Physics 312 – Problem Set 6
1. (Working out details from class) Use perturbation theory to find the probability that a hydrogen atom transitions
~
~ 0 e−t/τ . Show that your result
from a 1s state to a 2p state in the presence of an applied electric field E(t)
=E
recaptures the time-independent result in the appropriate limit.
2. Obtain the lifetime of the n = 2, l = 1 level of hydrogen against spontaneous decay by dipole radiation. What
is the lifetime against induced emission?
3. Rate of energy loss
In class we found an expression for the rate of energy loss by spontaneous emission of unpolarized dipole radiation
of frequency ωji in terms of Aji , the Einstein coefficient for spontaneous decay from level j to level i. Now
compare this to the case of a classical electron oscillator:
(a) Consider the one dimensional equation of motion
dx
d2 x
+γ
+ ω02 x = 0 ,
dt2
dt
where ω0 is the oscillation frequency and choose
γ=
1 2 e2 ω02
.
4π0 3 mc3
Show that γ is small compared to ω0 for atomic energies.
(b) Find a solution which is valid in this limit.
(c) Calculate the approximate rate of energy loss.
(d) Identify the classical analog of the dipole matrix element; that is, comparing your result to our expression
from class, what plays the role of the dipole matrix element?
4. Show that the multipole operator X̂ P̂y breaks up into an electric quadrupole piece and a magnetic dipole piece.
Then show that the ratio of the typical size of a magnetic dipole matrix element to the typical size of an
electric dipole matrix element is Zα, for a hydrogenic atom with Z protons. Find the ratio of a typical electric
quadrupole matrix element to a typical electric dipole matrix element. What does that tell you about the
relative size of the transition rates taking place through these operators?
5. Show that the relationship we obtained from imposing the condition that our atoms are in a steady state
of thermal equilibrium at temperature T and the applied radiation field is also in thermal equilibrium at
temperature T :
3
1
Bij πh̄ω
2 c3 eh̄ω/kT −1
Aji +
3
1
Bji πh̄ω
2 c3 eh̄ω/kT −1
=
gj −h̄ω/kT
e
gi
requires that
gi Bij = gj Bji and Aji =
h̄ω 3
Bji
π 2 c3
if the Einstein coefficients are independent of temperature.
6. C-TDL Ch. 13 problem 1
7. C-TDL Ch. 13 problem 2
8. C-TDL Ch. 13 problem 3
9. C-TDL Ch. 13 problem 5
10. Fermi’s Golden Rule application: Read Merzbacher’s Chapter 9, section 7, just past Eqn. 19.103. Fill in the
missing steps to obtain Eqn. 19.103.
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