Schrödinger equation

The Schrödinger equation is the master rule of quantum mechanics. It tells you how a wavefunction, the mathematical object that encodes everything knowable about a quantum system, changes over time. Where Newton's laws say how a ball's position and velocity evolve, Schrödinger's equation says how the wavefunction evolves — and from the wavefunction you read off the odds of every possible measurement outcome.

It is best thought of as a law of motion for probability amplitudes. Given the wavefunction now, the equation determines the wavefunction at every later moment with perfect, clockwork certainty; the evolution itself is smooth and deterministic. The randomness famous in quantum mechanics enters only when you make a measurement, not in how the wavefunction flows between measurements.

At its heart is the Hamiltonian, an operator that represents the system's total energy. Feed in the energies — kinetic and potential — and the equation grinds out the future. Almost everything in introductory quantum mechanics, from atoms and molecules to tunnelling and energy levels, is ultimately the Schrödinger equation solved in some particular situation.