In quantum mechanics, a quantum observable is a physical quantity that can be measured or observed in an experiment. It is associated with a mathematical operator that represents the observable in the formalism of quantum mechanics. These operators act on the quantum state of a system and produce a set of possible outcomes when a measurement is performed.
Quantum observables are represented by Hermitian operators, which are operators that are equal to their own adjoints. The eigenvalues and eigenvectors of these operators correspond to the possible outcomes and states of the system, respectively. When a measurement is made on a quantum system, the system collapses into one of its eigenstates, and the measurement yields one of the eigenvalues associated with the observable.
Examples of quantum observables include position, momentum, energy, angular momentum, and spin. Each of these observables has its own corresponding operator, such as the position operator, momentum operator, etc. The mathematical formalism of quantum mechanics allows for the calculation of the probabilities of obtaining different measurement outcomes for a given observable.
It's worth noting that the act of measurement in quantum mechanics can be probabilistic, meaning that even if a system is prepared in a well-defined state, the measurement outcome can be uncertain and can only be described by probabilities. This probabilistic nature is a fundamental characteristic of quantum mechanics and distinguishes it from classical physics.