Schrödinger equation
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In ordinary physics there is one master rule for motion: tell me the forces on a ball and Newton's law tells you how it moves from here on. Quantum mechanics needs its own master rule, because particles are described by spread-out waves rather than points. The Schrödinger equation is that master rule. Feed it the forces acting on an electron, and it tells you the shape of the electron's wave and which energies it can have.
More precisely, the Schrödinger equation is the central equation of quantum mechanics: it relates the wavefunction of a system to its energy and to the potential energy the particle feels. Solving it for a given situation yields the allowed wavefunctions and the allowed energy levels — and these come out quantized, in discrete steps, simply as a consequence of the maths. For chemistry, solving it for the hydrogen atom gives exactly the atomic orbitals (1s, 2p, and so on) that fill the periodic table.
The honest limitation is that the equation can be solved exactly only for very simple systems — a single particle in a box, the hydrogen atom, the harmonic oscillator. For anything with more than one electron, including every molecule, chemists must use clever approximations and computers. Still, the equation is the rock-solid foundation: every method in quantum chemistry is, at heart, an attempt to solve it well enough.
Set up the Schrödinger equation for the single electron in a hydrogen atom, with the electrical pull of the proton as the potential energy. Solving it does not give one orbit; it gives a family of standing-wave solutions — the 1s, 2s, 2p orbitals — each with its own fixed energy. These solutions match the hydrogen spectrum almost perfectly.
Solve it for hydrogen and the atomic orbitals fall out of the maths.
There are two common forms: the time-independent equation, which finds the steady allowed energies and orbital shapes used throughout chemistry, and the time-dependent equation, which describes how a wavefunction evolves in time. School and most chemistry use the time-independent version.