A γray creates an electron-positron pair. Show directly...

A $\gamma$ ray creates an electron-positron pair. Show directly that, without the presence of a third body to take up some of the momentum, energy and momentum cannot both be conserved. (Hint: Set the energies equal and show that this leads to unequal momenta before and after the interaction.)

Key Concepts

Pair Production

Pair production is the process by which a high-energy photon converts its energy into a particle-antiparticle pair, such as an electron and a positron. This process requires the energy of the photon to exceed the combined rest mass energies of the produced particles. Importantly, pair production cannot occur in a vacuum because additional particles or bodies are required to conserve momentum during the process.

Conservation of Energy

The conservation of energy is a fundamental principle stating that energy cannot be created or destroyed, only transformed. In any physical process, including pair production, the total energy before and after the event must remain constant. This principle plays a crucial role in determining whether a given reaction can occur, as the energy of the initial photon must match the sum of the rest mass energies and kinetic energies of the electron and positron produced.

Conservation of Momentum

The conservation of momentum is another fundamental law asserting that the total momentum of a closed system remains constant over time. In reactions like pair production, momentum conservation requires that the momentum carried by the photon before the interaction is exactly balanced by the vector sum of the momenta of the produced particles. Without a third body to absorb any excess momentum, it is mathematically impossible to satisfy momentum conservation along with energy conservation simultaneously for the pair production process.

Relativistic Energy-Momentum Relationship

The relativistic energy-momentum relationship, given by the equation E^2 = (pc)^2 + (mc^2)^2, describes how the energy, momentum, and rest mass of a particle are interconnected in the framework of special relativity. This relationship imposes strict conditions on the values of energy and momentum for particles created in high-energy interactions. In the context of pair production, this formula highlights the incompatibility between energy and momentum conservation if only a photon and two particles are involved, thereby necessitating the involvement of an additional body to balance the equations.

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