Dirac equationequation
The Dirac equation is a mathematical instruction that describes the behavior of electrons at very high speeds, close to the speed of light. If the ordinary Schrödinger equation works for slow particles, the Dirac equation is like a racing car, capable of taking into account both the speed and the spin of particles. It predicted that every particle has a counterpart from the anti-world — an antiparticle. That is how physicists learned about the existence of positrons.
How it works
The Dirac equation explains the fine structure of the hydrogen spectrum, takes into account the spin and magnetic moment of the electron naturally, without artificial additives. It also describes particles with zero mass — neutrinos, and formed the basis of quantum electrodynamics. Without it, we would not understand how semiconductor lasers work and why antimatter exists.
💡 Dirac initially did not believe in antiparticles, but when a few years later the positron was discovered in cosmic rays, he was called "genius madness." Dirac himself said that the equation turned out to be smarter than its creator.
(i\gamma^\mu \partial_\mu - m)\psi = 0
γ^μ are 4×4 Dirac matrices, ∂_μ is the spacetime derivative, m is the mass, ψ is a spinor wave function.
i\frac{\partial\psi}{\partial t} = \left(-i\vec{\alpha}\cdot\nabla + \beta m\right)\psi
α and β are Dirac matrices related to γ, ψ is a four-component spinor.