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 Demonstrate that the operators defined in Equations (5.11)(5.13) are Hermitian, and
satisfy
the commutation relations (5.1).
 Prove the BakerCampbellHausdorff lemma, (5.31).
 Let the
, for
, be the three spin
Pauli matrices. Demonstrate that
, and, hence, that
and
.
 Find the Pauli representations of the normalized eigenstates of (a)
and (b)
for
a spin
particle.
 Suppose that a spin
particle
has a spin vector that lies in the

plane, making an
angle
with the
axis. Demonstrate that a measurement of
yields
with probability
, and
with probability
.
 An electron is in the spinstate
in the Pauli representation. (a) Determine the constant
by normalizing
. (b) If a measurement of
is made, what values will be
obtained, and with what probabilities? What is the expectation
value of
? Repeat the previous calculations for (c)
and (d)
. [61]
 Consider a spin
system represented by the normalized spinor
in the Pauli representation, where
and
are real. What is the probability that a measurement of
yields
? [53]
 An electron is at rest in an oscillating magnetic field
where
and
are real positive constants.
 Find the Hamiltonian of the system.
 If the electron starts in the spinup state with respect to the
axis, determine the spinor
that represents the state
of the system in the Pauli representation at all subsequent times.
 Find the probability that a measurement of
yields
the result
as a function of time.
 What is the minimum value of
required to force a
complete flip in
?
 In the Schrödinger/Pauli representation, the generalization of Schrödinger's timedependent
wave equation for an electron moving in electromagnetic fields is written
where
the
vector potential,
the magnetic fieldstrength,
the scalar potential, and
the spinorwavefunction. The term involving the
Pauli matrices comes from the electron's intrinsic magnetic moment. (See Section 5.5.)
Demonstrate that this equation can also be written
The previous expression is known as the Pauli equation [80].
 Let
,
, and
be the Pauli matrices for a spin
particle.
 Show that
where
 Show that
where
 Show that
where
 Hence, deduce that the spinor matrices for rotations through an
angle
about the three Cartesian axes are
where
and
.
 Suppose that a spin
particle
has a spin vector that lies in the

plane, making an
angle
with the
axis. Demonstrate that a measurement of
yields
, 0
, and
with probabilities
,
,
and
, respectively.
Next: Addition of Angular Momentum
Up: Spin Angular Momentum
Previous: Spin Greater Than OneHalf
Richard Fitzpatrick
20160122