in 2 particles system, the Hamiltonian is formed by 2 Hilbert spaces, such that:
however, it can also be expressed in center of mass – Hilbert space and relative -Hilbert space:
since this is just another factorization, the 2 spaces are still independence. In practice, we have to define position vector for center of mass and relative position:
the corresponding momentum are:
thus, the example Hamiltonian becomes:
since the total momentum is just a translation motion of the system, we can safely omit it in most cases and this reduced the 2 body system into 1 body system with fixed center.