## B. Systems of Identical Particles

Consider a system of *N* identical particles with hamiltonian

H
N
=
∑
i=1
N
p
i
2
2m
+
∑
i<j=1
N(
N−1
)/2
V(
q
i
s
i
,
q
j
s
j
)
(B.1)

Indistinguishability of the particles means
that
H
N
and all measurable quantities are invariant
under permutation of the particle labels.
The eigenstates of
H
N
then must be representations of the
permutation group. According to the
spin-statistics theorem in quantum field theory, particles with integral spins
belong to the symmetric representation and are called bosons. Particles with half-integral spins belong to
the anti-symmetric representation and are called fermions.

B.1.
Position and Momentum
Eigenstates

B.2.
Symmetrized *N*-Particle Position and Momentum
Eigenstates

B.3.
The Number Representation