observables

An observable is a quantity that can be measured or observed about a system. In mathematics and physics, the term has related but distinct meanings that share a common idea: an observable yields numerical outcomes under appropriate conditions.

In probability theory, an observable is typically a measurable function from a sample space to the real numbers, i.e., a random variable. Observables are defined with respect to a sigma-algebra of events, and their distributions describe the likelihood of different measurement outcomes.

In quantum mechanics, an observable is represented by a self-adjoint operator on a Hilbert space. The possible measurement results are the operator’s eigenvalues, and the probabilities of outcomes are given by the system’s state via the Born rule. Observables that commute can be measured simultaneously, while non-commuting observables reflect fundamental limits on precision.

In classical mechanics, observables are real-valued functions on phase space, such as position, momentum, or energy. The set of observables interacts with dynamics through structures like the Poisson bracket, which encodes how observables change in time.

In statistics and data analysis, observables refer to quantities that can be measured in experiments and are modeled as random variables or features, with properties such as expectation, variance, and distribution.

In computer science, particularly reactive programming, an observable denotes a value or data stream that can be observed or subscribed to over time, enabling processing of asynchronous updates.