A photon with wavelength λis incident on a stationary particle with mass M, as shown in Fig. 𝐏 3

step 1 This problem involves pair production of an electron position pair. So for part A, we have energ

A photon with wavelength λis incident on a stationary...

A photon with wavelength $\lambda$ is incident on a stationary particle with mass $M,$ as shown in Fig. $\mathbf{P 3 8} .39 .$ The photon is annihilated while an electron-positron pair is produced. The target particle moves off in the original direction of the photon with speed $V_{M}$. The electron travels with speed $v$ at angle $\phi$ with respect to that direction. Owing to momentum conservation, the positron has the same speed as the electron. (a) Using the notation $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$ and $\gamma_{M}=\left(1-V_{M}^{2} / c^{2}\right)^{-1 / 2},$ write a relativistic expression for energy conservation in this process. Use the symbol $m$ for the electron mass. (b) Write an analogous expression for momentum conservation.

Key Concepts

Pair Production in Relativistic Collisions

Pair production is a quantum phenomenon in which a photon converts into a particle-antiparticle pair, such as an electron and a positron, typically in the presence of another particle to conserve momentum. In relativistic collisions, this process involves balancing the energy and momentum of the initial photon and the stationary particle with that of the resulting pair and any recoiling target, illustrating the interplay between quantum mechanics and special relativity.

Photon Energy and Momentum

Photons, although massless, carry energy and momentum, with their energy given by E = hc/? and momentum by p = E/c. These relations are fundamental in relativistic interactions involving light and matter, where a photon can be absorbed or annihilated, and its energy can be converted into the mass and kinetic energy of other particles, provided that conservation laws are satisfied.

Relativistic Momentum Conservation

In the realm of special relativity, momentum is defined as the product of mass, velocity, and the Lorentz factor, making it a vector quantity that must also be conserved in all processes. When particles interact or transform, the total vector sum of their momenta before the interaction must equal the total vector sum after the reaction. This is especially critical in processes where particles change directions, requiring careful accounting of momentum components.

Relativistic Energy Conservation

This principle states that the total energy, including both rest mass energy and kinetic energy (modified by the relativistic Lorentz factor), remains constant before and after an interaction. In high-energy processes, the energy of particles is typically expressed using gamma factors that account for relativistic speeds, and in reactions like particle creation, the sum of rest mass energies plus kinetic energies of all products equals the incoming energy, such as that from a photon or another particle.

Lorentz Factor

The Lorentz factor, denoted by gamma (?), is a key parameter in special relativity defined as ? = 1/?(1 - v²/c²), where v is the particle's speed and c is the speed of light. It quantifies how time, length, and relativistic mass change for objects moving at speeds close to the speed of light, and it is essential for accurately expressing both energy and momentum in high-speed processes.

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