Tensors for Physics

238 12 Integral Formulae and Distribution Functions

Multiplication of (12.144)byΨ∗′and subsequent integration overd^3 ryields non-
zero contributions only for′−=0,±2. The resulting quadrupole transition matrix
elements are

( 4 π)−^1

∫

Ψ∗+ 2 eλ̂kκ̂rλ̂rκΨd^3 r=

√

(+ 2 )(+ 1 )

( 2 + 5 )( 2 + 3 )

eν+ 1 ̂kν+ 2

×

∫

r^2 c∗ν 1 ···ν+ 1 ν+ 2 cν 1 ···νr^2 dr,

( 4 π)−^1

∫

Ψ∗eλ̂kκ̂rλ̂rκΨd^3 r=

2 

2 + 3

eλ̂kκ

∫

r^2 c∗λν 2 ···νcκν 2 ···νr^2 dr,

( 4 π)−^1

∫

Ψ∗− 2 eλ̂kκ̂rλ̂rκΨd^3 r=

√

(+ 1 )

( 2 + 1 )( 2 − 1 )

eν− 1 ̂kν

×

∫

r^2 c∗ν 1 ···ν− 2 cν 1 ···νr^2 dr.

The selection rule determines which angular momentum state can be reached in an
allowed transition. The strength of the transition rate is determined by the remaining
overlap integral

∫

···dr.
The relevant perturbation Hamiltonian fortwo-quantum absorptionprocesses is
proportional torλrκEκEλ, thus of second order in the radiation fieldE. Here the
quadrupole selection rules′−=0,±2 apply as well.