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 Demonstrate directly from the fundamental commutation relations for angular momentum, (4.11), that
(a)
, (b)
, and (c)
.
 Demonstrate from Equations (4.74)(4.79) that
where
,
are conventional spherical angles. In addition, show that
 A system is in the state
. Evaluate
,
,
, and
.
 Derive Equations (4.108) and (4.109) from Equation (4.107).
 Find the eigenvalues and eigenfunctions (in terms of the angles
and
) of
.
Express the
eigenfunctions in terms of the spherical harmonics.
 Consider a beam of particles with
. A measurement of
yields the result
. What
values will be obtained by a subsequent measurement of
, and with what probabilities? Repeat
the calculation for the cases in which the measurement of
yields the results 0
and
.
 The Hamiltonian for an axially symmetric rotator is given by
where
and
are the moments of inertia about the
axis (which corresponds to the symmetry axis), and about an axis
lying in the

plane, respectively.
What are the eigenvalues of
? [53]
 The expectation value of
in any stationary state is a constant.
Calculate
for a Hamiltonian of the form
Hence, show that
in a stationary state. This is another form of the Virial theorem. (See Exercise 9.)
[53]
 Use the Virial theorem of the previous exercise to prove that
for an energy eigenstate of the hydrogen atom whose principal quantum number is
.
 Suppose that a particle's Hamiltonian is
Show that
and
. [Hint: Use the Schrödinger representation.] Hence,
deduce that
[Hint: Use the Heisenberg picture.]
Demonstrate that if
, where
, then
 Let
where
is a nonnegative integer.
Show that
 Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
 Demonstrate that
for the hydrogen ground state. In addition, show that
 Show that the most probable value of
in the hydrogen ground state is
.
 Demonstrate that
where
denotes a properly normalized energy eigenket of the hydrogen atom
corresponding to the standard quantum numbers
,
, and
.
 Let
denote the expectation value of
for an energy
eigenstate of the hydrogen atom characterized by the standard quantum numbers
,
, and
.
 Demonstrate that
where
and
is a wellbehaved solution of the differential equation
 Integrating by parts, show that
and
as well as
 Demonstrate from the governing differential equation for
that
 Combine the final result of part (b) with the governing differential equation to prove that
 Combine the results of parts (c) and (d) to show that
Hence, derive Kramers' relation:
 Use Kramers' relation to prove that
 Let
, where
is a properly normalized radial hydrogen wavefunction corresponding to the conventional
quantum numbers
and
, and
is the Bohr radius.
 Demonstrate that
 Show that
in the limit
.
 Demonstrate that
 Hence, deduce that
for
.
Next: Spin Angular Momentum
Up: Orbital Angular Momentum
Previous: Energy Levels of Hydrogen
Richard Fitzpatrick
20160122