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Does an atom moving towards the light that is falling on it, absorb more energy than when it is stationary?

The atom should a see blue shift due to Doppler effect so the atom will see a light of greater frequency, causing it to absorb a greater amount of energy (or, equivalently, it will absorb less energy when it is moving away from the light due to red shift). Or will it absorb the same energy in all the cases? Does the atoms motion have any affect on the energy it absorbs?

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Yes it does have an effect as you correctly reasoned and is even used in cold atoms technology where it is known as the Doppler cooling technique.

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Yes it does. In a famous example, the Doppler effect was used to cancel out the gravitational redshift of high-energy photons in the Pound-Rebka experiment.

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