Beobachtbare

Beobachtbare is the German term for observables, the physical quantities whose values can be determined by measurement in a given system. In physics, and most notably in quantum mechanics, an observable is represented by a self-adjoint (Hermitian) operator acting on the system’s Hilbert space. The eigenvalues of this operator are the possible outcomes of a measurement.

Mathematically, an observable A can be decomposed spectrally as A = ∑ a_i P_i, where a_i are the eigenvalues and P_i are the corresponding projection operators. When the system is in a state described by a vector ψ, the probability of obtaining the outcome a_i is p(a_i) = ⟨ψ|P_i|ψ⟩. The expectation value of the observable in state ψ is ⟨A⟩ = ⟨ψ|A|ψ⟩.

A measurement generally updates the system’s state according to the projection postulate: if the result a_i is observed, the post-measurement state collapses to P_i ψ / ||P_i ψ||. If two observables A and B commute, they can be simultaneously measured with definite values; non-commuting observables exhibit uncertainty relations that limit the precision of simultaneous measurements.

Common examples of observables include position, momentum, and spin components. The Hamiltonian, which governs time evolution, is itself an observable. In classical physics, observables are real-valued functions on phase space and possess definite values independent of measurement. In quantum theory, generalized measurements extend this concept beyond projective measurements to positive operator-valued measures (POVMs).

Historically, the concept of observables emerged in the early 20th century and remains central to the formalism and interpretation of quantum mechanics.