Schrödinger's wave mechanics
Schrödinger's wave mechanics is the formulation of quantum theory that Erwin Schrödinger published in a burst of papers in 1926. Inspired by de Broglie's idea that matter has a wave nature, he sought the wave equation that such matter waves must obey, and the result was the equation that now bears his name. It recast the puzzles of the atom as a problem about standing waves.
Its first triumph was the hydrogen atom. Treating the electron as a wave bound by the nucleus, Schrödinger showed that only certain standing-wave patterns fit, and their energies reproduced exactly the spectral lines that had been measured for decades. The quantised energy levels that Bohr had simply postulated now fell out naturally as the allowed solutions of a single equation.
Wave mechanics turned out to be mathematically equivalent to Heisenberg's earlier matrix mechanics, and Schrödinger himself proved the two were just different languages for one theory. His version won wide affection because it spoke in the familiar tongue of waves and differential equations rather than abstract matrices, and it remains how most students first meet quantum mechanics today.
Matter waves obeying one equation explained the atom's sharp colours from first principles.
Schrödinger first pictured the wavefunction as a real, physical wave of charge spread through space. That reading does not survive: Born's probability interpretation, which Schrödinger disliked, is the one that works, and ψ is an amplitude of probability, not a substance.