##

Two Particles in Three Dimensions

The generalization of the Hamiltonian to three dimensions is simple.

We define the vector **difference between the coordinates** of the particles.
We also define the vector **position of the center of mass**.
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**

and the total mass
we get.
The Hamiltonian actually **separates into two problems**:
the motion of the **center of mass** as a free particle

and the **interaction between the two particles**.
This is exactly the same separation that we would make in classical physics.

Jim Branson
2013-04-22