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The Quanta
Before scientists could develop quantum atomic model of an atom, though, they had to find the right math. The mathematics that they were searching for came from a surprising source: a German physicist named Max Planck. Planck was the first to use the quanta, which he more or less stumbled on to in 1900. The quanta is the smallest amount of energy possible that can be emitted or absorbed by matter. The cause of the quanta comes from Bohr's model of a staircase of atoms. When an atom absorbs energy, it's electrons become excited and move up an energy level or two. The electrons can't absorb one and a half energy level's worth of energy, though. Conversely, the electrons can't fall one and a half energy levels and give off that much energy. It has to be an integer amount.
Planck had been working long and hard on an electrodynamics problem, and as a last resort, he used another scientist's thermodynamics equations to help him solve the problem. At the time, the connection between electrodynamics and thermodynamics had not been discovered, so there was no reason for Planck to do this, but he did. And he did it wrong. The thermodynamic equations he was using had several steps. The first step involved dividing the energy up into small chunks, much like quanta. After doing this, the mathematics could juggle the different chunks of energy, and then the last step put it all back together into a final solution with just one piece of energy. Planck, however, decided that the mathematics was working well halfway through the process, so he didn't bother doing the last step and didn't put the energy back together. Planck didn't apply the equations consistently either, yet he somehow got the right answer.
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