Tensors for Physics

236 12 Integral Formulae and Distribution Functions

12.5.2 Electric Dipole Transitions.

Electromagnetic waves, induce transitions between an ‘initial’ stationary state 1
and a ‘final’ state 2 with the energiesE 1 andE 2 , provided that the radiation has
the right frequencyω=(E 2 −E 1 )/. Furthermore, for weak fields, the transition
has to be ‘allowed’ by theselection rules. These rules follow from expressions for
the transition rate which, in turn, is proportional to the absolute square of a ‘matrix
element’〈Ψfinal|Hpert|Ψinitial〉, whereHpertstands for the time independent part of the
‘perturbation Hamiltonian’ which characterizes the interaction between the atom and
the electromagnetic field. In spatial representation, the matrix element is computed
as an integral over space.
The electric dipole transitions are associated with the perturbation Hamiltonian
Hpert=Hdip≡−pe·Ewith the electric dipole momentpe=qr, whereqis the
electric charge. The unit vector parallel to the electric fieldEis denoted bye.Now
Ψ, i.e. a state with a well defined magnitude of the angular momentum is chosen as
initial state. ThenHdip|Ψinitial〉is proportional to

eλ̂rλφν 1 ···νcν 1 ···ν

=

(√

+ 1

2 + 3

φν 1 ···νλ+

√



2 + 1

Δ()ν 1 ···ν,λκ 1 ···κ− 1 φκ 1 ···κ− 1

)

eλcν 1 ···ν. (12.140)

Here the (11.53) was used: the product of a vector with an irreducible tensor of rank
constructed from this vector yield an irreducible tensor of rank+1 and another
one of rank−1. This is already the essence of the selection rule for the electric
dipole transitions. Multiplication of the expression (12.140)byΨ∗′and subsequent
integration overd^3 ryields non-zero contributions only for

′=± 1.

The resulting dipole transition matrix elements are

( 4 π)−^1

∫

Ψast+ 1 eλrλΨd^3 r=

√

+ 1

2 + 3

eν+ 1

∫

rc∗ν 1 ···ν+ 1 cν 1 ···νr^2 dr,

( 4 π)−^1

∫

Ψ∗− 1 eλrλΨd^3 r=

√



2 + 1

eν

∫

rc∗ν 1 ···ν− 1 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 Cartesian indices characterize properties, e.g. the
direction of the electric field and the orientational sub-states of the initial and final
state.
The electric-field-induced transition from an=0toa=1 state prepares
an orientationally well defined state, depending on the polarization of the incident