So far we have been working with states of just one particle in one dimension.
extension to two different particles and to three dimensions
The coordinates and momenta of different particles and of the additional dimensions
commute with each other as we might expect from classical physics.
The only things that don't commute are a coordinate with its momentum, for example,
We may write states for two particles which are uncorrelated,
or we may write states in which the particles are correlated.
The Hamiltonian for two particles in 3 dimensions simply becomes
If two particles interact with each other, with no external potential,
the Hamiltonian has a translational symmetry,
and remains invariant under the translation
We can show that this
implies conservation of total momentum.
Similarly, we will show that rotational symmetry implies conservation of
and that time symmetry implies conservation of energy.
For two particles interacting through a potential that depends only on difference on the coordinates,
we can make the usual
transformation to the center of mass
made in classical mechanics
and reduce the problem to the CM moving like a free particle
plus one potential problem in 3 dimensions with the usual reduced mass.
So we are now left with a 3D problem to solve (3 variables instead of 6).