CHEM 442 Lecture 19 Problems
19-1. Show that the product of spherical harmonics Y
10
Y
10
is the linear combination of spherical harmonics Y
20
and Y
00
.
19-2. Show that the product of spherical harmonics Y
11
Y
11
is proportional to spherical harmonics Y
22
.
19-3. Extending the finding of 19-1, justify the selection rules of hydrogenic atomic transitions with z -polarized light:
D l
= ±
1;
D m l
=
0;
D m s
=
0 .
19-4. Extending the finding of 19-1 and 19-2, justify the selection rules of hydrogenic atomic transitions with nonpolarized or circularly polarized light:
D l
= ±
1;
D m l
=
0,
±
1;
D m s
=
0 .
19-5. List the orbitals to which a 4 s electron can make an optical transition.
19-6. The 3 d
3 z
2 r
2
orbital has the quantum numbers n
=
3, l
=
2, m l
=
0 . Consulting with the tables of radial and angular parts of the hydrogenic wave functions, show that it is
( z
2
r
2
) e
-
3 a
0 r
.
19-7. Using 19-6, show that the transition dipole moments in the Cartesian coordinates for the 1 s to 3 d
3 z
2 r
2
transition with z -polarized light is an odd function in the z coordinate and is forbidden. Show the same for x - and y -polarized lights.
19-8. Using 19-6, show that the transition dipole moments in the Cartesian coordinates for the 2 p z
to 3 d
3 z
2 r
2
transition with z -polarized light is an even function in all of the x , y , and z coordinates and is allowed. Show that the corresponding transition dipole moments for y - and z -polarized lights are forbidden.
19-9. Express the energy of a photon with wavelength of 400 nm in the units cm
−1
, eV,
K, and kcal/mol.