The Heisenberg uncertainty principle is one of the most famous — and most misunderstood — results in physics.
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Walk-through
How to Use This Calculator
3 steps▸
1
Pick position-momentum or energy-time
Use the Position-Momentum tab for Δx and Δp, or the Energy-Time tab for ΔE and Δt. Both follow the same relation: the product of the two uncertainties can never fall below ħ/2.
2
Enter one value, or both
Enter just one uncertainty to get the minimum possible value of its pair. Enter both to see the actual product and whether it satisfies the Heisenberg limit.
3
Use Minimum for a quick single-value solve
On the Minimum tab, pick which quantity to solve for (Δx, Δp, ΔE, or Δt), enter its known conjugate, and read the theoretical floor directly.
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Reference
Formula & Methodology
3 formulas▸
Position-momentum uncertainty
Δx · Δp ≥ ħ/2
The product of the uncertainty in a particle's position (Δx, meters) and the uncertainty in its momentum (Δp, kg·m/s) can never be smaller than half the reduced Planck constant, ħ/2 ≈ 5.273 × 10⁻³⁵ kg·m²/s.
Energy-time uncertainty
ΔE · Δt ≥ ħ/2
The product of the uncertainty in a system's energy (ΔE, joules) and the uncertainty in the time over which that energy is measured or the state persists (Δt, seconds) is likewise bounded below by ħ/2.
Reduced Planck constant
ħ = h / (2π) ≈ 1.055 × 10⁻³⁴ J·s
ħ ('h-bar') is Planck's constant h divided by 2π. It sets the universal scale of quantum uncertainty — every minimum-uncertainty calculation on this page traces back to this one number.
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Glossary
Key Terms Explained
6 terms▸
Uncertainty principle ↗Werner Heisenberg's 1927 result that certain pairs of physical properties — like position and momentum, or energy and time — cannot both be known to arbitrary precision simultaneously; sharpening one necessarily blurs the other.
Position (Δx) ↗The uncertainty, or spread, in where a particle can be found, measured in meters. A smaller Δx means the particle's location is more tightly pinned down.
Momentum (Δp) ↗The uncertainty in a particle's momentum (mass times velocity), measured in kg·m/s. A smaller Δp means the particle's motion is more precisely known.
Reduced Planck constant (ħ) ↗Planck's constant h divided by 2π, equal to about 1.055 × 10⁻³⁴ J·s. It is the fundamental constant that sets the scale of every quantum uncertainty limit.
Energy-time uncertainty ↗The uncertainty relation ΔE·Δt ≥ ħ/2 linking how precisely a system's energy can be known to how long that measurement (or the state itself) lasts — short-lived states necessarily have a broader spread in energy.
Quantum limit ↗The theoretical minimum uncertainty product (ħ/2) allowed by quantum mechanics — a floor set by nature itself, not by the precision of any instrument or measurement technique.
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Scenarios
Real-World Examples
3 worked examples▸
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Chemistry student
An electron confined to an atomic-scale box
Position uncertainty (Δx) 1 × 10⁻¹⁰ m
Δp ≥ ħ/(2·Δx) ≈ 5.3 × 10⁻²⁵ kg·m/s — confining an electron to roughly one atomic diameter forces its momentum uncertainty to be huge on the atomic scale, which is exactly why electrons don't sit still at fixed points inside an atom.
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Physics student
An excited atomic state with a 1-nanosecond lifetime
Time uncertainty (Δt) 1 × 10⁻⁹ s
ΔE ≥ ħ/(2·Δt) ≈ 5.3 × 10⁻²⁶ J (about 3.3 × 10⁻⁷ eV) — a state that only lives for a nanosecond cannot have a perfectly sharp energy; this natural linewidth is measurable in atomic spectroscopy.
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Curious skeptic
Why a thrown ball never looks 'fuzzy'
Position uncertainty (Δx) 1 × 10⁻⁵ m
Δp ≥ ħ/(2·Δx) ≈ 5.3 × 10⁻³⁰ kg·m/s — even pinning a macroscopic object's position down to a hair's-width leaves a momentum floor about 25 orders of magnitude smaller than any momentum you could ever measure, so quantum uncertainty is utterly negligible for everyday objects.
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Reference
Cite This Calculator
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Deep Dive
Understanding the Heisenberg Uncertainty Calculator
The Heisenberg uncertainty principle is one of the most famous — and most misunderstood — results in physics. It doesn't describe a limit on measurement technology; it describes a fundamental property of nature. This calculator turns the two standard uncertainty relations, position-momentum and energy-time, into a direct numeric tool.
How the Heisenberg Uncertainty Calculator works
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Both uncertainty relations share the same shape: the product of two paired ('conjugate') quantities can never fall below ħ/2, where ħ is the reduced Planck constant. Enter one uncertainty and the calculator returns the smallest the other can possibly be. Enter both, and it instead reports the actual product and whether that pair is physically consistent with the principle — a product below ħ/2 is not a measurement error, it is a value quantum mechanics forbids outright.
Position-momentum vs. energy-time
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Δx·Δp ≥ ħ/2 applies to a particle's location and its momentum at a single instant — it is the version most people learn first, and it explains why electrons in atoms don't have sharply defined orbits. ΔE·Δt ≥ ħ/2 instead links a system's energy spread to how long it persists (or how long it takes to measure that energy) — it explains why unstable particles and short-lived excited states have a natural 'width' to their energy rather than one exact value.
Why this never matters for everyday objects
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Because ħ is so small (about 1.055 × 10⁻³⁴ J·s), the uncertainty floor it sets is many orders of magnitude below anything a lab instrument, let alone a human eye, can detect for objects with everyday mass and size. The effect only becomes significant at atomic and subatomic scales — electrons, photons, and short-lived particles — which is exactly where quantum mechanics was first needed. For related atomic-scale calculations, see the de Broglie Wavelength Calculator and Photon Energy Calculator.
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Questions
Frequently Asked Questions
6 questions▸
What is the Heisenberg uncertainty principle formula?+
Δx·Δp ≥ ħ/2 for position and momentum, where Δx is the position uncertainty, Δp is the momentum uncertainty, and ħ is the reduced Planck constant (≈1.055 × 10⁻³⁴ J·s). The product of the two can never be smaller than ħ/2.
What is ħ (h-bar)?+
ħ is the reduced Planck constant, equal to Planck's constant h divided by 2π: ħ ≈ 1.055 × 10⁻³⁴ J·s. It is the fundamental constant that sets the scale for every quantum uncertainty limit.
What is the energy-time uncertainty relation?+
ΔE·Δt ≥ ħ/2, where ΔE is the uncertainty in a system's energy and Δt is the uncertainty in the time over which it is measured or the duration the state persists. Short-lived states necessarily have a broader energy spread.
Why don't we notice quantum uncertainty in everyday objects?+
Because ħ is extraordinarily small, the uncertainty floor it imposes on macroscopic objects (a thrown ball, a car, a person) is many orders of magnitude below any measurable quantity — it's mathematically real but practically negligible outside the atomic scale.
What units does the calculator use?+
Position (Δx) in meters, momentum (Δp) in kg·m/s, energy (ΔE) in joules, and time (Δt) in seconds — all SI units, consistent with how ħ is defined (in joule-seconds).
What does 'minimum uncertainty' mean on the Minimum tab?+
It's the equality case of the uncertainty relation, Δx·Δp = ħ/2 (or ΔE·Δt = ħ/2) — the smallest the conjugate quantity's uncertainty can theoretically be. Real physical states typically have a product somewhat larger than this floor, never smaller.
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