Mathematically precise and philosophically disputed.

The Schrödinger equation is the central equation of nonrelativistic quantum mechanics, describing how a quantum system’s wavefunction evolves and how its measurable states are determined. The time-dependent form tracks change, while the time-independent form identifies allowed energy states. Rather than predicting a single definite path, it governs a probability amplitude whose squared magnitude gives the likelihood of different outcomes. From atoms to molecules, it explains quantized energies, tunneling, interference, and much of microscopic behavior. Its solutions reveal a world governed by probability rather than classical trajectories.

The top equation of the design is the time-independent form, The bottom is the time-dependent form.