Position–momentum uncertainty lower bound
For position and its conjugate momentum component, quantum mechanics requires the product of their standard deviations to be at least ħ/2. The calculator solves the equality to report the smallest bound compatible with the entered uncertainty.
This is not a statement about poor instruments or disturbance alone. It concerns the statistical spread of measurement outcomes for a quantum state. Most states have an uncertainty product greater than the minimum.
How to use the Heisenberg uncertainty calculator
- Choose the unknown: Select position or momentum standard deviation.
- Enter a positive standard deviation: Use SI units and a strictly positive value.
- Calculate the bound: The result is the equality-case minimum, not a prediction for every state.
- Interpret the quantum state: Actual uncertainty can be larger depending on the wavefunction.
Formula and variables
The reduced Planck constant is ħ = 1.054571817 × 10⁻³⁴ J·s, an exact value under the SI definition of h.
ΔxΔp ≥ ħ/2; minimum Δp = ħ/(2Δx)- Δx — Position standard deviation
- Statistical spread of position along one coordinate (m)
- Δp — Momentum standard deviation
- Spread of the conjugate momentum component (kg·m/s)
- ħ — Reduced Planck constant
- Planck constant divided by 2π (J·s)
Atomic-scale localization
Position standard deviation is 1.0 × 10⁻¹⁰ m.
- Δx
- 1.0 × 10⁻¹⁰ m
- minimum Δp = 1.054571817 × 10⁻³⁴ / (2 × 10⁻¹⁰)
Result: Minimum momentum standard deviation is approximately 5.27286 × 10⁻²⁵ kg·m/s.
A state localized to that position spread cannot have a smaller conjugate momentum spread.
Understanding your results
The equality is a lower bound
The calculator returns the minimum uncertainty paired with the entered standard deviation.
- Gaussian minimum-uncertainty states can attain equality.
- Other states have ΔxΔp greater than ħ/2.
- The quantities are standard deviations, not simple resolution widths unless a definition connects them.
- Position and momentum components must refer to the same spatial axis.
Assumptions
- Δx and Δp are standard deviations for conjugate components in one quantum state.
- The canonical nonrelativistic position–momentum relation applies.
- The equality case is used only to calculate the lower bound.
Limitations
- Does not determine an actual wavefunction or measured distribution.
- Does not cover generalized uncertainty relations, angular variables, relativistic localization, or energy–time interpretations.
- Does not convert between momentum uncertainty and velocity uncertainty for a specified particle mass.
Common mistakes
- Interpreting the result as guaranteed actual uncertainty rather than a minimum.
- Using full widths or instrument resolution as standard deviations without conversion.
- Pairing position on one axis with momentum on another.
- Attributing the relation only to measurement disturbance.
Practical use cases
Quantum mechanics coursework
Check equality-bound calculations for position and momentum spreads.
Scale comparison
Explore how tighter localization raises the minimum conjugate momentum spread.
Frequently asked questions
Does the calculator give the actual momentum uncertainty?
No. It gives the minimum permitted by the inequality for the entered position standard deviation.
Can uncertainty be zero?
Not for either member of this conjugate pair in a normalizable state with finite uncertainty in the other.
Is this caused by inaccurate instruments?
No. The standard relation describes intrinsic statistical properties of quantum states, though measurement procedures have their own additional uncertainties.
Sources and review
- The Heisenberg Uncertainty Principle — OpenStax University Physics Volume 3. Accessed 2026-07-13.
- 2022 CODATA Recommended Values of the Fundamental Physical Constants — National Institute of Standards and Technology. Accessed 2026-07-13.
Reviewed 2026-07-13.