# Uncertainty principle <!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside --> ## Microsims — p5.js ### Uncertainty principle (p5.js) · `Δt·Δf ≥ 1/4π` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/A35AoTZqD" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Uncertainty principle — p5.js microsim"></iframe> </div> *You can't localize a signal arbitrarily well in both time and frequency at once — the fundamental trade of all analysis.* **Open in the editor:** [&#9654; fork this sketch](https://editor.p5js.org/sciencenibber/sketches/A35AoTZqD) · movement *IX · Foundations & the rest of the toolbox* · library `p5js` ### Related microsims Live sims on neighbouring articles — 3 of them inside this article's own Wikipedia link tree: - [[Frequency_domain]] *(in tree)* - [[Probability_density_function]] *(in tree)* - [[Time_domain]] *(in tree)* - [[Analog_signal]] - [[Autocorrelation]] - [[Bifurcation_theory]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.* <!-- MICROSIMGEN:END --> ## Links (Wikipedia order) <!-- injected from _registry/childlinks/Uncertainty_principle.json (2026-07-30T02:09:12Z) --> `Albert_Einstein` · `Aline_Bonami` · `American_Journal_of_Mathematics` · `Angular_momentum` · `Angular_resolution` · `Annals_of_Mathematics` · `Aperture` · `Arthur_Compton` · `Arthur_Zajonc` · `Arun_K._Pati` · `Bandlimiting` · `Bandwidth-limited_pulse` · `Basis_(linear_algebra)` · `Bell's_theorem` · `Bell_test` · `Bohr–Einstein_debates` · `Boris_Podolsky` · `Born_rule` · `Bra–ket_notation` · `Canonical_commutation_relation` · `Canonical_coordinates` · `Carl_Friedrich_von_Weizsäcker` · `Casimir_effect` · `Cauchy–Schwarz_inequality` · `Classical_mechanics` · `Coherent_state` · `Communications_in_Mathematical_Physics` · `Commutator` · `Complementarity_(physics)` · `Complex_conjugate` · `Complex_number` · `Conjugate_variables` · `Consciousness_causes_collapse` · `Conservation_of_energy` · `Consistent_histories` · `Continuum_limit` · `Copenhagen_interpretation` · `Correspondence_principle` · `Creation_and_annihilation_operators` · `Curvature` · `David_Bohm` · `Davisson–Germer_experiment` · `De_Broglie–Bohm_theory` · `Degenerate_energy_levels` · `Delayed-choice_quantum_eraser` · `Dennis_Gabor` · `Density_matrix` · `Determinant` · [[Differential_entropy]] · `Dirac_delta_function` · `Dirac_equation` · `Discrete_Fourier_transform` · `Double-slit_experiment` · `Earle_Hesse_Kennard` · `Edwin_C._Kemble` · `Ehrenfest_theorem` · `Einstein–Podolsky–Rosen_paradox` · [[Electron]] · `Elitzur–Vaidman_bomb_tester` · `Encyclopedia_of_Mathematics` · `Energy_level` · `Ensemble_interpretation` · `Entropic_uncertainty` · `Error_function` · `Erwin_Schrödinger` · `Eugene_Wigner` · `Evgeny_Lifshitz` · `Excited_state` · `Expectation_value_(quantum_mechanics)` · [[Expected_value]] · `Falsifiability` · `Fedor_Nazarov` · `Filter_(signal_processing)` · `Fock_state` · `Fourier_transform` · `Franck–Hertz_experiment` · `Free_particle` · [[Frequency_domain]] · `Full_width_at_half_maximum` · `Function_space` · `G._H._Hardy` · `Gabor_wavelet` · `Gaussian_function` · `Gibbs_paradox` · `Ginzburg–Landau_theory` · `Glossary_of_elementary_quantum_mechanics` · `Goodhart's_law` · `Gravitational_time_dilation` · `Gravitational_wave` · `Ground_state` · `Hamiltonian_(quantum_mechanics)` · `Harmonic_analysis` · `Heisenberg's_microscope` · `Heisenberg_picture` · `Helmut_Rechenberg` · `Hermann_Weyl` · `Hidden-variable_theory` · `Hilbert_space` · `History_of_quantum_field_theory` · `History_of_quantum_mechanics` · `Hugh_Everett_III` · `Identity_matrix` · `Igor_Tamm` · `Inequality_(mathematics)` · `Integral` · `Integration_by_parts` · `Interaction_picture` · `Interferometry` · `Interpretations_of_quantum_mechanics` · `Introduction_to_quantum_mechanics` · `Jagdish_Mehra` · `John_Stewart_Bell` · `Journal_d'Analyse_Mathématique` · `Karl_Popper` · `Klein–Gordon_equation` · `Küpfmüller's_uncertainty_principle` · `Lars_Hörmander` · `Lebesgue_measure` · `Leonid_Mandelstam` · `Lev_Landau` · `List_of_trigonometric_identities` · `Local_hidden-variable_theory` · `Logarithmic_Sobolev_inequalities` · `Mach–Zehnder_interferometer` · `Majorana_equation` · `Many-worlds_interpretation` · `Martin_Heisenberg` · `Mass–energy_equivalence` · `Mathematical_formulation_of_quantum_mechanics` · `Matrix_mechanics` · `Matter_wave` · `Max_Born` · `Maximum_entropy_probability_distribution` · `Measurement_in_quantum_mechanics` · `Measurement_problem` · `Michael_Benedicks` · `Momentum` · `Momentum_operator` · `Morton_Hamermesh` · `Moyal_product` · `Nanotechnology` · `Nathan_Rosen` · `Niels_Bohr` · `Nondimensionalization` · `Normal_distribution` · `Objective-collapse_theory` · `Observable` · `Observer_effect_(physics)` · `Old_quantum_theory` · `Operator_norm` · `Optics` · `POVM` · `Particle` · `Particle_decay` · `Particle_in_a_box` · `Particle_in_a_ring` · `Particle_physics` · `Partition_function_(statistical_mechanics)` · `Pascual_Jordan` · `Pauli_equation` · `Pergamon_Press` · `Phase-space_formulation` · `Phase_factor` · `Philosophical_realism` · [[Photon]] · [[Physics]] · `Physics_and_Beyond` · `Plancherel_theorem` · `Planck_constant` · `Plane_wave` · `Popper's_experiment` · `Prime_number` · `Princeton_University_Press` · `Principle_of_locality` · `Probability_amplitude` · [[Probability_density_function]] · `Propagator` · `Quantum_Fisher_information` · `Quantum_algorithm` · `Quantum_amplifier` · `Quantum_biology` · `Quantum_bus` · `Quantum_cellular_automaton` · `Quantum_channel` · `Quantum_chaos` · `Quantum_chemistry` · `Quantum_circuit` · `Quantum_complexity_theory` · [[Quantum_computing]] · `Quantum_cosmology` · `Quantum_cryptography` · `Quantum_decoherence` · `Quantum_differential_calculus` · `Quantum_dynamics` · `Quantum_engineering` · `Quantum_entanglement` · `Quantum_eraser_experiment` · `Quantum_error_correction` · `Quantum_field_theory` · `Quantum_finite_automaton` · `Quantum_fluctuation` · `Quantum_geometry` · `Quantum_gravity` · `Quantum_harmonic_oscillator` · `Quantum_image_processing` · `Quantum_imaging` · `Quantum_indeterminacy` · `Quantum_information` · `Quantum_information_science` · `Quantum_jump` · `Quantum_key_distribution` · `Quantum_limit` · `Quantum_logic` · `Quantum_logic_gate` · `Quantum_machine` · `Quantum_machine_learning` · [[Quantum_mechanics]] · `Quantum_metamaterial` · `Quantum_metrology` · `Quantum_mind` · `Quantum_mysticism` · `Quantum_network` · `Quantum_neural_network` · `Quantum_nonlocality` · `Quantum_number` · `Quantum_optics` · `Quantum_programming` · `Quantum_sensor` · `Quantum_simulator` · `Quantum_spacetime` · `Quantum_state` · `Quantum_statistical_mechanics` · `Quantum_stochastic_calculus` · `Quantum_superposition` · `Quantum_teleportation` · `Quantum_tunnelling` · `Rarita–Schwinger_equation` · `Relational_quantum_mechanics` · `Relativistic_Breit–Wigner_distribution` · `Relativistic_quantum_mechanics` · `Resonance_(particle_physics)` · `Richard_Feynman` · `Rudolf_Haag` · `Rutherford_scattering_experiments` · `Rydberg_formula` · `Scattering` · `Schrödinger's_cat` · [[Schrödinger_equation]] · `Schrödinger_picture` · [[Second_law_of_thermodynamics]] · `Self-adjoint_operator` · `Short-time_Fourier_transform` · `Signal_reconstruction` · `Sinc_function` · `Spectroscopy` · `Squeezed_coherent_state` · `Standard_deviation` · `Stationary_state` · `Stern–Gerlach_experiment` · `Stronger_uncertainty_relations` · [[Superconductivity]] · `Superdeterminism` · `Superposition_principle` · `Symmetry_in_quantum_mechanics` · `Terence_Tao` · `The_Logic_of_Scientific_Discovery` · `Thought_experiment` · [[Time_domain]] · `Timeline_of_quantum_computing_and_communication` · `Timeline_of_quantum_mechanics` · `Time–frequency_analysis` · `Transactional_interpretation` · `Transient_(acoustics)` · `Umdeutung_paper` · `Unbounded_operator` · `Uncertainty_principle_(disambiguation)` · `Universal_wave_function` · `Variance` · `Vector_space` · `Von_Neumann_entropy` · [[Wave]] · `Wave_function` · `Wave_function_collapse` · `Wave_interference` · `Wave_packet` · `Wavelength` · `Wavenumber` · `Wave–particle_duality` · [[Wayback_Machine]] · `Weak_measurement` · `Werner_Heisenberg` · `Weyl_equation` · `Wheeler's_delayed-choice_experiment` · `Wigner's_friend` · `Without_loss_of_generality` · `Wolfgang_Pauli` · `Yakir_Aharonov` · [[Zero-point_energy]] > Signal Processing concept · part of the Signal Processing Portal · movement IX · !09 解析 kaiseki.svg <!-- RENDER-THUMB:START --> !480 *Rendered from the live microsim (▶ motion).* <!-- RENDER-THUMB:END --> ## See it next [](Discrete_Fourier_transform) *→ Discrete Fourier transform* <!-- VISUAL-LINK:END --> --- Back to Signal Processing Portal · the room · Semiotic gateway ## What it is In signal processing the uncertainty principle (Gabor limit) states that a signal cannot be arbitrarily concentrated in both time and frequency at once: the product of its time spread and its bandwidth is bounded below. ## How it works / why it matters If a signal's spread in time is $\Delta t$ and in frequency is $\Delta f$, then $\Delta t\,\Delta f \ge \tfrac{1}{4\pi}$ (with equality for a Gaussian), a mathematical consequence of the Fourier transform identical in form to Heisenberg's quantum uncertainty relation. This trade-off means short-duration pulses necessarily have wide bandwidth, and it dictates the resolution limits of spectral estimation, the short-time Fourier transform, wavelets, and windowing in every time–frequency analysis. ## Signs & universals Instantiates: signal · frequency · transformation · measurement — a fundamental limit on jointly measuring a signal's time and frequency localization. ## Related It arises directly from the Discrete Fourier transform and Fourier duality between the [[Time_domain]] and frequency domain, constraining spectral resolution and window choice in analysis. The relation is stated rigorously within [[Functional_analysis]] over finite-energy Square-integrable functions and governs time–frequency methods throughout Signal Processing. <!-- VISUAL-LINK:START --> <!-- CRAFT-LINK:START g12 --> *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].* <!-- CRAFT-LINK:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Uncertainty_principle) : [Wikitube](https://en.wikitube.io/wiki/Uncertainty_principle) ## Previous hub tags Tree parent: [[Phase_space]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*