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},

(39)

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

(40)

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} .

(41)

Using the half-angle identities on the left gives exactly

\frac{1 + c o s ( θ _{1} + θ _{2} )}{c o s θ _{1} + c o s θ _{2}} = \frac{γ - 1}{γ + 1} .

(42)

Solved by gpt-5.6-sol high.

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2023
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