Putting these two vectors together is like adding toℓ= 1 states. We can get total angular momentum
quantum numbers 2, 1, and 0. Each vector has three components. Thedirect product tensor
has 9. Its another case of
3 ⊗3 = 5S⊕ (^3) A⊕ (^1) S.
Thetensor we make when we just multiply two vectors together canbe reduced into
three irreducible (spherical) tensors. These are the ones for which we can use the Wigner-
Eckart theorem to derive selection rules. Under rotations of the coordinate axes, the rotation matrix
for the 9 component Cartesian tensor will be block diagonal. It can be reduced into three spherical
tensors. Under rotations the 5 component (traceless) symmetric tensor will always rotate into
another 5 component symmetric tensor. The 3 component anti symmetric tensor will rotate into
another antisymmetric tensor and the part proportional to the identity will rotate into the identity.
(~k·~r)(ˆǫ(λ)·~pe) =
1
2
[(~k·~r)(ˆǫ(λ)·~p) + (~k·~p)(ˆǫ(λ)·~r)] +1
2
[(~k·~r)(ˆǫ(λ)·~p)−(~k·~p)(ˆǫ(λ)·~r)]Thefirst term is symmetric and the second anti-symmetricby construction.
Thefirst term can be rewritten.
1
2〈φn|[(~k·~r)(ˆǫ(λ)·~p) + (~k·~p)(ˆǫ(λ)·~r)]|φi〉 =1
2
~k·〈φn|[~r~p+~p~r]|φi〉·ˆǫ(λ)=
1
2
~k·im
̄h〈φn|[H 0 ,~r~r]|φi〉·ˆǫ(λ)= −
imω
2~k·〈φn|~r~r|φi〉·ǫˆ(λ)This makes the symmetry clear. Its normal toremove the trace of the tensor:~r~r→~r~r−δij 3 r^2.
The term proportional toδijgives zero because~k·ˆǫ= 0. The traceless symmetric tensor has 5
components like anℓ= 2 operator; The anti-symmetric tensor has 3 components; and the trace term
has one. This is the separation of the Cartesian tensor into irreducible spherical tensors. Thefive
components of the traceless symmetric tensor can be writtenas a linear combination
of theY 2 m.
Similarly, thesecond (anti-symmetric) term can be rewrittenslightly.
1
2[(~k·~r)(ˆǫ(λ)·~p)−(~k·~p)(ˆǫ(λ)·~r)] = (~k׈ǫ(λ))·(~r×~p)Theatomic state dependent part of this,~r×~p, is an axial vectorand therefore has three
components. (Remember and axial vector is the same thing as an anti-symmetric tensor.) So this
is clearly anℓ= 1 operator and can beexpanded in terms of theY 1 m. Note that it is actually a
constant times theorbital angular momentum operator~L.
So thefirst term is reasonably named the Electric Quadrupoleterm because it depends on
the quadrupole moment of the state. Itdoes not change parityand gives us the selection rule.
|ℓn−ℓi|≤ 2 ≤ℓn+ℓiThe second term dots the radiation magnetic field into the angular momentum of the atomic state,
so it is reasonably called themagnetic dipole interaction. The interaction of theelectron spin
with the magnetic field is of the same order and should be included together with the E2 and M1
terms.
e ̄h
2 mc
(~k׈ǫ(λ))·~σ