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schrodinger's equation gives the wave function that describes the states of a quantum mechanical system. It relates to orbitals in that once you square the function, the wave function represent the probability of finding an electron at a giving position. At a peak of the wave function there is a high probability of finding an electron(electron density) and therefore that location can be described as an orbital.
Basically, it's how scientists figured out the characteristics of orbitals in the first place and proved them. Schrodinger's equation show's us the wave function of an electron. Also, from the wave function squared, we see a probability plot which represents how electrons are present in the orbitals. For example, in the s orbital, it's graphed as a sphere, which means there is an equal probability of e- being anywhere in the s orbital. When you look at the p orbital and and see the "infinity sign" shape essentially (3d 2px, 2py, and 2pz), as you get closer to the nodes where the infinity sign crosses there is a lower probability of finding an electron. When you see a peak, there is a high probability of finding an electron. Hope this helps and references what you were asking about! Also, these wave functions have helped scientists figure out the quantum numbers, the first 3: n, l, and ml. Remember ms was separate from this!
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