Laser cooling is a modern technique to cool atoms to extremely low temperature. The key point is to tune the laser frequency slightly lower than the nominal frequency of an electronic transition in the atom. Derive the Doppler shift using Lorentz transformation in one dimension (8%). Prom the above derivation, explain why the atom can absorb the laser light (2%). Describe the phenomenon of spontaneous emission (3%). Explicitly show that after the spontaneous emission, the atom is cooled in the Lab frame (10%). What is the condition that the atom can keep its original energy (2%)?