The generalization of the Hamiltonian to three dimensions is simple.
We will use the chain rule to transform our Hamiltonian.
As a simple example, if we were working in one dimension we might
use the chain rule like this.
In three dimensions we would have.
Putting this into the Hamiltonian we get
Defining the reduced mass
The Hamiltonian actually separates into two problems:
the motion of the center of mass as a free particle
This is exactly the same separation that we would make in classical physics.
Jim Branson 2013-04-22