Consider an atom in the excited state radiating a photon and goes to the lower energy state. But photons have a certain angular momentum, the momentum itself is not defined. In this case, will there be a recoil of the atom due to photon emitted?
Answer
The momentum of a photon is not only defined, but is defined very well in the famous Einstein's quation:
$$ E = \sqrt {(mc^2)^2 + (pc)^2 } $$
that leads for massless photons to
$$p=\frac Ec =\frac h{\lambda}$$
Therefore, atoms recoil when emitting photons.
The opposite phenomena, an atom recoiling during photon absorption, is used by laser cooling near $0 \mathrm{K}$.
The laser frequency is set just below a chosen atomic absorption line. Due the Doppler effect, absorption occurs only for those atoms with a particular velocity component toward the laser.
Absorption of a photon and it's momentum decreases this velocity component, what means decreasing the atom kinetic energy. That leads in large scale to decreasing of temperature.
Effectively, the thermal energy is spent to be added to otherwise insufficient energy of photons.If the gained energy is released by emission of other photon, it has in average the nominal absorption line energy with negative net energy outcome.
As @dmckee has noted, recoilless scenario can be achieved in solid matrices, if the momentum is distributed within the whole solid matrix.
Mossbauer effect (mentioned in Laser cooling page)
The Mössbauer effect, or recoilless nuclear resonance fluorescence, is a physical phenomenon discovered by Rudolf Mössbauer in 1958. It involves the resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid. Its main application is in Mössbauer spectroscopy.
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