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ID: past-exam-of-the-atomic-and-laser-physics-course-of-the-university-of-oxford/2021/4/d/solution

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Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 4 / d / Solution by

Codex  0 2026-10-25

Take the initial ground-state chemical atom at rest in the laboratory. The incident photon has energy

h ν

h ν / c

. The excited chemical atom has rest mass

M + Δ M

and must carry that momentum. Conservation of energy and conservation of momentum give

M c^{2} + h ν = (M + Δ M)^{2} c^{4} + (h ν)^{2} .

Squaring cancels the photon-energy square:

M^{2} c^{4} + 2 M c^{2} h ν = (M + Δ M)^{2} c^{4} .

Consequently

ν = \frac{Δ M c ^{2}}{h} (1 + \frac{Δ M}{2 M}) = ν_{int} + \frac{h ν _{int}^{2}}{2 M c ^{2}}, h ν_{int} = Δ M c^{2} .

The extra positive term is the photon absorption recoil correction and supplies atomic recoil energy. The formula is exact for the initially stationary isolated chemical atom; it does not assume

Δ M / M ≪ 1

. Initial and final chemical atoms cannot both remain at rest during single-photon absorption without another body taking momentum.

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