General properties of many-body systems

Definition of individual system

So far we have studied the properties of single-body systems in prescribed external potentials. An individual physical system comprises a wave and a particle, both evolving in three-dimensional Euclidean space. Of course, isolating a system and representing the influence of other matter by an ‘external potential’ is an approximation to the real state of affairs in that we neglect the source of the potential. As a fundamental theory of matter quantum mechanics should apply to a closed many-body system (and ultimately to the universe as a whole) and reduce to a theory of systems of a few degrees of freedom as a special case under conditions where it is legitimate to neglect the ‘rest of the universe’. The extension of the quantum theory of motion to many-body systems is straightforward, although it displays some striking features not evident in the one-body case. To begin with, we define an individual n-body system as comprising:

1. (a) A wavefunction ψ = ψ(x1, …, xn, t) defined in a 3n-dimensional configuration space in which xl, …, xn provide a set of rectangular Cartesian coordinates.

2. (b) A set of n point particles pursuing trajectories xi(t) i = 1, …, n, in three-dimensional Euclidean space. A single configuration space trajectory is equivalent to n particle trajectories in Euclidean space.

When we speak of a ‘many-body system’ we mean then a single wavefunction together with a set of particles.