Forgotten energy in interactions of particles with opposite charge.

[KJW]

On 10/11/2023 at 10:38 AM, martillo said:

positron-electron pair creation. In the energies' balances are only considered the rest energies of the particles and not the energy between the two particles separated apart.

It appears to me that you are referring to the energy of the electromagnetic field surrounding the positron-electron pair. The Wikipedia article "Electromagnetic mass" may interest you. It doesn't entirely answer your question because it is about a classical notion. But the notion that the mass of an electron includes the mass equivalent of the energy of the surrounding electromagnetic field is an intriguing one and is connected to the modern notion of renormalization.

[martillo]

No, it is about the Electric Potential Energy between the electron and the positron which depends in their distance: kq₁q₂/r.

It does exist but the problem is that the distance r at which the annihilation takes place is unknown and the Heisenberg's uncertainty makes it uncertain so actually we cannot determine it. What I observe is that the energy tends to infinite as the distance r tends to zero.

Just for instance, if the annihilation distance would be about the Compton-wavelength of the electron (2.42631x10^-12 m) then such energy would be about 6x10^-17 Joules = 375 ev (electron-volts) what would be about a thousandth of the mass-energy of the electron (mc² approximately 500 Kev ) and would be negligible. 

Actually the annihilation distance and the associated Electric Potential Energy are unknown. Nowhere is mentioned about them.

Edited November 14, 2023 by martillo

[swansont]

51 minutes ago, martillo said:

No, it is about the Electric Potential Energy between the electron and the positron which depends in their distance: kq₁q₂/r.

It does exist but the problem is that the distance r at which the annihilation takes place is unknown and the Heisenberg's uncertainty makes it uncertain so actually we cannot determine it. What I observe is that the energy tends to infinite as the distance r tends to zero.

The positronium ground state energy is known, which allows for some knowledge of the potential energy.

[martillo]

6 hours ago, swansont said:

The positronium ground state energy is known, which allows for some knowledge of the potential energy.

From Wikipedia: "An electron and positron orbiting around their common centre of mass. An s state has zero angular momentum, so orbiting around each other would mean going straight at each other until the pair of particles is either scattered or annihilated, whichever occurs first. This is a bound quantum state known as positronium."

I think that for "ground state" you mean the possible configuration (considering even their spins) of minimum energy between them. They would stay orbiting for some time and they do not annihilate at this state. The state is unstable and the attractive force between the electron and the positron accelerates them towards each other until the annihilation takes place at some distance less than the orbiting radius.

Wikipedia article says the orbiting radius and binding energy can be roughly estimated through an analogy to the hydrogen atom but this is not the binding energy precisely at the annihilation which is what I'm referring to. The annihilation distance is less than the orbiting radius and the binding energy at the annihilation is higher than that while orbiting.

The binding energy increments as the distance diminishes and they both seems to remain still unknown at the final state of the pair real annihilation.

 

Edited November 14, 2023 by martillo

[KJW]

11 hours ago, martillo said:

No, it is about the Electric Potential Energy between the electron and the positron which depends in their distance: kq₁q₂/r.

The force between two charges corresponds to the increase or decrease in the total energy of the surrounding electromagnetic field as the distance between the charges is changed. I think what happens to the electromagnetic field of the electron and positron when they annihilate is quite relevant.

[swansont]

4 hours ago, martillo said:

From Wikipedia: "An electron and positron orbiting around their common centre of mass. An s state has zero angular momentum, so orbiting around each other would mean going straight at each other until the pair of particles is either scattered or annihilated, whichever occurs first. This is a bound quantum state known as positronium."

I think that for "ground state" you mean the possible configuration (considering even their spins) of minimum energy between them. They would stay orbiting for some time and they do not annihilate at this state. The state is unstable and the attractive force between the electron and the positron accelerates them towards each other until the annihilation takes place at some distance less than the orbiting radius.

Wikipedia article says the orbiting radius and binding energy can be roughly estimated through an analogy to the hydrogen atom but this is not the binding energy precisely at the annihilation which is what I'm referring to. The annihilation distance is less than the orbiting radius and the binding energy at the annihilation is higher than that while orbiting.

The binding energy increments as the distance diminishes and they both seems to remain still unknown at the final state of the pair real annihilation.

 

Energy is conserved, so the total energy of the system remains constant. Any decrease in PE would show up as an increase in KE. There is no change in the binding energy. The minimum energy of the system at annihilation is the ground state energy of positronium.

[martillo]

7 hours ago, swansont said:

Energy is conserved, so the total energy of the system remains constant. Any decrease in PE would show up as an increase in KE. There is no change in the binding energy. The minimum energy of the system at annihilation is the ground state energy of positronium.

Right. The total energy (PE plus KE) is then conserved and would be negligible in the case of the positronium.

I have made some calculations giving a total energy of about few 7.2 ev much less than the relativistic energy of the pair (2xmc²) about 1 Mev.

 

 

 

Edited November 14, 2023 by martillo