What mathematical solution maps out the probability distribution of an electron in an atomic orbital?

Schrödinger wave equation

The modern quantum mechanical description of an electron's location within an atom is fundamentally rooted in probability, not definite paths. The abstract, mathematical description of this probability distribution—the atomic orbital—is derived directly from the solutions to the Schrödinger wave equation. This equation governs the behavior of quantum systems, and its solution provides the wave function ($\ ext{Psi}$). When this wave function is squared ($\ ext{Psi}^2$), it yields the probability density function, which visually defines the shape and boundaries of the orbital region where the electron is most likely to be located. This mathematical framework replaced older, simpler models that incorrectly assumed electrons followed fixed orbits like planets.