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The equation is (usually?) used for an electron losing energy and consequently emitting a photon. Because the electron loses energy, its energy is "negative" while the energy for the photon is positive. Dr. Lavelle also said in lecture that the negative sign means that the bound electron has lower energy than the free electron, in the case of an electron popping off the atom. I could use some more clarification on that as well.
The negative sign is there because a bound electron has lower energy than a free electron. When e- is completely removed, E=0. This equation makes 0 a reference point, and indicates that when the e- gets closer to nucleus/lower in orbitals, the e-'s energy is becoming more negative (or in other words it is losing energy).
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