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I believe the reason that this equation is negative is because all free electrons (when they are not near something) have a value of zero joules and because bound electrons have a lower energy then free electrons the bound electrons have to be less than zero (or negative).
The negative in this equation is due to the idea of the answer being energy lost. This is also the same amount of energy that a photon has in this situation. A photon in the experiment would have the same amount of energy as the energy lost from the electron.
Dr. Lavelle answered this during lecture 3, which he said that if you start at a high energy level and you keep going down the different levels, energy is being lost until you get to n=0 where the electron is separated completely. So the negative stems from the loss of energy that occurs as you go from high energy levels to low energy levels.
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