I understand that radial probability is the probability of finding an electron at some distance from nucleus in a shell. But what does that exactly mean? What does it tell us about the atom and electrons?
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Probability density is probability per volume. Probability density at a given point means probability per volume in the limit that the volume is infinitesimally small.
Radial probability distribution at a given radius is the probability per distance that the event occurs in a infinitesimally thin spherical shell at that radius. (The distance being the thickness of the shell).
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Knowing the radial probability of finding an electron as a function of distance from the nucleus tells you a lot about the atom. For instance this electron density can be used to calculate the potential energy and to get an idea about the shell structure. You can estimate many different properties like cohesive energy, band gap, optical properties etc. Take a look on the Thomas$-$Fermi theory, the Hartree$-$Fock method or Density Functional Theory (DFT) to get an idea.
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$\begingroup$ The 1st question was answered by dave.The 2nd question was ' what the radial probability tells us about the electron/atom'. I tried to give some hints, little downvoter .. $\endgroup$ – OD IUM Nov 19 '16 at 16:52
