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What happens if a photon incident on an atom has greater energy than the difference of energies between successive energy levels corresponding to electronic excitation(say between ground state and next energy level) but smaller for multi energy-level hop? Will the photon be absorbed or not?

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If the photon energy coincides with the energy difference between two levels, and the initial level of that pair is populated, then that photon will excite the transition. If it doesn't, it won't.

This includes some amount of 'smudging':

  • The photon energy will typically span some nonzero bandwidth $\Delta\omega$, which is limited by the Heisenberg Uncertainty Principle (HUP) to be no smaller than $2\pi/T$, for $T$ the total duration of the experiment.

  • The electronic transitions will also have some nonzero width, which can come from multi-atom aspects ('inhomogeneous broadening' effects, like thermal Doppler shifts, which move the transition energy of each atom in the sample slightly up or down, by different amounts) and from intrinsic effects ('homogeneous broadening', also known as the 'natural linewidth', a HUP effect coming from the fact that transitions between non-ground-state effects cannot be held for unlimited times, since all excited states decay to the ground state after enough time).

If there is an overlap between these two bandwidths, the transition will happen. If there is no meaningful overlap, then no population transfers will take place.

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