Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 10C / b

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- Solution b

Solution

Momentum conservation makes the sum of the two photon momenta parallel to

u

, so all three vectors are coplanar. Put the photons on opposite sides of the incident direction. Transverse and longitudinal momentum and energy conservation give

E_{1} sin θ_{1} = E_{2} sin θ_{2},

(1)

E_{1} cos θ_{1} + E_{2} cos θ_{2} = γm u c, E_{1} + E_{2} = m c^{2} (γ + 1) .

(2)

Eliminating

E_{1}, E_{2}

yields

\frac{s i n ( θ _{1} + θ _{2} )}{s i n θ _{1} + s i n θ _{2}} = \frac{γ u / c}{γ + 1} = \frac{γ - 1}{γ + 1} .

(3)

Using the half-angle identities on the left gives exactly

\frac{1 + cos ( θ _{1} + θ _{2} )}{cos θ _{1} + cos θ _{2}} = \frac{γ - 1}{γ + 1} .

(4)

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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 10C / b

Solution

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