How close do electrons and positrons have to be to interact?

[Cato]

Electrons and positrons are assumed to be point masses. Two points presumably can never actually touch. How close do they have to be before they annihilate each other?

 

[Quantum9999]

When an electron and a positron collide an annihilation eventoccurs and gamma rays are produced. This is a frequent occurrence in PET scanning in hospitals.

The positron and electrons have opposite charges and so the overall charge before annihilation is zero. The resulting gamma rays have no charge. So charge is conserved in this collision.

In annihilation, the positron and electron collide head on moving at the same speed. The overall momentum is therefore zero. The resulting gamma rays move in opposite directions with equal and opposite momentum. So momentum is also conserved.

Einstein’s famous equation E = mc2 means that the mass of an object is a measure of its energy content, and that mass and energy can be converted into each other (mass energy).

In annihilation, the masses of both the positron and electron are converted into energy (gamma rays). The energy of the gamma rays is the same as the mass energy of the original positron and electron and so mass energy is also conserved.

 

[PeterDonis]

Electrons and positrons are assumed to be point masses.

Not really. The effective size of fundamental particles depends on the situation. In some situations there is no single number that describes their size.

How close do they have to be before they annihilate each other?

There is no single answer to this question because this is one of the situations where no single number describes the size of the particles. The effective cross section for electron-positron annihilation depends on the energies of the particles.

 

[PeterDonis]

In annihilation, the positron and electron collide head on moving at the same speed.

This is an approximate classical description of the process, which is good enough for, e.g., understanding how PET scanners work, but leaves out a lot.

 

[A. Neumaier]

How close do they have to be before they annihilate each other?

The standard scattering matrix elements give the probability density that an electron and a positron in a plane wave (thus delocalized with precise momentum but completely uncertain position) annihilate. This translates into a decay rate for streams of colliding pairs in two beams. For other states (strictly speaking also for beams - using the paraxial approximation) one has to take the appropriate superposition of momentum states to calculate the probability. Since there are no normalizable position states, there will always be some nonlocality (lack of precise position).