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The wave model of electrons properly predicts energy levels if electrons are modeled as circular standing waves around the nucleus. This is because all waves--circular or not--have differing energies. However, for a circular standing wave to exist, it must be continuous and have no discontinuities so that it can form a circle. Only certain energies of waves allow for the circular wave to be continuous and have no discontinuities. Most other wave energies would result in a circular wave that is not continuous. Thus, the specific energies which allow for circular standing waves to exist are the "energy levels" of electrons.
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