timeindependent

Time-independent is an adjective used to describe quantities, parameters, or systems whose explicit dependence on time is absent. In physics and mathematics, a system is time-independent when its governing equations or inputs do not explicitly include the time variable; any time variation arises only through other changing variables or through specific solutions.

In quantum mechanics, a time-independent Hamiltonian H has no explicit time dependence (∂H/∂t = 0). This leads to conservation of energy and allows solutions that can be written as stationary states. The time-independent Schrödinger equation, Hψ = Eψ, determines energy eigenstates. The full wave function evolves in time as ψ(x,t) = ψ_E(x) e^{-iEt/ħ}, so probabilities remain constant in time even though the phase factor evolves.

In classical mechanics, a time-independent potential V(q) means the Lagrangian L and the resulting equations of motion do not depend on time explicitly. By Noether’s theorem, time-translation symmetry leads to energy conservation. Time independence here implies the system’s rules stay the same over time, while the state can still evolve.

In differential equations, time-independent problems involve no explicit time variable, yielding steady-state or static solutions. Examples include boundary-value problems and Laplace’s equation, which describe configurations that do not change with time.

Other domains use the term to describe systems that are invariant under shifts in time. In signal processing, a linear time-invariant system responds to inputs independent of when they are applied, with a fixed impulse response.

Overall, time independence denotes the absence of explicit time dependence in the governing description, even as the system may exhibit time evolution through its internal dynamics or boundary conditions.