A Short Course on Quantum Mechanics and Methods of Quantization 11This construction can be easily generalized to higher dimensions, i.e. toH=
L^2 (Rn), by defining two sets of operators ˆxjand ˆpj(j=1, 2 ,···,n), such that:
[ˆxj,ˆpk]=ıδjkIand [ˆxj,ˆxk]=[ˆpj,ˆpk]=0.
Example 1.2.3. Fermionic (or two-level) systems.
In the finite-dimensional case, all self-adjoint operators are bounded and observ-
ables are represented byn×nHermitean matrices: O=Mn. For a two-level
system^7 , a basis of all Hermitean operators is given by the identityIand the set
of the three Pauli matricesσα,(α=1, 2 ,3):
I=(
10
01
)
,σ 1 =(
01
10
)
,σ 2 =(
0 −ı
ı 0)
,σ 3 =(
10
0 − 1
)
. (1.22)
It is not difficult to see that the operatorsSα=σα/2 yields the fundamental
representation (i.e. spin 1/2) of theSU(2) algebra since:
[Sα,Sβ]=i αβγSγ. (1.23)The canonical vectors
|+〉=(
1
0
)
,|−〉=
(
0
1
)
(1.24)
are the eigenstates of theS 3 operator, with eigenvalues±/2. It is convenient to
define the two ladder operators:
σ±=σ 1 ±iσ 2 , (1.25)satisfying the algebra commutators:
[
σ+,σ−
]
=σz,[
σz,σ±]
=2σ± (1.26)and the anti-commutation relations:
(
σ+
) 2
=
(
σ−) 2
=0,{σ+,σ−}=I. (1.27)From the latter, it follows immediately thatHis generated by the two states:
| 0 〉≡|−〉and| 1 〉≡σ+| 0 〉=|+〉, whileσ+| 1 〉= 0 as well asσ−| 0 〉=0.
The operatorsσ±satisfying (1.27) are calledfermionic creation/annihilation
operators, and| 0 〉is interpreted as the vacuum state.
We can also define thenumber operator: N ≡σ+σ−, for which| 0 〉,| 1 〉are
eigenvectors with eigenvalues 0,1 respectively. This represents an algebraic way of
encoding Pauli exclusion principle, which states that two fermions cannot occupy
the same state.
(^7) A two-level system is what is a called aqubitin the context of quantum information theory
[ 32 ].