Schrödinger equation

Schrödinger equation is the fundamental dynamical equation of quantum mechanics, governing how the wave function of a quantum system evolves over time. Formulated by Erwin Schrödinger in 1926, the equation is a linear partial differential equation that determines the future state of an isolated system given its present state and the Hamiltonian operator describing its total energy. It is deterministic and continuous: given exact initial conditions, the evolution is exact and unambiguous.

The time-dependent Schrödinger equation is iℏ ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator. The equation preserves the total probability of the wave function over time, ensuring that the Born rule remains consistent. Despite its mathematical elegance, the equation's physical interpretation is contested: it governs the evolution of the wave function, but the wave function itself has no universally agreed-upon physical meaning. The Schrödinger equation describes the quantum state's evolution perfectly, yet it does not explain why or how the state collapses upon measurement. This gap between the equation's deterministic evolution and the probabilistic outcomes of observation is the measurement problem that has defined quantum foundations for a century.