Tensors for Physics

12.5 Selection Rules for Electromagnetic Radiation 237

radiation. According to (12.140) the angle dependent part of the=1 wave function

isΦ 1 (̂r)=Cμφμ, withCμ=eμ≡E−^1 Eμ, thus

Φ 1 =Φ 1 (r)=

√

3 eλ̂rλ. (12.141)

Clearly,eis the unit vector parallel to the field. For linearly polarized light, propagat-

ing inz-direction,e=excan be chosen. The field orientation of circular polarized

light is described bye= 2 −^1 /^2 (ex±iey). Notice that the wave function is complex,

in this case. The vector and tensor polarizations in this exited state are

〈Lμ〉= 0 , 〈LμLν〉=exμexν, (12.142)

for the linearly polarized light. For circular polarization, one obtains

〈Lμ〉=ezμ, 〈LμLν〉=−ezμezν. (12.143)

The computations leading to these results are deferred to the Exercise12.5.

12.5 Exercise: Compute the Vector and Tensor Polarization for a= 1 State
Hint: use the wave function (12.141) witheμ=exμandeμ=(exμ+ie
y
μ)/

√

2for
the linear and circular polarized cases. For the angular momentum operator and its
properties see Sect.7.6.2.

12.5.3 Electric Quadrupole Transitions

The HamiltonianHquadinducing electric quadrupole transitions is proportional to

rλrκkκEλ, wherek=k̂kis the wave vector of the incident electric fieldE=Ee.

From (11.57) follows, that the application ofHquadon the wave functionΨyields

three contributionsΨ′with

′=, ± 2.

More specifically, one has

eλ̂kκ̂rλ̂rκφμ 1 ···μcμ 1 ···μ

=

(√

(+ 2 )(+ 1 )

( 2 + 5 )( 2 + 3 )

φμ 1 ···μλκ+

2 

2 + 3

Δ(,μ 1 μ^2 ,) 2 ···μ,λκ,ν 1 ν 2 ···νφν 1 ···ν

+

√

(− 1 )

( 2 + 1 )( 2 − 1 )

Δ()μ 1 μ 2 ···μ,νλ ν 1 ν 2 ···ν− 2 φν 1 ···ν− 2

)

eλ̂kκcμ 1 ···μ. (12.144)