B.3.b. Number Representation For Many States
Let the complete set of 1particle
eigenstates be arranged in a sequence according to some rules so that each of
them can be referred to by a single integer
α=0,1,⋯,∞
. For example, one can arrange them in the
order of increasing energies so that
α=0
is the ground state. The case of continuous quantum numbers can be
treated either by replacing a with real numbers or replacing the quantum numbers with their
discrete approximations. Thus, for the
case of the momentum and spin eigenstates, a given value of a will
represent a state with quantum numbers
k
α
=(
k
α
,
σ
α
)
.
The basis states in the number representation are written as

n
0
,
n
1
,⋯,
n
α
,⋯,
n
∞
⟩
where
n
α
is the number of particles in the 1particle
state a. The canonical (generalized
coordinate and momentum) operators in the number representation are the
creation
a
α
+
and annihilation
a
α
operators with
(
a
α
)
†
=
a
α
+
,
where
†
is the adjoint (hermitian conjugate)
operator. Note that we have suppressed
the ^ on both
a
α
+
and
a
α
since they are always used as operators. By definition, all dynamical operators of the
system can be written as functions of
a
α
+
and
a
α
. For example,
n
ˆ
α
=
a
α
+
a
α
.