Page 201 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

192 The hydrogen atom

magnetic ®eld into three levels. For d orbitals (l ˆ 2), the energies are split into
®ve levels.
This splitting of the energy levels by the magnetic ®eld leads to the splitting
of the lines in the atomic spectrum. The wave number ~ í of the spectral line
corresponding to a transition between the state jn 1 l 1 m 1 i and the state jn 2 l 2 m 2 i
is

jÄEj 2 1 1 ì B B
~ í ˆ ˆ RZ ÿ ‡ (m 2 ÿ m 1 ), n 2 . n 1 (6:91)
hc n 2 n 2 hc
1 2
Transitions between states are subject to certain restrictions called selection
rules. The conservation of angular momentum and the parity of the spherical
harmonics limit transitions for hydrogen-like atoms to those for which
Äl ˆÐ 1 and for which Äm ˆ 0, 1. Thus, an observed spectral line ~ í 0 in the
absence of the magnetic ®eld, given by equation (6.83), is split into three lines
with wave numbers ~ í 0 ‡ (ì B B=hc), ~ í 0 , and ~ í 0 ÿ (ì B B=hc).

Problems

6.1 Obtain equations (6.28) from equations (6.26).
^
^
^ ^
6.2 Evaluate the commutator [A ë , B ë ] where the operators A ë and B ë are those in
equations (6.26).
^
6.3 Show explicitly by means of integration by parts that the operator H l in equation
2
(6.18) is hermitian for a weighting function equal to r .
^
6.4 Demonstrate by means of integration by parts that the operator H9 l in equation
(6.36) is hermitian for a weighting function w(r) ˆ r.
^
^
6.5 Show that (A ë ‡ 1)S ë‡1,l ˆ a ë‡1,l S ël and that (B ë ‡ 1)S ël ˆ b ë‡1,l S ë‡1,l .
6.6 Derive equation (6.45) from equation (6.34).
6.7 Derive the relationship

1 1
2 2
a nl S nl S nÿ1,l r dr ÿ b n‡1,l S nl S n‡1,l r dr ˆ 1
0 0
ÿ1
6.8 Evaluate hr i nl for the hydrogen-like atom using the properties of associated
Laguerre polynomials. First substitute equations (6.22) and (6.55) into (6.69) for
k ˆÿ1. Then apply equations (F.22) to obtain (6.71).
6.9 From equation (F.19) with í ˆ 2, show that
2 3
1 2[3n ÿ l(l ‡ 1)][(n ‡ l)!]
2
r 2l‡3 ÿr 2l‡1 (r)] dr ˆ
e [L
n‡l
0 (n ÿ l ÿ 1)!
Then show that hri nl is given by equation (6.72).
6.10 Show that hri 2s ˆ 6a 0 =Z using the appropriate radial distribution function in
equations (6.66).
6.11 Set ë ˆ e9 in the Hellmann±Feynman theorem (3.71) to obtain hr ÿ1 i nl for the
hydrogen-like atom. Note that a 0 depends on e9.