Chapter 4
MANY PARTICLE SYSTEMS
The postulates of quantum mechanics outlined in previous chapters include no restrictions
as to the kind of systems to which they are intended to apply. Thus, although we have
considered numerous examples drawn from the quantum mechanics of a single particle,
the postulates themselves are intended to apply to all quantum systems, including those
containing more than one and possibly very many particles.
Thus, the only real obstacle to our immediate application of the postulates to a system
of many (possibly interacting) particles is that we have till now avoided the question
of what the linear vector space, the state vector, and the operators of a manyparticle
quantum mechanical system look like. The construction of such a space turns out to be
fairly straightforward, but it involves the forming a certain kind of methematical product
of di¤erent linear vector spaces, referred to as a
direct
or
tensor product
. Indeed, the
basic principle underlying the construction of the state spaces of manyparticle quantum
mechanical systems can be succinctly stated as follows:
The state vector
j
Ã
i
of a system of
N
particles is an element of the direct
product space
S
(
N
)
=
S
(1)

S
(2)

¢ ¢ ¢

S
(
N
)
formed from the
N
singleparticle spaces associated with each particle.
To understand this principle we need to explore the structure of such direct prod
uct spaces. This exploration forms the focus of the next section, after which we will return
to the subject of many particle quantum mechanical systems.
4.1
The Direct Product of Linear Vector Spaces
Let
S
1
and
S
2
be two independent quantum mechanical state spaces, of dimension
N
1
and
N
2
, respectively (either or both of which may be in…nite). Each space might represent that
of a single particle, or they may be more complicated spaces, each associated with a few
or many particles, but it is assumed that the degrees of mechanical freedom represented
by one space are independent of those represented by the other. We distinguish states in
each space by superscripts. Thus, e.g.,
j
Ã
i
(1)
represents a state in
S
1
and
j
Á
i
(2)
a state of
S
2
. To describe the combined system we now de…ne a new vector space
S
12
=
S
1

S
2
(4.1)
of dimension
N
12
=
N
1
£
N
2
which we refer to as the direct or tensor product of
S
1
and
S
2
. Some of the elements of
S
12
are referred to as direct or tensor product
states
, and are
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124
Many Particle Systems
formed as a direct product of states from each space. In other words, from each pair of
states
j
Ã
i
(1)
2
S
1
and
j
Á
i
(2)
2
S
2
we can construct an element
j
Ã; Á
i ´ j
Ã
i
(1)

j
Á
i
(2)
=
j
Ã
i
(1)
j
Á
i
(2)
2
S
12
(4.2)
of
S
12
;
in which, as we have indicated, a simple juxtaposition of elements de…nes the tensor
product state when there is no possibility of ambiguous interpretation.
By de…nition,
then, the state
j
Ã; Á
i
represents that state of the combined system in which subsystem
1
is de…nitely in state
j
Ã
i
(1)
and subsystem
2
is in state
j
Á
i
(2)
:
The linear vector space
S
12
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 Fall '10
 PaulE.
 mechanics, Vector Space, Hilbert space, direct product, Many Particle Systems

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