# How does the collision between two atoms work?

Considering the quantum mechanical model for an atom, what exactly happens when two atoms (say, two Ca2+ ions in a Brownian motion) collide with each other? As I know, this collision is not like a regular elastic or inelastic collision between two macroscopic objects. Is it mainly due to the coulombic repulsion between the electrons of the two atoms? And, how is the trajectory of the two atoms after collision determined, and what factors contribute to it? Are these trajectories and the angle by which the atoms get deviated deterministic, or fuzzy just as the atoms themselves?

• Hint: there's an entire subfield of quantum mechanics called scattering theory. – Emilio Pisanty Apr 21 '19 at 10:35

One can make a simplistic quantum model of the two atoms by treating them as point particles with appropriately fuzzy, but as high-information as possible, positions and momenta directed toward each other. You do not need relativistic quantum field theory in this case - though you might need at least a crude version if you want to also include photon emission, namely a coupled EM field, but not a full QFT for the electrons and such, just the EM, because if this is meant to simulate a chemical process, this is suitably low-energy that we aren't creating or destroying any known massive particles.

And you are right in your hunch: The collision will be, as you say, fuzzy. The available information in position falls steadily with time as the probability distributions broaden during approach and then even more in collision - if one starts out broader than the other it will spill over the other on each side. Moreover, the positions after collision will be entangled, or correlated: you cannot simply and totally faithfully write down two separate wave functions

$$\psi_\mathbf{r}(\mathbf{r}_\mbox{atom 1})$$

and

$$\psi_\mathbf{r}(\mathbf{r}_\mbox{atom 2})$$

to describe their fuzzy positions independently after collision. Instead, you need a six-dimensional wave function

$$\psi_{\mathbf{r}_1, \mathbf{r}_2}(\mathbf{r}_\mbox{atom 1}, \mathbf{r}_\mbox{atom 2})$$

for both and so can't also entirely faithfully picture the result. Nonetheless, if you finally then query the system at a suitably late time as to the outgoing directions of the atoms (thus increasing the information), your result will be quite random with the usual randomness of quantum queries.

There is a little animation that attempts to visualize something like this here: