
164 Commutation Relations, Normal Ordering, and Stirling Numbers
where {C, D} = CD + DC denotes the anticommutator.Furthermore,ifwedenoteby
ˆn
F
=
ˆ
f
†
ˆ
f the fermionic number operator, then one has [ˆn
F
,
ˆ
f
†
]=
ˆ
f
†
as well as [ˆn
F
,
ˆ
f]=−
ˆ
f.
Note that the above relations imply, in particular, (
ˆ
f
†
)
2
= 0. If we denote the ground state –
defined by
ˆ
f|0 =0–by|0, then we generate all other states by the action of
ˆ
f
†
. However,
due to (
ˆ
f
†
)
2
= 0 the state |1 is the only nontrivial state! Thus, the fermionic Fock space is
only two-dimensional (hence, isomorphic to C
2
). The operators
ˆ
f
†
and
ˆ
f can, therefore, be
represented by complex 2 ×2 matrices.
Remark 5.23 The ene ...