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Materials in Electronics/Confined Particles

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Schrödinger's Equation can be used to find the electron distributions in a range of situations. The easiest to solve, theoretically, is the potential well. This is a region of space with a lower potential than neigbouring regions. The simplest case is where the well is infinitely deep, so electrons cannot ever escape. This is the Infinite Potential Well:

If you add more electrons to the box, the states are filled according to the Aufbau Principle:

However, infinte wells don't exist and are just a convenient approximation. In real life, we deal with Finite Wells: