each occurring at time t with probability density 
.
In order to simplify the discussion in the following calculations, let 
, 
,
and write the distribution function in conjugate coordinates as 
.
Note that 
is positive semidefinite, and nonzero only on the surface 
.
In anticipation of a quantum Liouville equation, define a flow on the space 
by 
.
In a manner analogous to the derivation of the classical Liouville equation
of section 5
we may derive the quantum Liouville equation in terms of the conjugate
coordinates. To be technically precise, we must go through the algebra
using the coordinates 
,
and the result of doing this is
and the complex coordinates, it has the same form as that equation.
Given ,
the expected value of an operator Q is given by
where we note that we depend on the fact that the probability density
is nonzero only on the normalized surface of 's
to connect
and 
.
Equation 7.5
leads immediately to the coordinate-independent or invariant representation
of the expectation value by (note that the operator 
,
where 
)
where
is the density operator.
We move from the equation of motion of the probability density of states
to the equation of motion of the density operator. Take the equation of
motion of the probability density of states, equation 7.4,
multiply it on the left by ,
multiply it on the right by 
,
and then integrate over
and .
The term involving the partial time derivative of the density of states
is trivial to find, but the other terms involve more effort to work out.
The partial time derivative term is
while the other terms are found using the identity
and integrating by parts, noting that
is nonzero only on the normalized surface of 's.
Doing the algebra and substituting yields the coordinate-independent equation
of motion of the density operator
where 
is the operator commutator. (It is perhaps easiest to demonstrate this
result by showing that the ijth elements of the operators are equal
for all i,j). This relationship can also be derived by doing
a direct differentiation with respect to time of 
as defined in equation 7.9
on the one hand, and then on the other, noting that equation 7.4
shows that the probability density at a point moving with the Schrodinger
flow is constant, transforming to time zero coordinates, and doing the
time derivative on that form. The derivation done here shows how the coordinate-dependent
and coordinate-independent descriptions of the motion interrelate, demonstrates
how to begin with the coordinate independent quantum Liouville equation
and then derive the coordinate independent quantum Liouville equation,
and prepares the reader for the quantum reduced density equations of motion,
directly analogous to the classical BBGKY equations of section 5.2.