and
(σ 1 −iσ 2 )(
1
0
)
= 2
(
0
1
)
(σ 1 −iσ 2 )(
0
1
)
=
(
0
0
)
(σ 1 +iσ 2 ) is called a “raising operator”: on eigenvectors ofσ 3 it either
increases the eigenvalue by 2, or annihilates the vector. (σ 1 −iσ 2 ) is called
a “lowering operator”: on eigenvectors ofσ 3 it either decreases the eigenvalue
by 2, or annihilates the vector. Note that these linear combinations are not
self-adjoint and are not observables, (σ 1 +iσ 2 ) is the adjoint of (σ 1 −iσ 2 ) and
vice-versa.
3.1.2 Exponentials of Pauli matrices: unitary transforma-
tions of the two-state system
We saw in chapter 2 that in theU(1) case, knowing the observable operatorQon
Hdetermined the representation ofU(1), with the representation matrices found
by exponentiatingiθQ. Here we will find the representation corresponding to
the two-state system observables by exponentiating the observables in a similar
way.
Taking the identity matrix first, multiplication byiθand exponentiation
gives the diagonal unitary matrix
eiθ^1 =(
eiθ 0
0 eiθ)
This is exactly the case studied in chapter 2, for aU(1) group acting onH=C^2 ,
with
Q=(
1 0
0 1
)
This matrix commutes with any other 2 by 2 matrix, so we can treat its action
onHindependently of the action of theσj.
Turning to the other three basis elements of the space of observables, the
Pauli matrices, it turns out that since all theσjsatisfyσ^2 j= 1 , their exponentials
also take a simple form.
eiθσj= 1 +iθσj+1
2
(iθ)^2 σ^2 j+1
3!
(iθ)^3 σj^3 +···= 1 +iθσj−1
2
θ^21 −i1
3!
θ^3 σj+···= (1−1
2!
θ^2 +···) 1 +i(θ−1
3!
θ^3 +···)σj= (cosθ) 1 +iσj(sinθ) (3.1)Asθgoes fromθ= 0 toθ= 2π, this exponential traces out a circle in the
space of unitary 2 by 2 matrices, starting and ending at the unit matrix. This
circle is a group, isomorphic toU(1). So, we have found three differentU(1)