and have dimensionN+1. The corresponding quantum systems are said to have
“spinN/2”. The caseN= 0 is the trivial representation onCand the case
N= 1 is the case of this chapter. In the limitN→ ∞one can make contact
with classical notions of spinning objects and angular momentum, but the spin
1
2 case is at the other limit, where the behavior is purely quantum-mechanical.
3.1 The two-state quantum system
3.1.1 The Pauli matrices: observables of the two-state
quantum system
For a quantum system with two dimensional state spaceH=C^2 , observables are
self-adjoint linear operators onC^2. With respect to a chosen basis ofC^2 , these
are 2 by 2 complex matricesMsatisfying the conditionM=M†(M†is the
conjugate transpose ofM). Any such matrix will be a (real) linear combination
of four matrices:
M=c 01 +c 1 σ 1 +c 2 σ 2 +c 3 σ 3
withcj∈Rand the standard choice of basis elements given by
1 =
(
1 0
0 1
)
, σ 1 =(
0 1
1 0
)
, σ 2 =(
0 −i
i 0)
, σ 3 =(
1 0
0 − 1
)
where theσjare called the “Pauli matrices”. This choice of basis is a convention,
with one aspect of this convention that of taking the basis element in the 3-
direction to be diagonal. In common physical situations and conventions, the
third direction is the distinguished “up-down” direction in space, so often chosen
when a distinguished direction inR^3 is needed.
Recall that the basic principle of how measurements are supposed to work
in quantum theory says that the only states that have well-defined values for
these four observables are the eigenvectors for these matrices, where the value
is the eigenvalue, real since the operator is self-adjoint. The first matrix gives
a trivial observable (the identity on every state), whereas the last one,σ 3 , has
the two eigenvectors
σ 3(
1
0
)
=
(
1
0
)
and
σ 3(
0
1
)
=−
(
0
1
)
with eigenvalues +1 and−1. In quantum information theory, where this is
the qubit system, these two eigenstates are labeled| 0 〉and| 1 〉because of the
analogy with a classical bit of information. When we get to the theory of spin in
chapter 7, we will see that the observable^12 σ 3 corresponds (in a non-trivial way)
to the action of the groupSO(2) =U(1) of rotations about the third spatial