How do electrons and positrons collide if both have undefined position

[Sam M]

Homework Statement

Hi, this isn't a homework question but one that I think has bothered not just me...

According to most sources out there an electron and a positron annihilate when they 'collide' or 'encounter' each other, however we also know that the position of an electron or positron is undefined, and the probability of the electrons and positrons having exactly the same position is zero ∫(ψ^2) dx between x and x is zero, they can only be found within a certain non zero distance of each other. So, in terms of the wave functions of the electron and positron, what constitutes a collision? do the ψ^2 graphs have to line up exactly? or does it have something to do with the overlap of those graphs?

I understand this answer may be somewhat over my head!

Homework Equations

we know the wave function ψ for both the electron and the positron, and I suppose we may assume that we know x values between which both integrals for ψ^2 are normalized.

The Attempt at a Solution

given how they tend to annihilate when in close proximity I am making a blind guess that the probability of annihilation is the area of overlap of the two ψ^2 graphs, (one for the electron and one for the positron), this area decreases as they move further apart, before becoming zero at infinite distance away. This probability becomes 1 as the ψ^2 functions become perfectly aligned. this consistency check is not mathematically rigorous, but it is the best my feeble mind can come up with. A definitive answer would be much appreciated.

 

[ehild]

Hi Sam, welcome to PF.

You do not know the exact position of an electron or a positron, but you can make beams of both kinds of particles and direct them against each other. The probability of a positron and electron in the beams getting close has quite high probability. You are right, the wavefunction of the individual electron and that of the individual positron have to overlap so as they can interact. But they have the Coulomb attraction which is a long-distance interaction. Positron and electron attract each other and start to move towards each other. They make a system and that system can be similar as a hydrogen atom.The particles orbit about their common centre of mass, as if a single particle of reduced mass me/2 orbit around a centre. Such an "atom" is called positronium. It is not stable, it decays with the creation of photons. Read more here http://en.wikipedia.org/wiki/Positronium

ehild