Dirac equation

In 1928, Paul Dirac formulated a quantum theory for electrons that is consistent with special relativity. Earlier quantum descriptions worked well for atomic energy levels but were not fully relativistic. In a commonly used form, the Dirac equation is iℏ ∂ψ/∂t = (c α·p + βmc²)ψ, where ψ is a four-component wavefunction, p is momentum, m is the particle's mass, c is the speed of light, and α and β are matrices acting on the components. The equation describes how the quantum state changes over time while incorporating relativistic energy and momentum.

An important consequence of Dirac's formulation is that electron spin emerges within the relativistic theory rather than being added as an unrelated extra assumption. The equation's structure accounts for the electron's spin-½ and its magnetic behavior. It also provides a more complete account of relativistic electron states than the nonrelativistic Schrödinger equation.

The equation admits solutions associated with states that initially seemed puzzling, including negative-energy solutions. Dirac's interpretation led to the prediction of an anti-electron with the same mass as an electron but opposite electric charge. Carl Anderson's 1932 cosmic-ray observations established the positron, the electron's antiparticle. The prediction became a landmark example of mathematical theory pointing to a previously unknown particle.

Dirac's work helped lay foundations for relativistic quantum theory and quantum electrodynamics, the theory of how light interacts with charged particles. In modern particle physics, related Dirac equations describe spin-½ fermion fields, including electrons and quarks, with details depending on the particle and the theory. The equation is not a complete theory of every particle or interaction by itself; it is a fundamental tool used within broader quantum field theories.