Usikkerhetsrelationen

Usikkerhetsrelationen, commonly referred to as the Heisenberg uncertainty principle, is a fundamental concept in quantum mechanics that imposes a limit on the simultaneous precision with which certain pairs of observables can be known. The archetypal example is position and momentum, whose operators do not commute. The relation expresses a property of quantum states rather than a simple limitation of measurement devices.

For any quantum state, the standard deviations Δx and Δp satisfy the inequality Δx Δp ≥ ħ/2. More generally, for any two observables A and B with a nonzero commutator [A,B], the Robertson-Schrödinger form gives ΔA ΔB ≥ (1/2) |⟨[A,B]⟩|, with refinements that also involve the covariances of A and B. These bounds are intrinsic to the state and the mathematical structure of quantum theory.

Historically, the concept was introduced by Werner Heisenberg in 1927 and was later formalized by Robertson (1936) and Schrödinger, who clarified the role of commutators and covariances. It is important to distinguish the preparation aspect of the uncertainty from the measurement-disturbance interpretation often discussed in popular accounts; the principle reflects fundamental constraints on quantum states rather than practical measurement error alone.

Equality in the uncertainty bound is achieved by certain minimum-uncertainty states, notably Gaussian states for position and momentum. The uncertainty relation also applies to other conjugate pairs and has important implications in quantum metrology, information, and optics, where it informs the limits of precision and the design of squeezed states. The energy-time relationship is related but treated differently in standard quantum mechanics because time is not an operator in the same way as position.