Δx·Δp ≥ ℏ/2: minimum Δp from Δx, minimum Δx from Δp, or a pair check with satisfaction ratio. Units m–pm and kg·m/s / g·cm/s / eV/c; optional mass derives Δv.
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Key facts
Category
Science & Education
Input types
select, number
Output type
text
Sample coverage
4
API ready
Yes
Overview
The Heisenberg Uncertainty Principle Calculator determines the quantum limits of position and momentum using the fundamental relation Δx·Δp ≥ ℏ/2. It calculates the theoretical minimum momentum uncertainty (Δp) from a given spatial confinement (Δx), finds the minimum position uncertainty from momentum, or evaluates whether an experimental (Δx, Δp) pair satisfies quantum mechanics with an exact margin ratio, including optional particle mass to derive velocity spread (Δv).
When to use
Calculating the minimum momentum spread or required velocity uncertainty when a particle is confined to a microscopic boundary like an atom or quantum dot.
Finding the theoretical spatial resolution limit given a particle's known momentum uncertainty.
Checking whether a measured pair of position and momentum uncertainties satisfies quantum limits and comparing macroscopic versus microscopic quantum margins.
How it works
1Select your calculation mode to solve for minimum Δp, minimum Δx, or check a given (Δx, Δp) pair.
2Enter the known uncertainty values and select their respective measurement units (m, mm, µm, nm, or pm for position; kg·m/s, g·cm/s, or eV/c for momentum).
3Optionally provide the particle's rest mass in kilograms to convert momentum uncertainty into velocity uncertainty (Δv = Δp/m) and assess relativistic fraction (% of c).
4The tool applies the fundamental lower bound ℏ/2 (≈ 5.27286 × 10⁻³⁵ J·s) to compute the corresponding quantum bound or ratio.
Use cases
Quantum physics homework and coursework solving for ground-state confinement limits and electron momentum spreads.
Semiconductor physics calculations estimating minimum velocity dispersion of charge carriers in nanometer-scale quantum wells.
Demonstrating why quantum mechanical uncertainty bounds are imperceptible in classical macroscopic mechanics.
Examples
1. Electron Confinement in an Atom
Physics Student
Background
Studying atomic physics and evaluating why an electron confined to an atomic radius (~100 pm) cannot remain stationary.
Problem
Determine the minimum momentum uncertainty and corresponding velocity spread for an electron localized within 100 pm.
How to use
Select 'Minimum Δp from Δx', enter 100 with unit 'pm', select 'kg·m/s' for output momentum unit, and enter the electron mass 9.10938e-31 kg.
Outcome
Calculates a minimum Δp of 5.27 × 10⁻²⁵ kg·m/s (≈ 986.6 eV/c) and a minimum velocity spread of ~578.8 km/s (0.193% of c).
2. Classical vs Quantum Scale Boundary Check
Science Educator
Background
Preparing lecture slides to show students why Heisenberg uncertainty is negligible for macroscopic engineering measurements.
Problem
Verify whether a dust particle with Δx = 1 µm and Δp = 10⁻⁶ kg·m/s violates quantum mechanical bounds.
How to use
Select 'Check a (Δx, Δp) pair', set Δx to 1 µm, and set Δp to 0.000001 kg·m/s.
Outcome
FAQ
What is the formula used in this calculator?
It uses Heisenberg's position-momentum uncertainty relation Δx·Δp ≥ ℏ/2, where ℏ is the reduced Planck constant (h / 2π ≈ 1.05457 × 10⁻³⁴ J·s).
What units are supported for position and momentum?
Position uncertainty supports meters (m), millimeters (mm), micrometers (µm), nanometers (nm), and picometers (pm). Momentum uncertainty supports kg·m/s, g·cm/s, and eV/c.
How does providing a particle mass change the output?
When mass is provided, the tool calculates the velocity uncertainty Δv = Δp/m and displays it as both an absolute speed in m/s and a percentage of the speed of light.
What does the pair check mode evaluate?
It multiplies your input Δx and Δp, compares the product to ℏ/2, and outputs whether the pair is physically permissible along with the exact margin factor.
Why do macroscopic objects show large margin factors in check mode?
Everyday objects have masses and dimensions many orders of magnitude larger than quantum scales, making their product Δx·Δp exceed ℏ/2 by 20 to 30 orders of magnitude.