Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2014

Define the 4-momentum of a particle and describe briefly the principle of conservation of 4-momentum.

A photon of angular frequency ω\omega is absorbed by a particle of rest mass mm that is stationary in the laboratory frame of reference. The particle then splits into two equal particles, each of rest mass αm\mathrm{\alpha m}.

Find the maximum possible value of α\alpha as a function of μ=ω/mc2\mu=\hbar \omega / m c^{2}. Verify that as μ0\mu \rightarrow 0, this maximum value tends to 12\frac{1}{2}. For general μ\mu, show that when the maximum value of α\alpha is achieved, the resulting particles are each travelling at speed c/(1+μ1)c /\left(1+\mu^{-1}\right) in the laboratory frame.

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