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Minimum-Product Position and Momentum Uncertainty

Find the saturated position–momentum uncertainty boundary, or relate its momentum width to rest mass and low-speed velocity width.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For position width 1 nm, the minimum momentum width is about 5.27286 × 10⁻²⁶ kg·m/s. With one atomic mass unit, the low-speed velocity width is about 31.7539 m/s.

σx σp = h/(4π) at the minimum-product boundary. For nonrelativistic motion σp = m σv; h = 6.62607015 × 10⁻³⁴ J·s exactly.

Worked example

Enter these known values and leave the other values blank.

Position standard deviation
1 nm
Rest mass for low-speed relation
1 u
Momentum standard deviation (kg·m/s)
52.73 × 10⁻²⁷ kg⋅m/s
Low-speed velocity standard deviation
31.754 m/s

Assumptions and limitations

  • This page solves the saturated, minimum-product boundary σxσp = ħ/2. The general Heisenberg relation is σxσp ≥ ħ/2; ordinary states may have a larger product. Do not use the equality to certify measured uncertainty.
  • Widths are standard deviations of one position component and its conjugate momentum component in a specified quantum state. They are not instrument resolution or simultaneous measurement error bars.
  • The rest-mass and velocity-width branch uses nonrelativistic momentum p = mv. It is suitable only for a low-speed state with momentum spread well below mc; mean motion must also be nonrelativistic. A small velocity width alone cannot establish that assumption.
  • Position and momentum widths alone do not require a mass or the nonrelativistic approximation. Their bound uses the canonical position and momentum relation.
  • All four quantities remain editable and all 47 unit choices are retained. Momentum exponent selectors scale values in kg·m/s, while mass selectors include kg, customary pounds and particle mass multiples.
  • Planck’s constant is exact in SI. Particle and atomic mass selectors use 2022 CODATA central values: me = 9.1093837139×10⁻³¹ kg, mp = 1.67262192595×10⁻²⁷ kg, mn = 1.67492750056×10⁻²⁷ kg and u = 1.66053906892×10⁻²⁷ kg. Their experimental uncertainties are not propagated.
  • All widths and mass must be strictly positive. Zero position width would require unbounded momentum width and is outside this finite calculator. Missing values remain incomplete; mutually inconsistent locked quantities produce a conflict.
  • Doubling position width halves its boundary momentum width. At fixed mass it also halves the modeled velocity width. These scaling relationships describe the minimum-product model, not an arbitrary quantum state.

Common questions

Does uncertainty always equal h divided by four pi?

No. The general position–momentum uncertainty product is at least h/(4π). Equality applies to a minimum-product state, which is the boundary solved here.

Can I solve the bound without a mass?

Yes. Enter a positive position width or momentum width; its conjugate boundary width follows without using mass.

What does velocity uncertainty mean here?

It is the standard deviation of one velocity component within a nonrelativistic state, related to the momentum width by σp = mσv.

Why does zero position width fail?

With a nonzero lower product, zero position width has no finite compatible momentum width. This calculator rejects zero instead of displaying infinity.

References

The position–momentum bound uses the exact SI Planck constant. Equality describes the minimum-product boundary; arbitrary quantum states can have a larger product. All widths and rest mass must be positive. Particle mass selectors use 2022 CODATA central values without uncertainty propagation. The velocity-width relationship assumes a nonrelativistic state and rejects widths at or above c.

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