In the winter of 1925–26, the Austrian physicist Erwin Schrödinger — vacationing in the Alps with a mistress whose identity has never been confirmed — wrote down the wave equation of quantum mechanics. He was looking for an equation whose solutions would be the de Broglie waves of matter, just as Maxwell's equations have electromagnetic waves as solutions. iℏ ∂ψ/∂t = Ĥψ became the central equation of quantum mechanics, and ψ — the wavefunction — became its central object. The equation is deterministic; the physical meaning of ψ is not. Max Born proposed that |ψ|² is a probability density, and quantum mechanics has been irreducibly probabilistic at its foundations ever since.
The time-dependent Schrödinger equation iℏ ∂ψ/∂t = Ĥψ has ψ(x, t) as the wavefunction, ℏ as the reduced Planck constant, and Ĥ as the Hamiltonian — the quantum translation of total energy. It is first-order in time, second-order in space, and linear, so solutions superpose. The time-independent form Ĥψ = Eψ is an eigenvalue equation: stationary states with quantized energies E. Solving it for the hydrogen atom (Schrödinger himself did this weeks after writing the equation) reproduces every spectral line and its relative intensity — the triumph that made quantum mechanics suddenly persuasive.
The mathematical setting is linear algebra on infinite-dimensional Hilbert spaces: wavefunctions are vectors, observables are Hermitian operators, measurement projects onto eigenspaces. Position x̂ and momentum p̂ = −iℏ ∂/∂x do not commute — x̂p̂ − p̂x̂ = iℏ — and that non-commutation is the origin of the Heisenberg uncertainty principle: observables whose operators fail to commute cannot be simultaneously measured to arbitrary precision. The time-dependent solutions describe unitary evolution that preserves total probability, until measurement. What happens at measurement — the collapse of the wavefunction, in Copenhagen language — is not described by the Schrödinger equation. The equation is silent on what observation does, and the various interpretations of quantum mechanics differ precisely on this point.
Quantum chemistry is applied Schrödinger equation: every chemical bond, every reaction rate, every spectroscopic line is in principle a solution for the relevant electrons, with density functional theory and coupled-cluster methods supplying the approximations. Solid-state physics solves the equation in periodic potentials; band structure, semiconductor electronics, and superconductivity follow. Quantum computing manipulates qubits through unitary gates engineered as Schrödinger-evolution solutions. Quantum sensing — atomic clocks, magnetometers, gravimeters, gravitational-wave detectors — uses ultra-stable transitions. Quantum cryptography uses the measurement-disturbance feature as a security primitive. The little equation Schrödinger wrote down on holiday is one of the most-exercised in human science.