Heisenberg Uncertainty Calculator - Solve Δx·Δp and ΔE·Δt

The Heisenberg Uncertainty Calculator applies the quantum mechanics relation Δx · Δp ≥ ℏ/2 — and the related energy-time form ΔE · Δt ≥ ℏ/2 — to find the minimum spread in position, momentum, energy, or time for a quantum object.

• • Atomic-scale confinement: Find the minimum momentum an electron confined to a 1 Å box must carry.
• • Slit and microscope diffraction: Turn a slit width or microscope resolution into the minimum momentum spread and the matching de Broglie wavelength.
• • Excited-state lifetime: Convert an atomic-state lifetime into the minimum energy linewidth ΔE.
• • Pedagogy and textbook checks: Reproduce textbook worked examples for an electron in 1 Å and a baseball at 10 m/s with the exact 2022 CODATA ℏ.

The position-momentum form is a bound on simultaneous knowledge of two conjugate variables. It is a property of the wave function, not a measurement disturbance or observer effect.

The energy-time form ΔE · Δt ≥ ℏ/2 looks similar but is structurally different. Δt is the characteristic timescale of change of the system, not a measurement uncertainty on time.

The calculator uses the 2022 CODATA value ℏ = 1.054571817 × 10⁻³⁴ J·s, exact after the 2019 SI redefinition fixed h = 6.62607015 × 10⁻³⁴ J·s.

Experimental spread on a single measurement is covered by the absolute uncertainty calculator.

Pick the relation (position-momentum, energy-time, or momentum-from-mass-and-speed), choose the variable to solve for, and the calculator returns the matching spread plus a check of the product against ℏ/2.

• Δx: Standard deviation of position along one axis, in metres.
• Δp: Standard deviation of momentum along the same axis, in kg·m/s.
• ΔE: Standard deviation of energy in joules.
• Δt: Characteristic timescale of change in seconds.
• ℏ (hbar): Reduced Planck constant. Default 1.054571817 × 10⁻³⁴ J·s from 2022 CODATA.
• m: Particle mass in kg. Default is the electron rest mass (9.1093837015 × 10⁻³¹ kg).

Solving for Δx uses Δx = ℏ / (2 · Δp); solving for Δp uses Δp = ℏ / (2 · Δx). The momentum-from-mass-and-speed mode adds the de Broglie wavelength λ = h / p as a diagnostic.

The check mode takes both Δx and Δp and returns the product against ℏ/2. If the product is below ℏ/2 the chosen pair is flagged, a useful sanity check for single-particle measurements.

According to NIST CODATA 2022 - reduced Planck constant, ℏ = h / (2π) = 1.054571817 × 10⁻³⁴ J·s, exact after the 2019 SI redefinition.

The de Broglie wavelength that the Heisenberg Uncertainty Calculator reports uses the same h and ℏ constants as the compton wavelength calculator.

Four ideas cover the language of any quantum uncertainty problem.

These four ideas cover the technical vocabulary most quantum mechanics textbooks use around the uncertainty principle.

Four steps take you from a quantum problem statement to the matching spread.

1. 1 Pick the mode: Choose position-momentum for Δx · Δp ≥ ℏ/2, energy-time for ΔE · Δt ≥ ℏ/2, or momentum-from-mass-and-speed for mass and velocity inputs.
2. 2 Choose the variable to solve for: Select Δx, Δp, or the check mode that computes the product.
3. 3 Enter the known value: Use SI units. Atomic-scale inputs are easier in scientific notation, for example 1e-10 m for 1 Å.
4. 4 Read the result and the limit: The primary panel returns the solved spread, the product, the limit ℏ/2, and (in mass-and-speed mode) the momentum magnitude and de Broglie wavelength.

Electron confined to a 1 Å region in an STM: switch to position-momentum, set 'solve for Δp', enter Δx = 1e-10 m, and the calculator returns Δp ≈ 5.27 × 10⁻²⁵ kg·m/s with de Broglie wavelength λ ≈ 1.26 × 10⁻⁹ m.

For the standard deviation of an ensemble of momentum readings, the standard deviation calculator works through the same dataset step by step.

A purpose-built tool avoids the most common pitfalls in a hand calculation.

• • Exact 2022 CODATA ℏ: The reduced Planck constant and electron mass are pre-loaded with the 2022 CODATA values, so you never have to memorise 1.054571817 × 10⁻³⁴ J·s.
• • All four directions: Solve for Δx, Δp, ΔE, or Δt, or use check mode to verify that a pair of inputs satisfies the principle.
• • Built-in physicality checks: Δx = 0, Δp = 0, ΔE = 0, and Δt = 0 each surface a clear status flag instead of returning zero or infinity silently.
• • Side-by-side diagnostics: Momentum magnitude, de Broglie wavelength, kinetic energy, and v/c appear with the result.
• • Energy-time variant included: The energy-time relation is treated as a first-class mode rather than a footnote.

When the same inputs need to be converted into joules, electronvolts, or kilowatt-hours, the work, energy, and power calculator handles the unit conversions.

Five factors change the numerical answer, plus two limitations worth knowing before you trust a number.

• • The non-relativistic momentum p = m · v breaks down above about 0.1 c. For relativistic electrons use p = γ m v and re-derive the de Broglie wavelength from γ.
• • The energy-time relation is not a commutation relation. Δt is the timescale of change, not a measurement uncertainty on time.

The principle holds for macroscopic objects too: a 0.145 kg baseball with Δv = 1 m/s gives a minimum Δx ≈ 3.6 × 10⁻³⁴ m, roughly 30 orders of magnitude smaller than an atomic nucleus.

According to NIST CODATA 2022 - Planck constant, h is exactly 6.62607015 × 10⁻³⁴ J·s since the 2019 SI redefinition, so ℏ = h / (2π) is exact in SI units.

According to Wikipedia - Uncertainty Principle, Δx · Δp ≥ ℏ/2 with equality reached only for Gaussian minimum-uncertainty wave packets.

For an experimentalist who wants to compare a measured position spread to a theoretical Δx, the relative standard deviation calculator helps quantify how close the measurement is to the theoretical minimum.

After a measurement run, the percent error calculator turns the deviation between an experimental Δx and the theoretical ℏ/(2 Δp) bound into a percent-error number.