The formal part of this course deals almost exclusively with systems of one particle. While we have not yet discussed all the the physics of multiple particle systems, the general considerations we have laid out so far are sufficient for us to formulate and give a valid proof the virial theorem in the general case of more than one particle.
Within our framework, a systems of N particles is described by the
coordinates of the particles in the system 
.
Measurements of the positions of the particles in the system will
lead to a distribution of results, where some probability
function 
describes the probability of finding a
particle in the small volume 
around 
and another
particle in the volume 
about 
and so forth for all
of N particles in the system. For a given quantum state 
of
the system, this probability is given by the square of a quantum
probability amplitude 
.
The wavefunction of the 
pure state of energy will be an
eigenstate of the Hamiltonian (total energy) operator,
where the Hamiltonian 
is the sum of the total kinetic and
potential energy operators,
There is only one other fact about multiple particle systems, the
Pauli Exclusion Principle, which puts additional constraints on the
beyond those imposed by the Time Independent
Schrödinger Equation (25). We will discuss this
principle in a later note. All that we will need for now to
demonstrate the virial theorem is the knowledge that 
satisfies condition (25).
