XI. Identical Particles
a. The Product Basis
We have already dealt with multiple particles implicitly. For example,
when we were doing inelastic scattering calculations, our basis states
involved specifying the state of the electron
and
the molecule,
n
;
ψ
.
Although we did not stress it at the time, these basis states are
equivalent to
products
: a state for the electron times a state for the
molecule:
n
n
ψ
ψ
=
;
To verify this, we act on the right hand state with the zeroth order
Hamiltonian:
(
)
(
)
(
)
n
n
n
n
n
H
k
ψ
ω
ψ
ω
ψ
2
1
2
2
1
2
2
1
0
2
ˆ
ˆ
+
+
=
+
+
∇

=
and we see that these are, indeed, eigenfunctions of
0
ˆ
H
as
advertised.
This is a general rule: one can conveniently build a many
particle wavefunction that describes the state of particles A, B, C,
D,… by considering a product of states for A, B, C, D,…
individually:
...
...
D
C
B
A
D
C
B
A
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
=
≡
Ψ
where from here on out we will use capitol Greek letters to denote
many particle states and lower case Greek letters when representing
oneparticle states. The particles A, B, C, D,… can be
any
set of
distinct particles (e.g. A=H
2
O, B=He+, C= H
2
O+, D=He,…).
Then
A
ψ
would be a wavefunction that represents the state of the water
molecule,
B
ψ
would represent the state of the He+, etc.
What do these product states mean? To look at this question, lets
assume the Hamiltonian can be decomposed into a Hamiltonian that
acts just on A plus one for B plus one for C …:
...
ˆ
ˆ
ˆ
ˆ
ˆ
+
+
+
+
=
D
C
B
A
h
h
h
h
H
Many Particle
ψ
Single Particle
ψ
’s
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This Hamiltonian describes independent particles (because it lacks
coupling terms
AB
h
ˆ
that depend on A and B simultaneously). Now, the
eigenstates of this independent particle Hamiltonian can always be
chosen to be product states
, for which:
(
)
(
)
Ψ
=
+
+
+
+
=
+
+
+
+
=
Ψ
E
e
e
e
e
h
h
h
h
H
D
C
B
A
D
C
B
A
D
C
B
A
D
C
B
A
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
...
...
ˆ
ˆ
ˆ
ˆ
ˆ
.
Thus, product states describe particles that are independent of, or
uncorrelated with, one another.
One consequence of this fact is that
a measurement on A will not affect the state of B and vice versa. For
this reason, product states are also sometimes called “independent
particle” states, which is appropriate since they describe the state of
each particle
independently
.
Note that just because product states happen to describe many
particles, it does not follow that
every
many particle wavefunction can
be written in this form.
In fact, it is easy to build a many particle
wavefunction that is
not
of product form. For example:
D
C
B
A
D
C
B
A
c
c
2
2
2
2
2
1
1
1
1
1
2
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
+
≡
Ψ
cannot be written as a product of one particle wavefunctions. In
particular, for this state, a measurement on A influences the outcome
of measurements on B, C and D: if we find that A is in state 1, B,C
and D
must collapse to
state 1, while A in state 2 implies B, C and D
will collapse to state 2.
This strange correlation between different
particles is called
entanglement
and plays a very important role in
manyparticle quantum mechanics. Meanwhile, the state
(
)
(
)
(
)
(
)
D
D
C
C
B
B
A
A
c
c
c
c
c
c
c
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 Spring '04
 RobertField
 Electron, Mole, Quantum Field Theory, Pauli exclusion principle, Fermion, Boson

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