What is the name of the equation that describes how a quantum wave function evolves?

The equation that describes how a quantum wave function evolves is the Schrödinger equation. For a non-relativistic quantum system, it determines how the wave function changes with time when the system’s Hamiltonian and initial state are known.

Erwin Schrödinger proposed the equation in 1925 and published it in 1926. It became one of the central mathematical foundations of quantum mechanics, providing a quantum counterpart to the role Newton’s laws play in classical mechanics. The wave function contains the system’s quantum state, while its squared magnitude gives probability information about measurement outcomes.

For an isolated system undergoing ordinary time evolution, the Schrödinger equation produces continuous and deterministic change. Measurement introduces a separate interpretive issue: in standard textbook treatments, the state may be updated discontinuously and probabilistically when an observation occurs.

The equation named in the question is the ordinary non-relativistic one. Relativistic systems may require the Klein–Gordon equation or Dirac equation, while Maxwell’s equations describe classical electromagnetism. Those are neighboring theories, not the general answer to this question.