# Why do we only consider relative motion of hydrogenic atoms?

In my lectures on quantum mechanics, it was said that the Hamiltonian for a hydrogenic atom could be split into relative motion and translational motion, as follows:

$$\hat{H} = - \frac{\hbar^2}{2m}\nabla^2_{cm} - \frac{\hbar^2}{2\mu}\nabla^2 - \frac{Ze^2}{4\pi\epsilon_0r}$$

and that one only need consider the relative motion:

$$\hat{H} = - \frac{\hbar^2}{2\mu}\nabla^2 - \frac{Ze^2}{4\pi\epsilon_0r}$$

This statement is reiterated in Atkins' Physical Chemistry (10th Edition), as well as his Molecular Quantum Mechanics (4th Edition), and various other online resources, but I am yet to find an explanation as to why.

Why can we, and why do we, neglect the translational motion of the atom?

• I understand separability but the author goes on to write $- \frac{\hbar^2}{2\mu}\nabla^2 - \frac{Ze^2}{4\pi\epsilon_0r} = E\Psi$, declaring it the Schrödinger equation for a hydrogenic atom. What is the physical intuition behind seemingly totally discarding the translational motion? (i.e. how can you disregard translational motion and still set the result equal to $E$?) – Jacob Nov 14 '17 at 12:11
• This is $E$ of relative motion. Translational motion is associated with its own kinetic energy; it can be anything, so why bother. – Ivan Neretin Nov 14 '17 at 12:37