2.2 Transitions 31whereAπdepends on the component of the electric field along thez-
axis and the component in thexy-plane is written as a superposition of
two circular polarizations with amplitudesAσ+andAσ−(rather than
in terms of linear polarization in a Cartesian basis).^27 Similarly, the^27 We will see that the labelsπ,σ+and
σ−refer to the transition that the radi-
ation excites; for this it is only impor-
tant to know how the electric field be-
haves at the position of the atom. The
polarization state associated with this
electric field, e.g. whether it is right-
or left-handed circularly-polarized radi-
ation, also depends on the direction of
propagation (wavevector), but we shall
try to avoid a detailed treatment of the
polarization conventions in this discus-
sion of the principles. Clearly, however,
it is important to have the correct po-
larization when setting up actual exper-
iments.
classical motion of the electron was written in terms of three eigenvectors
in Section 1.8: an oscillation along thez-axis and circular motion in the
xy-plane, both clockwise and anticlockwise.
From the expression for̂rin terms of the angular functionsYl,m(θ, φ)
withl= 1 we find that the dipole induced on the atom is proportional
to^28
(^28) The eigenvectors have the following
properties:
̂ex+îey
√
2
·
̂ex−îey
√
2
=1
and
̂ex√±îey
2
·̂ex√±îey
2
=0.
̂r·̂erad∝Aσ−Y 1 ,− 1 +AzY 1 , 0 +Aσ+Y 1 ,+1. (2.34)
The following sections consider the transitions that arise from these three
terms.^29
(^29) In spherical tensor notation
(Woodgate 1980) the three vector
components are writtenA− 1 ,A 0 and
A+1, which is convenient for more
general use; but writing eqn 2.34 as
given emphasises that the amplitudes
Arepresent the different polarizations
of the radiation and the spherical
harmonics come from the atomic
response (induced dipole moment).
π-transitions
The component of the electric field along thez-axisAzinduces a dipole
moment on the atom proportional tôerad·̂ez=cosθand the integral
over the angular parts of the wavefunctions is
Iπang=∫ 2 π0∫π0Yl∗ 2 ,m 2 (θ, φ)cosθYl 1 ,m 1 (θ, φ)sinθdθdφ. (2.35)To determine this integral we exploit the symmetry with respect to ro-
tations about thez-axis.^30 The system has cylindrical symmetry, so the
(^30) Alternative methods are given below
and in Exercise 2.9.
value of this integral is unchanged by a rotation about thez-axis through
an angleφ 0 :
Iangπ =ei(m^1 −m^2 )φ^0 Iangπ. (2.36)
This equation is satisfied if eitherIangπ =0orml 1 =ml 2 .Forthis
polarization the magnetic quantum number does not change, ∆ml=0.^31
(^31) We usemlto distinguish this quan-
tum number fromms, the magnetic
quantum number for spin angular mo-
mentum that is introduced later. Spe-
cific functions of the spatial variables
such asYl,mand e−imφdo not need
this additional subscript.
σ-transitions
The component of the oscillating electric field in thexy-plane excitesσ-
transitions. Equation 2.34 shows that the circularly-polarized radiation
with amplitudeAσ+excites an oscillating dipole moment on the atom
proportionalY 1 , 1 ∝sinθeiφ, for which the angular integral is
Iσ+
ang=∫ 2 π0∫π0Yl∗ 2 ,m 2 (θ, φ)sinθeiφYl 1 ,m 1 (θ, φ)sinθdθdφ. (2.37)Again, consideration of symmetry with respect to rotation about thez-
axis through an arbitrary angle shows thatIσ
- ang=0unlessml 1 −ml 2 +
1 = 0. The interaction of an atom with circularly-polarized radiation of
the opposite handedness leads to a similar integral but with eiφ→e−iφ;
this integralIσ
−
ang= 0 unlessml 1 −ml 2 −1 = 0. Thus the selection rule
for theσ-transitions is ∆ml=±1.
We have found the selection rules that govern ∆mlfor each of the
three possible polarizations of the radiation separately. These apply