Zustandvektor

Zustandvektor is the term used for the vector that encodes the essential information about a system at a given time, allowing its future behavior to be predicted from a model. The concept appears in different fields, notably quantum mechanics and classical control theory, with different interpretations and mathematical structures, but the common idea is to capture the current state in a compact form.

In quantum mechanics, the state vector, denoted |ψ⟩, is a unit vector in a Hilbert space that describes a pure quantum state. Normalization ⟨ψ|ψ⟩ = 1 ensures probabilistic interpretation, and only the vector up to an overall phase e^{iφ} is physically meaningful. Observables are represented by Hermitian operators, and measurement probabilities are given by the Born rule, ⟨ψ|P|ψ⟩ for a projection P. Time evolution follows the Schrödinger equation iħ d|ψ⟩/dt = H|ψ⟩, yielding unitary evolution. For composite systems, the joint state resides in a tensor product space and can exhibit entanglement. Mixed states are described by density operators ρ, which can be interpreted as ensembles of state vectors.

In classical control theory and dynamical systems, a Zustandvektor refers to the state vector x(t) that collects the system’s internal variables. In a state-space model, continuous-time form is dx/dt = Ax + Bu and y = Cx + Du; discrete-time form is x[k+1] = Ax[k] + Bu[k], y[k] = Cx[k] + Du[k]. The state vector enables compact, implementable modeling of the system and prediction of outputs from inputs. State estimation methods, such as the Kalman filter, aim to infer x(t) from noisy measurements.

See also: quantum state, density operator, state-space representation.