Exploring Matter-Antimatter Annihilation: Distance and Probability Factors

[Funestis]

Hello all. I had some questions on some of the specifics of matter-antimatter annihilation. I've tried looking this up but haven't had much success. If you guys know of any textbooks or journal articles that dig deep into the mechanics I'd be grateful if you'd post them.

Anyway, my basic question is does anyone know how close a particle and antiparticle have to be to one another for annihilation to occur? Does that distance depend on the specific particles involved? I'm assuming that there's some probabilities involved as well (e.g. if a particle/antiparticle pair are x meters apart they have a 10% chance of annihilation, if they are x - y meters apart they have a 50% chance of annihilation, etc.) but I'm uncertain. Any info you guys have would be much appreciated.

~thanks

 

[mfb]

Anyway, my basic question is does anyone know how close a particle and antiparticle have to be to one another for annihilation to occur?

So close that quantum mechanics makes that concept meaningless. Their waves functions have to overlap.

 

[Funestis]

So close that quantum mechanics makes that concept meaningless. Their waves functions have to overlap.

I figured that much, but how much overlap is there? If there's only a slight amount of overlap is annihilation absolutely guaranteed? What about protonium - I have no idea what the distance is which the proton and antiproton orbit one another, but given that a protonium atom can exist for a short time is it reasonable to ask how long annihilation takes? As I understand it, proton-antiproton annihilation occurs via the strong force, so is it a valid assumption then to say that an upper bound on the distance between particles before an annihilation event could occur would be the distance over which the strong force dominates (1 femtometer)? What about electron-positron annihilation?

 

[BvU]

does anyone know how close a particle and antiparticle have to be to one another for annihilation to occur

In high energy physics we have the notion of cross section for a given process such as $\ \ e^+e^-\rightarrow X\ \$ to help in imagining the transition probability as a kind of collision. Much better than trying to get your head around overlapping wave functions. See e.g. particle data group

 

[arivero]

I figured that much, but how much overlap is there?

In some formulae for quark-antiquark aniquilation inside a meson, they use the expresion $\Psi(0)$, which seem to imply that the two quarks must be in the same point. But again, given that both particles are spreaded along the wavefunction, I guess we should integrate across all the possible points.

 

[mfb]

If there's only a slight amount of overlap is annihilation absolutely guaranteed?

No.

but given that a protonium atom can exist for a short time is it reasonable to ask how long annihilation takes?

It does not take time. In any time frame you consider, it either happens or does not happen (to a very good approximation, as protons are not elementary particles).

As I understand it, proton-antiproton annihilation occurs via the strong force, so is it a valid assumption then to say that an upper bound on the distance between particles before an annihilation event could occur would be the distance over which the strong force dominates (1 femtometer)?

No, the wave function can be spread out much more, so "distance" becomes meaningless earlier.

 

[snorkack]

The probability of wave functions of two particles overlapping is exactly zero whenever they possesses a nonzero orbital momentum relative to each other.

Is annihilation of particles with orbital angular momentum completely and unconditionally impossible, i. e. particles always have to reach a state with zero angular momentum separately, before they can annihilate?

 

[ChrisVer]

The probability of wave functions of two particles overlapping is exactly zero whenever they possesses a nonzero orbital momentum relative to each other.

Why is that?

 

[mfb]

Imagine a measurement of both positions at the same time. If the two particles are at the same place, no matter what their (not well-defined) velocity is, how can they have orbital angular momentum?
I'm sure there is some higher order effect taking this orbital angular momentum with a virtual photon or whatever, but that doesn't look like a frequent process. States with orbital angular momentum are not the ground state, they can decay quickly.

 

[arivero]

well, in electromagnetic transitions in the hydrogen atom of course the orbitals are orthogonal, but we do not project a wavefunction over the other, we have also a interaction factor, the photon field in this case, between both.

 

[ZealScience]

For phenomenological descriptions, this is a scattering problem and in general characterised by a scattering cross-section, which is a Lorentz invariant quantity. The notion of distance is not as useful, since it is not invariant, so there is no 'universal distance' of scattering. I think any decent particle physics textbook contains this kind of information.