Material absorbs light power, which makes the material all excited. Problem is, material wants to relax, so it “works off” the extra power by letting the extra power back out as light.
More complex: It’s all to do with electron energy states. Phosphorescent materials (like Zinc Sulphide) have multiple energy states that have a lot to do with the electron energy itself. Remember, things want to be in the lowest stable energy. By adding energy of a particular wavelength (For example, most “green” glow in the dark materials need to be charged by blue or higher light) we can push those electrons up higher. When they drop back down, the energy MUST be released somehow, and in this case it’s released as a photon of light. Coincidentally, the requirement for energy is why if you shine a red laser on a “green” glow object it doesn’t stay glowing anywhere near as long.
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