32 The hydrogen atom
when the polarized light interacts with an atom that has a well-defined
orientation, e.g. an atom in an external magnetic field. If the light
is unpolarized or there is no defined quantisation axis, or both, then
∆ml=0,±1.Example 2.1 Longitudinal observation
Electromagnetic radiation is a transverse wave with its oscillating elec-
tric field perpendicular to the direction of propagation,̂erad·k=0.
Thus radiation with wavevectork=k̂ezhasAz=0andπ-transitions
do not occur.^32 Circularly-polarized radiation (propagating along thez-(^32) Similar behaviour arises in the clas-
sical model of the normal Zeeman ef-
fect in Section 1.8, but the quantum
treatment in this section shows that
it is a general feature of longitudinal
observation—not just for the normal
Zeeman effect.
axis) is a special case for which transitions occur with either ∆ml=+1
or ∆ml=−1, depending on the handedness of the radiation, but not
both.
2.2.2 Integration with respect toθ
In the angular integral the spherical harmonic functions withl=1(from
eqn 2.34) are sandwiched between the angular momentum wavefunctions
of the initial and final states so thatIang∝∫ 2 π0∫π0Yl 2 ∗,m 2 Y 1 ,mYl 1 ,m 1 sinθdθdφ. (2.38)(^33) See the references on angular mo- To calculate this angular integral we use the following formula: 33
mentum in quantum mechanics; the
reason why the magnetic quantum
numbers add is obvious from Φ(φ).
Y 1 ,mYl 1 ,m 1 =AYl 1 +1,m 1 +m+BYl 1 − 1 ,m 1 +m, (2.39)
whereAandBare constants whose exact values need not concern us.
(^34) We have Thus from the orthogonality of the spherical harmonics (^34) we find
∫ 2 π
0
∫π
0 Yl′,m′Yl,msinθdθdφ
=δl′,lδm′,m. This reduces to the nor-
malisation in Table 2.1 whenl′=land
m′=m.
Iang∝Aδl 2 ,l 1 +1δm 2 ,m 1 +m+Bδl 2 ,l 1 − 1 δm 2 ,m 1 +m.
The delta functions give the selection rule found previously, namely
∆ml=m,wherem=0,±1 depending on the polarization, and also
∆l=±1. In the mathematics the functions withl= 1 that represent
the interaction with the radiation are sandwiched between the orbital
angular momentum eigenfunctions of the initial and final states. Thus
therule∆l=±1 can be interpreted as conservation of angular momen-
tum for a photon carrying one unit of angular momentum,(Fig. 2.8
(^35) This argument applies only for elec- illustrates this reasoning for the case of total angular momentum). (^35) The
tric dipole radiation. Higher-order
terms, e.g. quadrupole radiation, can
give ∆l>1.
changes in the magnetic quantum number are also consistent with this
picture—the component of the photon’s angular momentum along the
z-axis being ∆ml=0,±1. Conservation of angular momentum does not
explain why ∆l= 0—this comes about because of parity, as explained
below.
2.2.3 Parity
Parity is an important symmetry property throughout atomic and molec-
ular physics and its general use will be explained before applying it to