Prove that a single photon (isolated from any other...

Prove that a single photon (isolated from any other particles) cannot convert spontaneously into an $\mathrm{e}^{+} \mathrm{e}^{-}$ pair: $$ \gamma \longrightarrow \mathrm{e}^{+}+\mathrm{e}^{-} $$ [Hint: Use the Pythagorean relation ( $2.23$ ) to relate the energy and momentum on each side of this reaction, and then show that the reaction cannot conserve both energy and momentum. This result explains the need for the nucleus in Eq. (18.1); the recoil of the nucleus allows energy and momentum to be conserved.]

Prove that a single photon (isolated from any other particles) cannot convert spontaneously into an $\mathrm{e}^{+} \mathrm{e}^{-}$ pair:
$$
\gamma \longrightarrow \mathrm{e}^{+}+\mathrm{e}^{-}
$$
[Hint: Use the Pythagorean relation ( $2.23$ ) to relate the energy and momentum on each side of this reaction, and then show that the reaction cannot conserve both energy and momentum. This result explains the need for the nucleus in Eq. (18.1); the recoil of the nucleus allows energy and momentum to be conserved.]

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Key Concepts

Conservation Laws

In all physical interactions, energy and momentum must be conserved simultaneously. In relativistic particle processes, this means that the sum of the energy and momentum before the interaction must equal the sum after the interaction, placing strict requirements on the kinematics of the particles involved.

Relativistic Energy-Momentum Relation

The energy and momentum of a particle are related by the equation E² = (mc²)² + (pc)², which connects a particle's rest mass, energy, and momentum in a relativistic framework. For a photon, which is massless, this simplifies to E = pc, while for particles with mass, the rest energy plays a critical role in determining the allowed energy and momentum values.

Role of an External Body in Pair Production

Pair production, where a photon converts into an electron-positron pair, cannot satisfy both energy and momentum conservation on its own. An additional body, such as a nucleus, is needed to absorb the excess momentum through recoil. This external interaction permits the overall conservation laws to hold, enabling the process to occur within the framework of relativistic physics.

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