PHY4604
R. D. Field
Department of Physics
Chapter5_2.doc
University of Florida
The Exchange Operator
Two Identical Particles in a OneDimensional Box:
We see that for two
particles in a box
2
2
2
2
2
2
1
2
)
(
)
(
)
(
mL
E
E
E
β
α
π
αβ
+
=
+
=
h
where
α
= 1, 2, 3, .
..
and
β
= 1, 2, 3, .
..
.
and the wavefunctions are given by
)
/
sin(
)
/
sin(
2
)
(
)
(
)
,
(
2
1
2
1
2
1
L
x
L
x
L
x
x
x
x
βπ
απ
ψ
=
=
The Hamiltonian is
m
p
m
p
H
op
op
op
2
)
(
2
)
(
2
2
2
1
+
=
and is
symmetric under the
interchange of the two particles
(
as it must be since the two particles are
indistinguishable
).
Exchange Operator:
The exchange operator is the operator that
interchanges the two particles (
1
↔
2)
as follows
)
,
(
)
,
(
1
2
2
1
x
x
x
x
P
ex
=
with
1
2
=
ex
P
The eigenvalue of the
P
ex
operator are
+1
and
1
and the with eigenfunctions
given by
()
)
,
(
)
,
(
2
1
)
,
(
1
2
2
1
2
1
x
x
x
x
x
x
S
+
=
(
α
≠
β
symmetric under 1
↔
2
)
)
,
(
)
,
(
2
1
)
,
(
1
2
2
1
2
1
x
x
x
x
x
x
S
αα
+
=
(
α
=
β
symmetric under 1
↔
2
)
)
,
(
)
,
(
2
1
)
,
(
1
2
2
1
2
1
x
x
x
x
x
x
A
−
=
(
antisymmetric under 1
↔
2
)
For identical particles the Hamiltonian is invariant under
1
↔
2
and hence
[P
ex
,H
op
] = 0
and
P
ex
is a constant of the motion (
it is conserved
).
Also,
note that
)
,
(
)
,
(
)
,
(
2
1
1
2
2
1
x
x
x
x
x
x
P
ex
βα
=
=
Mixed State:
The following superposition is also a solution of
Schrödinger’s equation (for arbitrary angle
θ
):
)
,
(
sin
)
,
(
cos
)
,
(
2
1
2
1
2
1
x
x
x
x
x
x
S
A
mix
θψ
+
=
.
The
probability density
for finding one particle at
x
1
and the other at
x
2
is
( )
∗
+
+
=
S
A
S
A
mix
x
x
θ
ρ
Re
)
2
sin(


sin


cos
)
,
(
2
2
2
2
2
1
.
It appears that we have lost our ability to predict the probability density
since it depends on the arbitrary angle
θ
!