In quantum physics, which equation describes how a nonrelativistic quantum state changes with time?

In quantum physics, the Schrödinger equation describes how a nonrelativistic quantum state changes with time.

Erwin Schrödinger introduced the equation in 1926 while developing wave mechanics. Its time-dependent form uses a wave function and a Hamiltonian, the operator representing the system’s total energy. Solving it gives the state’s evolution and allows physicists to calculate measurable probabilities.

For a time-independent Hamiltonian, the equation can be separated into a stationary form. That version is used to find allowed energy levels in systems such as atoms, quantum wells, and molecules. The hydrogen atom is a classic example: its solutions produce discrete energy states that help explain atomic spectra.

The equation is nonrelativistic, so it is not the final description for particles moving near light speed. Paul Dirac later developed a relativistic wave equation for spin-½ particles. Another common mistake is calling the wave function itself a physical water-like wave; its squared magnitude gives probability density instead.