referred to as a tensor operator.
Obvious examples are the components of the electric or magnetic dipole-moment
operators. The Wigner–Eckart theorem introduces a symmetry factorization, which
simplifies the evaluation of matrix elements.
Theorem 12 A matrix element, involving a tensor operator, may be factorized into
a product of an intrinsic scalar part and an appropriate 3Γ coupling coefficient.
The scalar constant is denoted by ψ Ω O Λ φ Γ , and is called the reduced
matrix element.
ψ
Ω
ω
O
Λ
λ |φ
Γ
γ
=
ψ
Ω O
Λ φ
Γ
Ωω|ΛλΓ γ
(6.46)
To prove this theorem, one first considers the coupling of two ingredients of the matrix element, and then compares the result with the third one. We thus first consider
the coupling of the operator and the ket. The transformation of their product does
indeed correspond to the super matrix which is due to the direct product Λ × Γ .
ˆ
R
O
Λ
λ |φ
Γ
γ
= ˆ
R
O
Λ
λ | ˆ
R
−1 ˆ
R|φ
Γ
γ
=
λ ′
γ ′
D
Λ
λ ′ λ (R)D
Γ
γ ′ γ (R)|O
Λ
λ ′
φ
Γ
γ ′
(6.47)
This means that we couple the tensor operator and the ket to form product entities:
(Oφ)
Π
π
=
λ ′
γ ′
O
Λ
λ ′ |φ
Γ
γ ′
Λλ
′ Γγ
′ |Ππ
(6.48)
We now invert this equation, using the unitary properties of the coupling coefficients, to yield:
O
Λ
λ |φ
Γ
γ
=
Π ′
π ′
(Oφ)
Π ′
π ′
Π
′ π
′ |ΛλΓ γ
(6.49)
Then we combine this expression with the bra.
ψ
Ω
ω |O
Λ
λ |φ
Γ
γ
=
Π ′
π ′
ψ
Ω
ω |(Oφ)
Π ′
π ′
Π
′ π
′ |ΛλΓ γ
(6.50)
The matrix elements on the right-hand side are now in fact reduced to an overlap integral where the direct product irreps are compared with the irrep of the bra. Hence,
the selection rules for the overlap integrals apply:
Ωω|(Oφ)
Π ′
π ′
= δ Ω,Π ′ δ ωπ ′
1
dim(Ω)
ω ′
Ωω
′
(Oφ)
Ω ′
ω ′
≡
ψ
Ω O
Λ φ
Γ
(6.51)
The trace summation in this equation is identified as a scalar interaction constant,
which is represented by the reduced matrix element.
127
Note that the general form of the operator |O Λ
λ | refers to a component of an irreducible set. Such a set of operators is usually referred to as a tensor operator.
Obvious examples are the components of the electric or magnetic dipole-moment
operators. The Wigner–Eckart theorem introduces a symmetry factorization, which
simplifies the evaluation of matrix elements.
Theorem 12 A matrix element, involving a tensor operator, may be factorized into
a product of an intrinsic scalar part and an appropriate 3Γ coupling coefficient.
The scalar constant is denoted by ψ Ω O Λ φ Γ , and is called the reduced
matrix element.
ψ
Ω
ω
O
Λ
λ |φ
Γ
γ
=
ψ
Ω O
Λ φ
Γ
Ωω|ΛλΓ γ
(6.46)
To prove this theorem, one first considers the coupling of two ingredients of the matrix element, and then compares the result with the third one. We thus first consider
the coupling of the operator and the ket. The transformation of their product does
indeed correspond to the super matrix which is due to the direct product Λ × Γ .
ˆ
R
O
Λ
λ |φ
Γ
γ
= ˆ
R
O
Λ
λ | ˆ
R
−1 ˆ
R|φ
Γ
γ
=
λ ′
γ ′
D
Λ
λ ′ λ (R)D
Γ
γ ′ γ (R)|O
Λ
λ ′
φ
Γ
γ ′
(6.47)
This means that we couple the tensor operator and the ket to form product entities:
(Oφ)
Π
π
=
λ ′
γ ′
O
Λ
λ ′ |φ
Γ
γ ′
Λλ
′ Γγ
′ |Ππ
(6.48)
We now invert this equation, using the unitary properties of the coupling coefficients, to yield:
O
Λ
λ |φ
Γ
γ
=
Π ′
π ′
(Oφ)
Π ′
π ′
Π
′ π
′ |ΛλΓ γ
(6.49)
Then we combine this expression with the bra.
ψ
Ω
ω |O
Λ
λ |φ
Γ
γ
=
Π ′
π ′
ψ
Ω
ω |(Oφ)
Π ′
π ′
Π
′ π
′ |ΛλΓ γ
(6.50)
The matrix elements on the right-hand side are now in fact reduced to an overlap integral where the direct product irreps are compared with the irrep of the bra. Hence,
the selection rules for the overlap integrals apply:
Ωω|(Oφ)
Π ′
π ′
= δ Ω,Π ′ δ ωπ ′
1
dim(Ω)
ω ′
Ωω
′
(Oφ)
Ω ′
ω ′
≡
ψ
Ω O
Λ φ
Γ
(6.51)
The trace summation in this equation is identified as a scalar interaction constant,
which is represented by the reduced matrix element.