kvanttitila

Kvanttitila is the quantum state of a system, a description that contains all information needed to determine the probabilities of outcomes for any measurement performed on the system. In quantum mechanics, kvanttitila can be represented in two main formalisms: as a pure state, given by a normalized vector |ψ⟩ in a Hilbert space (the wavefunction is a common representation in position space), or as a density operator ρ that describes statistical ensembles and mixed states.

In the density-operator formalism, ρ encodes both quantum superpositions and classical mixtures. The density operator satisfies Tr(ρ) = 1. Pure states have ρ^2 = ρ and Tr(ρ^2) = 1, while mixed states have Tr(ρ^2) < 1. Observables are represented by Hermitian operators A, and the Born rule gives measurement probabilities: p(a) = Tr(ρ P_a), where P_a is the projector onto the eigenstate corresponding to outcome a. For a pure state, p(a) = ⟨ψ|P_a|ψ⟩.

Time evolution of the kvanttitila is unitary for closed systems, described by iħ d|ψ⟩/dt = H|ψ⟩ or, in density form, dρ/dt = -i/ħ [H, ρ]. Open systems require additional dissipative terms to account for interaction with the environment.

For composite systems, kvanttitila may be entangled; the state of subsystems is obtained by partial tracing over other parts. In the Bloch-sphere representation of a qubit, ρ = (1/2)(I + r·σ) with the Bloch vector r satisfying |r| ≤ 1. Kvanttitila underpin predictions in quantum information, spectroscopy, and many-body physics, and are central to concepts such as superposition, measurement, decoherence, and entanglement.