Identical Particles Revisited
Michael Fowler
Introduction
For two identical particles confined to a onedimensional box, we established earlier that the
normalized twoparticle wavefunction
( )
12
,
x
x
ψ
, which gives the probability of finding
simultaneously one particle in an infinitesimal length
dx
1
at
x
1
and another in
dx
2
at
x
2
as
()
2
,
x
xd
x
d
x
, only makes sense if
2
21
,
2
,
x
xx
ψψ
=
x
, since we don’t know which of
the two indistinguishable particles we are finding where.
It follows from this that there are two
possible wave function symmetries:
( ) ( )
,,
x
x
=
or
(
)(
,
)
,
x
x
=−
.
It turns
out that if two identical particles have a symmetric wave function in some state, particles of that
type always have symmetric wave functions, and are called
bosons
.
(If in some other state they
had an antisymmetric wave function, then a linear superposition of those states would be neither
symmetric nor antisymmetric, and so could not satisfy
2
,
2
,
x
=
x
)
,
.)
Similarly,
particles having antisymmetric wave functions are called
fermions
.
(Actually, we could in
principle have
(
,
i
x
xe
x
x
α
=
, with
α
a constant phase, but then we wouldn’t get back
to the original wave function on exchanging the particles twice.
Some twodimensional theories
used to describe the quantum Hall effect do in fact have excitations of this kind, called
anyons
,
but all ordinary particles are bosons or fermions.)
To construct wave functions for three or more fermions, we assume first that the fermions do not
interact with each other, and are confined by a spinindependent potential, such as the Coulomb
field of a nucleus.
The Hamiltonian will then be symmetric in the fermion variables,
( ) ( ) ( )
222
123
1
2
3
/2
H p mp mp m
Vr Vr Vr
=+++
+
+
+
+
GGG
G
G
G
……
and the solutions of the Schrödinger equation are products of eigenfunctions of the single
particle Hamiltonian
( )
2
Hp mV
r
=+
GG
,
.
However, these products, for example
() ( ) ()
ab
c
do not have the required antisymmetry property.
Here
a
,
b
,
c
, … label the
singleparticle eigenstates, and 1, 2, 3, … denote both space and spin coordinates of single
particles, so 1 stands for
(
.
The necessary antisymmetrization for the particles 1, 2 is
achieved by subtracting the same product wave function with the particles 1 and 2 interchanged,
so
)
11
,
rs
G
c
is replaced by
( ) ( ) ( ) ( ) ( ) ( )
213
c
a
bc
−
, ignoring overall
normalization for now.
But of course the wave function needs to be antisymmetrized with respect to
all
possible particle
exchanges, so for 3 particles we must add together all 3! permutations of 1, 2, 3 in the state
a
,
b
,
c
, with a factor

1 for each particle exchange necessary to get to a particular ordering from the
original ordering of 1 in
a
, 2 in
b
, and 3 in
c
.
In fact, such a sum over permutations is precisely
the
definition
of the determinant, so, with the appropriate normalization factor: