Exercises for Chapter 7 149(7.6)The properties of some transitions in hydrogen
The table gives the values ofA 21 for transitions
from then= 3 shell of hydrogen to lower levels.
(Spin and fine structure are ignored.)
Transition A 21 (s−^1 )
2p–3s 6. 3 × 106
1s–3p 1. 7 × 108
2s–3p 2. 2 × 107
2p–3d 6. 5 × 107(a) Draw an energy-level diagram for then=1, 2
and 3 shells in hydrogen that shows the al-
lowed electric dipole transitions between the
orbital angular momentum levels. (Neglect
transitions for whichndoes not change, e.g.
2s–2p.)
(b) Calculate the lifetimes of the 3s, 3p and 3d
configurations. What fraction of atoms that
start in 3p end up in the 2s configuration?
(c) An electron in the 2p configuration has a life-
time of only 1.6 ns. Why is this shorter than
for the 3p configuration?
(d) Calculate the radial matrix elementsD 12 in
units ofa 0 for the transitions in the table, and
for 1s–2p.
(e) CalculateIsatfor the 2p–3s and 1s–3p transi-
tions.(7.7)The classical model of atomic absorption
(a) A simple classical model of absorption as-
sumes that an electron (in an atom) behaves
like a damped simple harmonic oscillator of
charge−eand massmedriven by the oscil-
lating electric field of the radiation:E 0 cosωt.
The electron’s equation of motion has the form
of eqn 7.55 with a driving force of constant
amplitudeF 0 =−eE 0. Find a solution of this
equation in the formx=Ucosωt−Vsinωt
(UandVare not functions of time in the case
considered here).
(b) Show that the displacement of the electron has
an amplitude
√
U^2 +V^2 =
F 0 /m
√
(ω^2 −ω^20 )^2 +(βω)^2
F 0
2 mω{
(ω−ω 0 )^2 +
β^2
4}− 1 / 2
.Show that the angular frequencyωat which
this amplitude is maximum is very close toω 0
for a narrow resonance.
(c) Show that the phase is given bytanφ=
V
U
=
βω
ω^2 −ω 02
.How does this phase vary as the angular fre-
quencyωincreases fromωω 0 toω ω 0?
(d) For frequencies close to the atomic resonance
(ωω 0 ), show that your expressions forU
andVcanbewritteninanapproximateform
that agrees with eqns 7.58 and 7.59 (that were
derived using the slowly-varying envelope ap-
proximation in which the amplitude of the
driving force may change slowly over time).
(e) Show that in the steady state the powerPab-
sorbed by the electron is a Lorentzian function
ofω:
P∝
1
(ω−ω 0 )^2 +(β/2)^2.(7.8)Oscillator strength
This question shows the usefulness of a dimen-
sionless parameter called the absorption oscillator
strength, denoted byf 12.(a) Show that for the cross-section in eqn 7.79 we
have
∫∞−∞σ(ω)dω=2π^2 r 0 cf 12 , (7.96)where r 0 =2. 8 × 10 −^15 mandf 12 =
2 meωD 122 /(3).
(b) From the simple model of the atom as an os-
cillating electron in Exercise 7.7, find the clas-
sical absorption cross-sectionσcl(ω) in terms
ofβ,ω 0 and fundamental constants.
(c) Without the driving electric field, the oscilla-
tor undergoes damped harmonic motionx=
x 0 e−βt/^2 cos (ω′t−φ). The power radiated by
an oscillating dipole leads to a decay rate given
by eqn 1.23 (from classical electromagnetism).
Determineβ.
(d) Show thatσcl(ω) integrated over all frequen-
cies gives 2π^2 r 0 c.
Comment. This classical value is the maxi-
mum value for any transition, sof 12 1 .The
absorption oscillator strength is a fraction of
the integrated cross-section associated with a
given transition.