schrödingerequatie

Schrödingerequatie is a fundamental equation in non-relativistic quantum mechanics that describes how the quantum state of a system evolves in time. It governs the evolution of the wavefunction ψ, a complex-valued function of position and time, from which probabilities of measurement outcomes are derived.

There are two common forms: the time-dependent and time-independent Schrödingerequatie. For a single non-relativistic particle of mass m in a potential V(r,t): iħ ∂ψ/∂t = [ -ħ^2/(2m) ∇^2 + V(r,t) ] ψ. If V is time‑independent, a separation of variables gives the time-independent form: [ -ħ^2/(2m) ∇^2 + V(r) ] φ(r) = E φ(r).

The Hamiltonian operator H encodes total energy. The equation is linear, yielding deterministic evolution of ψ given initial conditions, while measurement outcomes are inherently probabilistic with probability density |ψ(r,t)|^2.

In the Schrödingerequatie, states evolve in time while operators may be time-independent; the equation relates to other formalisms such as the Heisenberg and interaction pictures. The equation connects to classical mechanics in the limit ħ → 0 via Ehrenfest's theorem, where expectation values follow Newtonian trajectories on average.

Relativistic generalizations include the Dirac equation for spin-1/2 particles and the Klein-Gordon equation for spinless particles. For many-body systems, exact solutions are rare and computational methods such as Hartree-Fock and density functional theory are used. The Schrödingerequatie remains central to chemistry, condensed matter, and quantum simulations, providing the foundational description of non-relativistic quantum dynamics.