Tidsevolution

Tidsevolution is the change of a system's state as time progresses. In physics, it refers to how observables or the underlying state vector or probability distribution evolve. The precise description depends on the theory: closed (isolated) systems follow deterministic laws, while open systems interact with environments and may exhibit stochastic or dissipative dynamics.

In classical mechanics, a system's state is given by positions and momenta in phase space. Time evolution is determined by Hamilton's equations, and the flow preserves phase-space volume (Liouville's theorem). For statistical ensembles, the Liouville equation governs how probability densities evolve.

In quantum mechanics, the state is a wavefunction or density operator. Time evolution is generated by the system's Hamiltonian via the Schrödinger equation. The unitary time-evolution operator U(t,t0) = exp(-i H (t-t0)/ħ) advances states in time. In the Heisenberg picture, observables evolve while states remain fixed; in the Schrödinger picture it's the opposite.

For open quantum systems, the evolution is non-unitary due to environment interactions. Master equations, such as the Lindblad form, describe how the density operator changes, including decoherence and dissipation.

In numerical practice, time evolution is implemented through discretization schemes such as Euler, Runge-Kutta, or more advanced symplectic and unitary integrators. The choice depends on whether the system is conservative, stochastic, or dissipative, and on the desired accuracy and stability.

Applications span many areas, including molecular dynamics, quantum information processing, cosmology, and statistical mechanics. Time evolution also clarifies the relationship between different formulations (e.g., Schrödinger vs. Heisenberg representations) and underpins simulations of dynamical processes.