Quantumobservables

Quantumobservables are quantities in quantum mechanics that can be measured. In standard quantum theory, each quantumobservable is represented by a self-adjoint operator on a Hilbert space. The possible measurement outcomes are the operator’s eigenvalues, and the system’s state is described by a state vector or a density operator.

Measurement statistics follow the Born rule: the probability of obtaining a given eigenvalue o for a system in state |ψ⟩ is |⟨o|ψ⟩|^2, with continuous spectra described by a probability density. The expectation value is ⟨ψ|O|ψ⟩, and the spectral decomposition writes O = ∑k o_k P_k + ∫ o dP(o), where P_k are projection operators onto the corresponding eigen-subspaces.

If two quantumobservables O1 and O2 commute, they have a common eigenbasis and can be measured simultaneously; otherwise, their measurements are subject to uncertainty, as quantified by the Heisenberg uncertainty principle. After a projective measurement of O, the system collapses to the eigenstate associated with the observed eigenvalue (projection postulate). More general measurement schemes are described by POVMs, which extend the framework to include imperfect or indirect measurements.

In dynamics, observables may evolve in time: in the Schrödinger picture states evolve via unitary time evolution while observables are fixed, and in the Heisenberg picture observables carry the time dependence O(t) = U†OU. The operational meaning of quantumobservables arises from concrete measurement models, and, in principle, every self-adjoint operator represents a possible observable, though practical measurement can be challenging; domains and unbounded operators add technical nuances in infinite dimensions.