# Electron-Positron Annihilation Conservation Laws

• says
In summary: Two photons are created in electron-positron annihilation and this is why energy conservation is conserved.
says
What ensures conservation of energy and conservation of momentum in Electron-Positron annihilation?

If e=mc2 and momentum = mass*velocity, couldn't the energy equivalence of the electron and the positron be converted to make one photon, or more that 2 photons? I've read that two photons are created and this is why conservation of energy is conserved, and they got in opposite directions, which conserves momentum.

says said:
What ensures conservation of energy and conservation of momentum in Electron-Positron annihilation?
The laws of physics.
says said:
If e=mc2
That is true for particles at rest only.
says said:
momentum = mass*velocity
That is an approximation for nonrelativistic particles only.

One photon would violate energy-momentum conservation. More than 2 photons are possible, but rare.

mfb said:
The laws of physics.
Ahhhhhhh

mfb said:
That is true for particles at rest only.
Do particles not have energy-equivalence when they are in motion? Once the electron and positron collide couldn't we think of them at rest?

mfb said:
That is an approximation for nonrelativistic particles only.
One photon would violate energy-momentum conservation. More than 2 photons are possible, but rare.

Can't one photon have the combined energy of the electron and positron?

says said:
Can't one photon have the combined energy of the electron and positron?

If it did, it would break momentum conservation. It cannot simultaneously have the same momentum and energy as the electron-positron pair.

says
says said:
Do particles not have energy-equivalence when they are in motion? Once the electron and positron collide couldn't we think of them at rest?

You can go to the reference frame where the electron-positron system is at rest. They (individually) will have some momentum ##\vec{p}_-## for electron and ##\vec{p}_+## for positron. The Center-of-Mass frame is given by ##\vec{p}_- = - \vec{p}_+## (or the particles have equal and opposite momenta,so that the total momentum is zero). In that frame you have that the initial 4-momentum is:
$(p_+ + p_-)^\mu = \begin{pmatrix} 2E_e \\ \vec{0} \end{pmatrix}$
since the energies of positron/electron are equal (they have the same mass and the same momentum magnitude).
What happens for the photon in that frame? Well the conservation of energy (in that frame) would imply that the photon has energy ##2E_e## (so far so good)... what about the momentum? The momentum of the photon will have to be zero.
That is impossible for a real photon, since the real photon should have equal momentum magnitude and energy -so that $E^2 - |\vec{p}|^2= m_\gamma^2 =0$.

That's why 1 photon alone cannot conserve energy/momentum.

If you have 2 photons you can assign their momenta/energy so that they cancel out ...that's why each photon would have a four momentum $p_\mu^{(\gamma1)}= \begin{pmatrix}E_e \\ \vec{p}_\gamma \end{pmatrix}$ and $p_\mu^{(\gamma2)}= \begin{pmatrix}E_e \\ -\vec{p}_\gamma \end{pmatrix}$ and so $p^{\mu~(\gamma1)} + p^{\mu~(\gamma2)}= p_+^\mu + p_-^\mu$. So in the center of mass frame of the electron/positron the photons go in opposite directions.

In a similar manner you can have more than 2 photons, however at each photon emission you get one factor of $\alpha_{em}$ (fine-structure constant for electromagnetism), which suppresses the interaction by approximately a factor of 100.

Last edited:
says
says said:
Do particles not have energy-equivalence when they are in motion?
They have, but you need the more general expression$$E^2 = m^2 c^4 + p^2 c^2$$
Once the electron and positron collide couldn't we think of them at rest?
They can be, but the photon cannot.

## 1. What is electron-positron annihilation?

Electron-positron annihilation is a process in which an electron and a positron (the antiparticle of an electron) collide and are converted into energy in the form of photons.

## 2. How does the conservation of energy apply to electron-positron annihilation?

According to the law of conservation of energy, energy cannot be created or destroyed, only converted from one form to another. In the case of electron-positron annihilation, the combined mass of the electron and positron is converted into energy in the form of photons.

## 3. What is the conservation of momentum in electron-positron annihilation?

The law of conservation of momentum states that the total momentum of a system remains constant unless acted upon by an external force. In electron-positron annihilation, the net momentum before and after the collision must be equal, meaning the total momentum of the electron and positron must be equal to the total momentum of the resulting photons.

## 4. How does the conservation of charge apply to electron-positron annihilation?

The law of conservation of charge states that the total amount of electric charge in a closed system must remain constant. In electron-positron annihilation, both the electron and positron have an electric charge of -1 and +1 respectively, and the total charge before and after the collision must be equal.

## 5. What other conservation laws are applicable to electron-positron annihilation?

In addition to energy, momentum, and charge, other conservation laws that apply to electron-positron annihilation include conservation of mass-energy (E=mc^2), conservation of lepton number, and conservation of baryon number.

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