Specification of the state of a many particle system

Consider the time evolution of and .
This can be visualized by plotting
the point (, ) in the
- plane. This plane is generally known as *phase-space*. In general,
the point (, ) will trace out some very complicated pattern in
phase-space.
Suppose that we divide phase-space into rectangular cells of uniform dimensions
and .
Here, is the intrinsic error in the position measurement, and
the intrinsic error in the momentum measurement. The ``area'' of each cell
is

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Let us now consider a single spinless particle moving in three dimensions. In order
to specify the state of the system we now need to know three - pairs:
*i.e.*, -, -, and -. Incidentally,
the number of - pairs needed to specify the state of the
system is usually called the *number of degrees of freedom* of the
system. Thus, a single particle moving in one dimension constitutes a one
degree of freedom system, whereas a single particle moving
in three dimensions constitutes a three degree of freedom system.

Consider the time evolution of and , where
, *etc*.
This can be visualized by plotting
the point (, ) in the six dimensional
- phase-space.
Suppose that we divide the - plane
into rectangular cells of uniform dimensions
and , and do likewise for the - and
- planes. Here, and are again the intrinsic
errors in our measurements of position and momentum, respectively.
This is equivalent to dividing phase-space up into
regular six dimensional cells of volume .
The coordinates and can now be
conveniently specified by indicating
the cell in phase-space into which they plot at any given time.
Again, this procedure automatically ensures that we do not attempt to specify
and to an accuracy greater than our experimental error.

Finally, let us consider a system consisting of spinless particles moving
classically in three dimensions.
In order to specify
the state of the system, we need to specify a large number of - pairs.
The requisite
number is simply the number of degrees of freedom, .
For
the present case,
. Thus, phase-space (*i.e.*, the space of all the - pairs)
now possesses dimensions.
Consider a particular pair of conjugate coordinates, and
.
As before, we divide the -
plane into rectangular cells of uniform dimensions
and .
This is equivalent to dividing phase-space into regular dimensional cells of volume . The state of the
system is specified by indicating which cell it occupies in phase-space
at any given time.

In principle, we can specify the state of the system to arbitrary
accuracy by taking the limit
. In reality, we know from
quantum mechanics that it is impossible to simultaneously
measure a coordinate and
its conjugate momentum to greater accuracy than
.
This implies that

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In quantum mechanics we can specify the state of
the system by giving its wave-function at time ,

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In reality, this approach does not work because the Hamiltonian of the system is
only
known approximately. Typically, we are dealing with a system consisting of many
*weakly interacting* particles. We usually know the Hamiltonian for completely
non-interacting particles, but the component of the Hamiltonian associated
with particle interactions is either impossibly complicated, or not very well
known (often, it is both!). We can define approximate stationary eigenstates
using the Hamiltonian for non-interacting particles. The state of the system is
then specified by the quantum numbers identifying these eigenstates. In the absence
of particle interactions, if the system starts off
in a stationary state then it
stays in that state for ever, so its quantum numbers never change. The interactions
allow the system to make transitions between different ``stationary''
states, causing its quantum numbers to change in time.