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 Consider the twostate system examined in Section 8.3. Suppose that
where
,
,
, and
are real. Show that
where
,
, and
Hence, deduce that if the system is definitely in state 1 at time
then the probability of finding it in state 2
at some subsequent time,
, is
 Consider an atomic nucleus of spin
and
factor
placed in the magnetic field
where
. Let
be a properly normalized simultaneous eigenket of
and
, where
is the nuclear spin. Thus,
and
, where
. Furthermore, the instantaneous nuclear
spin state is written
where
.
 Demonstrate that
for
, where
and
.
 Consider the case
. Demonstrate that if
and
then
 Consider the case
. Demonstrate that if
and
then
 Consider the case
. Demonstrate that if
and
then
 Derive Equation (8.45).
 Derive Equation (8.83).
 If
is the probability that a system initially in state
at time
is still in that
state at some subsequent time, deduce that the mean lifetime of the state is
.
 Demonstrate that
when
, where
is the momentum operator, and
is a real function of the position operator,
.
Hence, show that the Hamiltonian (8.128) is Hermitian.
 Demonstrate that
where the average is taken over all possible directions of the unit vector
.
 Demonstrate that
where
is specified by Equation (8.165).
[Hint: Write
where
.]
 Demonstrate that a spontaneous transition between two atomic states of zero orbital
angular momentum is absolutely forbidden. (Actually, a spontaneous transition between
two zero orbital angular momentum states is possible via the simultaneous emission of two photons, but takes place at a very slow rate [56,15,102].)
 Find the selection rules for the matrix elements
and
to
be nonzero. Here,
denotes an energy eigenket of a hydrogenlike
atom corresponding to the conventional quantum numbers,
,
, and
.
 The Hamiltonian of an electron in a hydrogenlike atom is written
where
. Let
denote a properly normalized energy eigenket belonging to the eigenvalue
. By calculating
, derive the ThomasReicheKuhn sum rule,
where
and
, and the sum is over all energy eigenstates. Here,
stands for either
,
, or
. (See Exercise 6.)

 Show that the only nonzero
electric dipole matrix elements for the
hydrogen atom
take the values
where
is the Bohr radius.
 Likewise, show that the only nonzero
electric dipole matrix elements
take the values
 Finally, show that the only nonzero
electric dipole matrix elements
take the values

 Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any
state to a
state
of a hydrogen atom is
where
is the fine structure constant.
Hence, deduce that the relative natural width of the associated spectral line is
where
denotes wavelength.
 Likewise, show that the spontaneous decay rate from any
state to a
state
is
Hence, deduce that the relative natural width of the associated spectral line is
 Demonstrate that the net oscillator strength for
transitions in a hydrogen atom is
irrespective of the polarization of the radiation.
 Likewise, show that the net oscillator strength for
transitions is
irrespective of the polarization of the radiation.
 Taking electron spin into account, the wavefunctions of the
states of a hydrogen
atom are written
where
. Here, the
are
standard radial wavefunctions, and the
are spinangular functions. Likewise, the
wavefunctions of the
states are written
, where
.
Finally, the wavefunctions of the
states are written
, where
,
,
,
. As a consequence of spinorbit interaction, the four
states lie
at a slightly higher energy than the two
states. Demonstrate that the spontaneous decay
rates, due to electric dipole transitions, between the various
and
states are
where
Here, the states have been labeled by their spinangular functions.
Hence, deduce that for an ensemble of hydrogen atoms in thermal equilibrium, the spectral line associated with
transitions is twice as bright as that associated with
transitions.
 Taking electron spin into account, the wavefunctions of the
states of a hydrogen
atom are written
where
. The wavefunctions of the
states are specified in the previous exercise.
Demonstrate that the spontaneous decay
rates, due to electric dipole transitions, between the various
and
states are
where
Here, the states have been labeled by their spinangular functions.
 Consider the
electric dipole transition of a hydrogen atom. Show that the angular distribution of spontaneously emitted photons is
where
, and the photon's direction of motion is parallel to
.
 Consider the
electric dipole transition of a hydrogen atom. Show that the angular distribution of spontaneously emitted photons is
 Hence, show that for an ensemble of hydrogen atoms in thermal equilibrium, the angular distribution of
spontaneously emitted photons associated with the
electric dipole transition is
 The properly normalized
triplet and singlet state kets of a hydrogen atom can be written
and
respectively. Here, the kets on the left are
kets, where
and
are the conventional
quantum numbers that determine the overall angular momentum, and the projection of the overall
angular momentum along the
axis, respectively. Finally,
and
are the
properly normalized spinup and spindown states for the proton and the electron, respectively.
As explained in Section 7.9, the triplet states have slightly higher energies than the
singlet state, as a consequence of spinspin coupling between the proton and the electron. Spontaneous
magnetic dipole transitions between the triplet and singlet states occur via the interaction of the
magnetic component of the emitted photon and the electron magnetic moment. (The magnetic moment
of the proton is much smaller than that of the electron, and consequently does not play a significant role
in this transition.) In the following, the initial state,
, corresponds to one of the triplet states, and the final
state,
, corresponds to the singlet state.
 Demonstrate that
where
and
are the energies of the initial and final states, and
is the proton
factor.
 Let
be the magnetic dipole matrix element, where
is the electron spin.
Show that
if the initial state is the
state,
if the initial state is the
state,
and
if the initial state is the
state.
 Deduce that the angular distribution of the spontaneously emitted photon is
if the initial state is either the
state or the
state.
Show that the angular distribution is
if the initial state is the
state. Here, the photon's direction of motion is
parallel to
.
 Finally, show that the overall spontaneous transition rate is
 Consider the
electric quadrupole transition of a hydrogen atom.
 Show that the angular distribution of spontaneously emitted photons can be written
 Demonstrate that the only nonzero
electric quadrupole matrix elements takes the values
,
, and
 Hence, deduce that
and
Here, the spontaneously emitted photon's direction of motion is
parallel to
.
 Let
where
is a nonnegative integer.
Demonstrate that
and
Hence, deduce that
and
 Treating
as a small parameter, and neglecting terms of order
, show that
 Repeat the calculation of Section 8.15 for the case where the electric polarization vector
of the incident photon is
. Hence, show that, in this case, the differential
photoionization crosssection is
Deduce that the photoionization crosssection for unpolarized electromagnetic radiation propagating in the
direction
is
 Consider the photoionization of a hydrogen atom in the
state. Demonstrate that formula (8.249)
is replaced by
 Consider the inverse process to the photoionization of a hydrogen atom investigated in Section 8.15.
In this process, which is known as radiative association, an unbound electron of energy
is captured by a (stationary) proton to form a hydrogen
atom in its ground state, with the emission of a photon of energy
where
. Here,
is (negative) hydrogen groundstate energy, and
is
the (positive) groundstate ionization energy. Radiative association can be thought of as a form of
stimulated emission in which the initial state is unbound. Thus, according to the analysis of Section 8.9,
where
is the electron momentum,
is the wavevector of the emitted photon, and
is the number of photon states whose angular frequencies lie between
and
, and whose
direction of motion lie in the range of solid angles
. Here,
are two independent
unit vectors normal to
.
 Show that
where the initial electron and final photon are both assumed to be contained in a periodic
box of volume
.
 Hence, demonstrate that
where
, and
.
 Finally, show that
where
.
Next: Identical Particles
Up: TimeDependent Perturbation Theory
Previous: PhotoIonization
Richard Fitzpatrick
20160122