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Zatman
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Homework Statement
An atom of rest mass M in an excited state of energy Q0 (measured in its rest frame) above the ground state moves towards a counter with speed v. The atom decays to its ground state by emitting a photon of energy Q (as recorded by the counter), coming completely to rest as it does so. Show that
[itex]Q=Q_0[1+\frac{Q_0}{2Mc^2}][/itex]
Homework Equations
[itex]E^2=p^2c^2 + m_0^2c^4[/itex]
[itex]E = \gamma m_0c^2[/itex]
[itex]\gamma = (1-\frac{v^2}{c^2})^{-\frac{1}{2}}[/itex]
The Attempt at a Solution
In the lab frame, the atom is initially moving with speed v. When the photon is emitted, the atom is at rest. Hence using conservation of energy:
[itex]\gamma Q_0 = Mc^2 + Q[/itex]
By conservation of momentum, the photon's momentum must equal the initial momentum of the atom. This gives:
[itex]\gamma Mv = \frac{Q}{c}[/itex]
There is sufficient information here to eliminate v and γ (since γ depends on v). However, when I do so, I get nothing close to the required answer. So I guess my interpretation of the context is wrong. Please could I have a hint?