What does it mean for an atom to be formed?

Question

Confusion arised out of reading this answer, it says

You can immediately describe the 2 particles by their center of mass description (an atom) plus their individual attributes (i.e. what the particles are doing within the atom). Assuming the 2 particles start off at rest, then they are in a bound state already because they can't escape each other (go off to infinite separation) due to lack of energy.

It would seem to me that, that we could, for instance, we could pair any proton in the universe with an electorn anywhere else and call that as an atom. This is at least consistent with Bohr Theory as we can take radius as large as we want.

So, would a question asking "if a proton and electron can combine" to form an atom be non-sensical? or, is my understanding of what an "atom" means wrong?

Answer 1

You've missed a key phrase "assuming the two particles start off at rest" most electrons are not moving at the same velocity as most protons in the universe.

You cannot make two particles be at rest by choosing a particular reference frame.

Take a very slight approximation where we assume the center of mass is determined by the proton. Then we choose the center of mass frame, and the proton is not moving. So we now consider the electron's velocity with respect to the proton. If $$\frac{m_ev^2}{2}-\frac{1}{4\pi\epsilon_0}\frac{e^2}{r}>0$$ Then the electron has posive energy and is not bound to the proton.

If you consider also quantum mechanics, it becomes more dire. If the electron is localized so it has some uncertainty in its velocity this is another way for the electron to be unbound. If it's position uncertainty is $\delta_x$, then $\hbar/2\delta_x$ is roughly its momentum uncertainty, and $(\hbar/2\delta_x)^2/(2m)$ is roughly its kinetic energy expectation value. So even if $\langle v\rangle=0$, the electron may not be bound by the proton if it is reasonably localized.

Not to mention the issue that if there is lots of other stuff between the electron and the proton, one cannot approximate the system as only a force between an electron and a proton. The proton may not even be relevant for determining the motion of the electron.