Electron shell bombardment

Question

If you bombard an electron shell with a photon below the critical level to promote the electron to a higher state, will the shell absorb nothing and the photon get deflected with the same amount of frequency/energy that it came with initially?

Answer 1

The answer is yes. What you describe is the quantum mechanical picture behind the elastic Rayleigh scattering. For the actual calculation, see R. Loudon, The quantum theory of light, Chapter 8.8.