Heisenberg Uncertainty Principle Calculator (Position-Momentum)
Find the minimum momentum uncertainty for a given position uncertainty using Δx·Δp ≥ ħ/2.
🔬 What is the Heisenberg Uncertainty Principle Calculator?
The Heisenberg uncertainty principle calculator finds the smallest possible momentum uncertainty for a particle confined to a given region of space. Enter a position uncertainty and a particle mass, and it returns the minimum momentum uncertainty, the equivalent velocity uncertainty, and the Δx·Δp product itself.
The uncertainty principle, published by Werner Heisenberg in 1927, is one of the defining results of quantum mechanics. It says that a particle's position and momentum cannot both be known to arbitrary precision at the same instant: the product of their uncertainties is bounded below by ħ/2, where ħ is the reduced Planck constant. This is not a statement about clumsy instruments, it is a fundamental property of quantum states.
The key relationship is inverse: shrink Δx and Δp must grow, shrink Δp and Δx must grow. Because momentum uncertainty translates into velocity uncertainty through Δv = Δp / m, light particles like electrons are affected far more than heavy ones like protons for the same confinement. This is exactly why electrons cannot be confined inside an atomic nucleus, the implied velocity uncertainty would exceed the speed of light.
This calculator is useful for physics students working through introductory quantum mechanics problems, since it handles the tiny constant ħ and the unit conversions automatically and shows the working at every step.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Electron confined to an atom (0.1 nm)
Example 2 - Electron confined to a nucleus (5 femtometres)
Example 3 - Proton confined to a nucleus (1 femtometre)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Heisenberg uncertainty principle?
The Heisenberg uncertainty principle states that the position and momentum of a particle cannot both be known with unlimited precision at the same time. Mathematically, the product of their uncertainties obeys Δx times Δp is greater than or equal to ħ/2, where ħ is the reduced Planck constant (about 1.0546 x 10^-34 J·s).
How do you calculate the minimum momentum uncertainty?
Rearrange the inequality to Δp minimum equals ħ divided by (2 times Δx). Enter the position uncertainty Δx in nanometres and the calculator returns the smallest possible momentum uncertainty, in kilogram-metres per second, that is consistent with quantum mechanics.
Why does a smaller Δx give a bigger Δp?
Because Δp minimum equals ħ divided by (2·Δx), the two uncertainties are inversely related. Confining a particle to a smaller region of space forces its momentum to become less certain, which is a fundamental feature of quantum mechanics, not a limitation of measuring instruments.
What is the difference between Δp and Δv?
Δp is the momentum uncertainty in kilogram-metres per second. Δv is the velocity uncertainty, found by dividing Δp by the particle's mass (Δv = Δp / m). For the same Δp, a lighter particle such as an electron has a much larger Δv than a heavier one such as a proton.
Why can't electrons exist inside an atomic nucleus?
A nucleus is only a few femtometres across. Confining an electron to that space with the uncertainty principle implies a minimum velocity uncertainty that exceeds the speed of light, which is physically impossible. This is one of the classic arguments showing that nuclei contain protons and neutrons, not electrons.
Is the uncertainty principle about measurement disturbance?
No. It is often explained as measurement disturbing the particle, but the modern understanding is that Δx and Δp are intrinsic properties of the particle's quantum state itself. Even a perfect, disturbance-free measurement cannot beat the ħ/2 limit, because a particle simply does not possess a precise position and momentum simultaneously.
What is ħ, the reduced Planck constant?
ħ (h-bar) is Planck's constant h divided by 2π, equal to about 1.054571817 x 10^-34 joule-seconds. It appears throughout quantum mechanics wherever angular quantities are involved, including the uncertainty principle, angular momentum, and the Schrödinger equation.
Does the uncertainty principle apply to everyday objects?
Yes, but the effect is unmeasurably small. For a 1-gram object with a position uncertainty of 1 micrometre, the minimum velocity uncertainty is around 10^-23 metres per second, far below anything detectable. Quantum uncertainty only becomes significant for particles as light as electrons or protons.
Are there other forms of the uncertainty principle?
Yes. The same mathematical structure applies to energy and time (ΔE·Δt ≥ ħ/2), which limits how precisely an unstable state's energy can be known over a short lifetime, and to other pairs of non-commuting quantum observables such as angular momentum components.
What are typical Δx and Δp values for atoms versus everyday objects?
For an electron confined to an atom (Δx ≈ 0.1 nm), the minimum Δp is around 5 x 10^-25 kg·m/s, comparable to the electron's actual momentum in an orbit. For a macroscopic object like a 1-gram marble confined to 1 micrometre, the minimum Δp is around 10^-28 kg·m/s, utterly negligible next to the marble's normal momentum, which is why quantum uncertainty is invisible in daily life.