Heisenberg uncertainty principle

A quantum principle stating that certain pairs of physical quantities, such as position and momentum, cannot both have arbitrarily precise values in the same state.

Position–momentum uncertainty relation: The inequality stating that position and momentum spreads satisfy ΔxΔp ≥ ħ/2. This is the best-known quantitative expression of the principle.
Wavefunction: A complex-valued function encoding the probability amplitudes of a quantum system. Its spatial and momentum representations expose the spreads constrained by the principle.
Measurement disturbance: A change in a system caused by its interaction with a measuring apparatus. Disturbance can occur, but it is not the fundamental basis of uncertainty relations.
Zero-point energy: The lowest energy a quantum system retains even in its ground state. Confinement and uncertainty prevent many systems from having both zero motion and zero position spread.
Werner Heisenberg: A German theoretical physicist who helped establish quantum mechanics and formulated its uncertainty principle. He introduced the principle in a 1927 paper.
Robertson–Schrödinger uncertainty relation: A general uncertainty inequality for two observables that includes both commutators and correlations. It extends the principle to arbitrary pairs of quantum observables.
Fourier transform: A mathematical transformation expressing a function in terms of its frequency components. Position and momentum wavefunctions are Fourier-related, producing a spread tradeoff.
Hidden-variable theory: A class of theories supplementing quantum mechanics with additional variables to account for outcomes. These theories challenge interpretations of quantum probabilities, not the mathematical uncertainty bound.
Quantum harmonic oscillator: A quantum system with a quadratic potential and equally spaced energy levels. Its ground state illustrates the minimum allowed position–momentum uncertainty.
Matrix mechanics: An early formulation of quantum mechanics using arrays of transition quantities and noncommuting operations. Its noncommuting quantities supplied the mathematical setting for Heisenberg's argument.

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