Show that the creation of an electron-positron pair (or any...

Name: Problem 47, Chapter 2: Modern Physics
Uploaded: 2021-11-08T19:04:34-08:00
Duration: 42 s
Channel: Keshav Singh
Description: Show that the creation of an electron-positron pair (or any particle-antiparticle pair, for that matter) by a single photon is not possible in isolation, that is, that additional mass (or radiation) must be present. (Hint: Use the conservation laws.)

Show that the creation of an electron-positron pair (or any particle-antiparticle pair, for that matter) by a single photon is not possible in isolation, that is, that additional mass (or radiation) must be present. (Hint: Use the conservation laws.)

Key Concepts

Conservation_of_Energy

This principle states that the total energy before and after any physical interaction must remain constant. In the context of particle interactions, the energy carried by the incoming photon must equal the combined energy (including the rest mass energy and kinetic energy) of the created pair, ensuring that energy is neither created nor destroyed in the process.

Conservation_of_Momentum

Momentum must be conserved in any isolated system, meaning the total momentum of the photon must be equal to the sum of the momenta of the electron and positron after the interaction. Since a photon has momentum related to its energy (p = E/c) and particles with mass have rest frames, satisfying momentum conservation without an additional body to absorb excess momentum is problematic.

Photon_Momentum_Energy_Relationship

For photons, energy and momentum are intrinsically linked by the relation E = pc, where c is the speed of light. This relationship poses a challenge in pair production because a photon, being massless, can only have energy and momentum values that satisfy this linear relation, making it impossible, on its own, to produce two massive particles that need to share this energy and momentum without violating conservation laws.

Lorentz_Invariance_and_Four_Momentum

The concept of four-momentum in special relativity combines energy and momentum into a single four-vector. This invariant quantity must be conserved in all inertial frames of reference. During the pair production process, achieving conservation of the four-momentum with only a single photon is impossible, as there is no inertial frame where the photon can have a rest mass equivalent, implying that an additional body (like a nucleus) is required to absorb the momentum imbalance.

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