Heisenberg Uncertainty Principle Calculator (Δx·Δp ≥ ħ/2)

Understanding the Heisenberg uncertainty principle calculator — concept and principle

This calculator applies Heisenberg's uncertainty principle to find the smallest possible momentum spread of a particle confined within a given distance, together with the matching speed spread and a kinetic-energy scale. It works for electrons, protons, neutrons or any mass you enter, and shows why the principle matters for atoms but not for everyday objects.

How it is calculated

Δx × Δp ≥ ħ ÷ 2, so Δp_min = ħ ÷ (2 Δx)

Δv = Δp ÷ m; energy scale = (Δp)² ÷ 2m (in eV)

Δp in eV/c = Δp × c ÷ e

• ħ = h ÷ 2π = 1.0546 × 10⁻³⁴ J·s; Δx can be entered in pm, nm, µm, mm or m.

The principle

In quantum mechanics a particle is described by a wave. A wave squeezed into a small region must be built from many wavelengths, and since momentum is linked to wavelength by p = h/λ, a narrow position spread forces a wide momentum spread. This is a property of waves, not a limit of measuring instruments. Δx and Δp are standard deviations, and the product of the two can never be smaller than ħ/2.

Worked example

Default inputs: Δx = 0.1 nm, electron.

1. Δp = 1.0546 × 10⁻³⁴ ÷ (2 × 10⁻¹⁰) = 5.273 × 10⁻²⁵ kg·m/s.
2. Δv = 5.273 × 10⁻²⁵ ÷ 9.109 × 10⁻³¹ = 578,838 m/s, which is 0.1931% of light speed.
3. Energy scale = (5.273 × 10⁻²⁵)² ÷ (2 × 9.109 × 10⁻³¹) = 1.526 × 10⁻¹⁹ J = 0.9525 eV.
4. In eV/c: Δp × c ÷ e = 986.635 eV/c.

How to read the result

0.1 nm is about the size of an atom, and the result shows that an electron confined that tightly must have a momentum spread corresponding to around an electronvolt of energy. That is the right order of magnitude for atomic energies, which helps explain why atoms do not collapse. For a 1 g object confined to 1 µm, Δv is far below anything measurable.

Common mistakes

• Using h instead of ħ, or forgetting the factor of 2.
• Treating Δv as the particle's actual speed; it is the spread.
• Applying the non-relativistic formulas when Δv approaches a large fraction of c.

Frequently asked questions

Does the principle come from measurement disturbance?

No. It follows from the wave nature of matter and holds before any measurement.

Why do we not notice it in daily life?

ħ is tiny; for everyday masses the velocity spread is immeasurably small.

Is there an energy–time version?

A related relation ΔE Δt ≳ ħ/2 is often quoted, but time is not an observable in the same sense.

Sources: OpenStax, University Physics Volume 3 (2016), chapter 6 key equations