Quantum Mechanics for Mathematicians

The energies of the energy eigenstates| 0 〉and| 1 〉will then be±^12 since

H| 0 〉=−

1

2

| 0 〉, H| 1 〉=

1

2

| 1 〉

Note that the quantum system we have constructed here is nothing but our
old friend the two-state system of chapter 3. Taking complex linear combinations
of the operators
aF,a†F,NF, 1

we get all linear transformations ofHF=C^2 (so this is an irreducible repre-
sentation of the algebra of these operators). The relation to the Pauli matrices
is

a†F=

1

2

(σ 1 +iσ 2 ), aF=

1

2

(σ 1 −iσ 2 ), H=

1

2

σ 3

27.2 Multiple degrees of freedom

For the case ofddegrees of freedom, one has this variant of the canonical
commutation relations (CCR) amongst the bosonic annihilation and creation
operatorsaBjandaB†j:

Definition(Canonical anticommutation relations).A set of 2 doperators

aFj, aF†j, j= 1,...,d

is said to satisfy the canonical anticommutation relations (CAR) when one has

[aFj,aF†k]+=δjk 1 , [aFj,aFk]+= 0, [aF†j,aF†k]+= 0

In this case one may choose as the state space the tensor product ofNcopies
of the single fermionic oscillator state space

HF= (C^2 )⊗d=C︸^2 ⊗C^2 ︷︷⊗···⊗C^2 ︸

dtimes

The dimension ofHFwill be 2d. On this space an explicit construction of the
operatorsaFjandaF†jin terms of Pauli matrices is

aFj=σ 3 ⊗σ 3 ⊗···⊗σ 3

︸ ︷︷ ︸

j−1 times

⊗

(

0 0

1 0

)

⊗ 1 ⊗···⊗ 1

aF†j=σ 3 ⊗σ 3 ⊗···⊗σ 3

︸ ︷︷ ︸

j−1 times

⊗

(

0 1

0 0

)

⊗ 1 ⊗···⊗ 1

The factors ofσ 3 are there as one possible way to ensure that

[aFj,aFk]+= [aF†j,aF†k]+= [aFj,aF†k]+= 0