Dirac equation
The Dirac equation, written down by Paul Dirac in 1928, is the wave equation that successfully unites quantum mechanics with special relativity for an electron. The Schrödinger equation treats time and space on unequal footing and cannot cope with high speeds; Dirac sought an equation that did, and the only way he found to make it work was to allow the wavefunction to have several components woven together, rather than a single value.
The reward was a string of triumphs that Dirac had not set out to find. The extra structure of the equation produced the electron's spin automatically, explaining as a relativistic necessity what had previously been bolted on by hand. It also predicted the electron's magnetic strength with startling accuracy, and it forecast the fine details of the hydrogen spectrum that earlier theories could only approximate.
Most dramatically, the equation carried solutions of negative energy that refused to go away. Dirac's bold interpretation was that these correspond to antiparticles, leading directly to the prediction of the positron. The Dirac equation thus stands as a landmark: it is correct relativistic quantum mechanics for a single electron and the historic gateway to quantum field theory, where its insights find their full and consistent home.
A relativistic wave equation whose multi-component solution forces spin and predicts antimatter.
As a single-particle equation the Dirac equation has trouble: its negative-energy states are paradoxical until reinterpreted. The fully consistent reading treats it as an equation for the electron field in quantum field theory, where particles and antiparticles coexist cleanly.