Heisenberg Uncertainty Calculator
The Heisenberg uncertainty principle is a fundamental limit on knowledge, not a measurement limitation. A particle cannot simultaneously have a precise position and momentum — it is inherent in wave-particle duality.
What the uncertainty principle actually says
The Heisenberg uncertainty principle is not a statement about clumsy measuring instruments. It says that position and momentum are jointly undefined in quantum mechanics: a particle simply does not possess a sharp value of both at once. Squeeze the spread in one and the spread in the other necessarily grows.
The relationship is an inequality, not an equation. It sets a floor on the product of the two uncertainties. Real experiments almost always land above that floor — only a minimum-uncertainty state (a Gaussian wave packet) sits exactly on it.
The same structure applies to energy and time, which is why short-lived excited states have measurably broadened spectral lines. That link between lifetime and linewidth is one of the most directly observable consequences of the principle.
Formula Reference Table
Δx · Δp ≥ ℏ / 2 and ΔE · Δt ≥ ℏ / 2| Symbol | Meaning |
|---|---|
| Δx | uncertainty in position (m) — the standard deviation, not a measurement error bar |
| Δp | uncertainty in momentum (kg·m/s) |
| ΔE | uncertainty in energy (J) |
| Δt | characteristic time over which the state changes (s) |
| ℏ | reduced Planck constant = h / 2π = 1.054571817 × 10⁻³⁴ J·s (exact by definition of h) |
You will also see this written Δx · Δp ≥ h / 4π. That is the identical statement — h/4π and ℏ/2 are the same number, 5.2728 × 10⁻³⁵ J·s. Older texts sometimes quote Δx · Δp ≥ ℏ, a looser order-of-magnitude version.
3 Worked Examples
An electron is localised inside an atom, so its position is known to roughly the atomic diameter, Δx = 1.0 × 10⁻¹⁰ m. How fast is it forced to move?
- Δp ≥ ℏ / (2Δx) = (1.0546 × 10⁻³⁴) / (2 × 1.0 × 10⁻¹⁰)
- Δp ≥ 5.27 × 10⁻²⁵ kg·m/s
- Δv = Δp / mₑ = (5.27 × 10⁻²⁵) / (9.109 × 10⁻³¹ kg)
This is why electrons cannot simply sit still in a nucleus-sized region — confining them harder would push the velocity spread relativistic. It is also a rough justification for the size of atoms.
A grain of dust of mass 1.0 × 10⁻⁹ kg has its position pinned to Δx = 1.0 × 10⁻⁶ m under a microscope.
- Δp ≥ ℏ / (2Δx) = (1.0546 × 10⁻³⁴) / (2 × 1.0 × 10⁻⁶) = 5.27 × 10⁻²⁹ kg·m/s
- Δv = Δp / m = (5.27 × 10⁻²⁹) / (1.0 × 10⁻⁹)
At that speed the grain would take longer than the age of the universe to drift one atomic diameter. The principle applies to every object; it is only consequential when mass is tiny.
An atomic excited state has a lifetime of 1.0 × 10⁻⁸ s. What is the minimum spread in its energy?
- ΔE ≥ ℏ / (2Δt) = (1.0546 × 10⁻³⁴) / (2 × 1.0 × 10⁻⁸)
- ΔE ≥ 5.27 × 10⁻²⁷ J
- Convert: (5.27 × 10⁻²⁷) / (1.602 × 10⁻¹⁹ J/eV)
Shorter-lived states give broader lines. This is a real, measured effect in spectroscopy, and it is why very short-lived particles have a mass 'width' rather than a single mass.
Constants used
| Constant | Symbol | Value |
|---|---|---|
| Planck constant | h | 6.62607015 × 10⁻³⁴ J·s (exact) |
| Reduced Planck constant | ℏ = h/2π | 1.054571817 × 10⁻³⁴ J·s |
| ℏ/2 (the uncertainty floor) | ℏ/2 | 5.27285908 × 10⁻³⁵ J·s |
| Electron mass | mₑ | 9.1093837015 × 10⁻³¹ kg |
| Proton mass | m_p | 1.67262192369 × 10⁻²⁷ kg |
| Electronvolt | eV | 1.602176634 × 10⁻¹⁹ J (exact) |
Which form of the inequality to use
| Form | Right-hand side | When you'll see it |
|---|---|---|
| Δx·Δp ≥ ℏ/2 | 5.273 × 10⁻³⁵ | Modern standard. Correct for Gaussian wave packets. Use this unless told otherwise. |
| Δx·Δp ≥ h/4π | 5.273 × 10⁻³⁵ | Identical value, common in intro textbooks that avoid ℏ. |
| Δx·Δp ≥ ℏ | 1.055 × 10⁻³⁴ | Older order-of-magnitude estimate. Twice as strict; fine for scaling arguments. |
| Δx·Δp ≥ h | 6.626 × 10⁻³⁴ | Heisenberg's original 1927 heuristic. Do not use for numeric problems. |
If a homework answer is out by a factor of 2 or 4π, it is nearly always a mismatch between these forms.
Typical uncertainty scales
| System | Δx | Minimum Δv |
|---|---|---|
| Electron in an atom | 10⁻¹⁰ m | ≈ 5.8 × 10⁵ m/s |
| Electron in a nucleus | 10⁻¹⁵ m | ≈ 5.8 × 10¹⁰ m/s (impossible — electrons are not nuclear constituents) |
| Proton in a nucleus | 10⁻¹⁵ m | ≈ 3.2 × 10⁷ m/s |
| Buckyball (C₆₀, ≈1.2 × 10⁻²⁴ kg) | 10⁻⁹ m | ≈ 4.4 × 10⁻² m/s |
| Dust grain (10⁻⁹ kg) | 10⁻⁶ m | ≈ 5.3 × 10⁻²⁰ m/s |
Common Mistakes to Avoid
Δx is the standard deviation of the quantum probability distribution — an intrinsic property of the state. A perfect instrument does not reduce it. Averaging many measurements does not reduce it either.
If you use h = 6.626 × 10⁻³⁴ you must divide by 4π. If you use ℏ = 1.055 × 10⁻³⁴ you divide by 2. Using h with /2 gives an answer 6.28× too large.
The formula gives a minimum. A real state can, and usually does, have a much larger product. Only a Gaussian wave packet achieves equality.
In the ΔE·Δt form, Δt is the timescale on which the system's state appreciably changes — a lifetime, not how long you watched. Time is not an operator in standard quantum mechanics, so this relation has a different footing from the position–momentum one.
Frequently Asked Questions
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Formula Explorer connections
Interpretation: This relationship connects quantized energy, wavelength, probability, nuclear mass or radioactive change. Assumption: Use the correct particle, quantum state, nuclide and energy units. Idealized potentials, nonrelativistic motion or single decay channels may be assumed.