Distance between positron and electron

[Einstein's Cat]

Is there a way in determining the distance between an electron and a positron as they anhilliate? If this figure has been determined already, what is it? Thank you very much for your help!

 

[mfb]

Things like a distance (as single value) are not meaningful in quantum mechanics.

 

[jfizzix]

Is there a way in determining the distance between an electron and a positron as they annihilate? If this figure has been determined already, what is it? Thank you very much for your help!

Experimentally, the best you can hope for is to find out what the (average) initial positions of the two gamma ray photons released as the total product of the annihilation, and use that as a ballpark estimate of how far apart the electron and positron were (on average) just before annihilation.

Theoretically, the mean distance between the electron and positron before annihilation will be no less than twice the Bohr radius (though possibly more depending on the quantum state of the electron-positron pair).

To see how this works, consider the following:
Positronium is a bound state of an electron and a positron. They are two opposite charges orbiting their collective center of mass, just like the electron and proton in a hydrogen atom.
If we just consider them as a pair of opposite charges (and forget annihilation for the time being), we can calculate their joint measurement statistics from the Schrödinger equation just like they can be for the hydrogen atom.
The minimum average distance between the electron and positron will be when the positronium atom is in its ground state, just like the electron will have a minimum average distance to the proton in the corresponding state of a hydrogen atom.

Unlike atoms of hydrogen, positronium is unstable, and in a rather short time (on the order of tenths of nanoseconds), the positronium atom will decay usually into a pair of gamma ray photons.
That being said, those bound states still exist before annihilation, so you can calculate the mean distance between the electron and positron in such a system.

Electrons and positrons can annihilate from higher energy states, which will be from on average farther away, but such annihilations are less likely, and have a longer lifetime before decay. Since positronium can self-annihilate, no matter what state it's in, any distance is possible, but the most likely distances are of the order of tenths of nanometers.

 

[Einstein's Cat]

Experimentally, the best you can hope for is to find out what the (average) initial positions of the two gamma ray photons released as the total product of the annihilation, and use that as a ballpark estimate of how far apart the electron and positron were (on average) just before annihilation.

Theoretically, the mean distance between the electron and positron before annihilation will be no less than twice the Bohr radius (though possibly more depending on the quantum state of the electron-positron pair).

To see how this works, consider the following:
Positronium is a bound state of an electron and a positron. They are two opposite charges orbiting their collective center of mass, just like the electron and proton in a hydrogen atom.
If we just consider them as a pair of opposite charges (and forget annihilation for the time being), we can calculate their joint measurement statistics from the Schrödinger equation just like they can be for the hydrogen atom.
The minimum average distance between the electron and positron will be when the positronium atom is in its ground state, just like the electron will have a minimum average distance to the proton in the corresponding state of a hydrogen atom.

Unlike atoms of hydrogen, positronium is unstable, and in a rather short time (on the order of tenths of nanoseconds), the positronium atom will decay usually into a pair of gamma ray photons.
That being said, those bound states still exist before annihilation, so you can calculate the mean distance between the electron and positron in such a system.

Electrons and positrons can annihilate from higher energy states, which will be from on average farther away, but such annihilations are less likely, and have a longer lifetime before decay. Since positronium can self-annihilate, no matter what state it's in, any distance is possible, but the most likely distances are of the order of tenths of nanometers.

Thank you very much for your help!

 

[ontodva]

That's tantalising, and I never considered it before. I just supposed they would come into contact to annihilate. Do they actually annihilate from a distance, or does one tunnel into the other, or something?

 

[jfizzix]

I'd like to hear what a proper particle physicist has to say on the subject, but I think it's not so much contact, but rather that the closer the particles are, the more likely they are to annihilate, which would be a quantum transition from one set of particles to another, not unlike when a photon is absorbed by an atom.