Mathematics · Quantum Mathematics
Heisenberg Uncertainty Calculator
Calculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Use ℏ/2=5.272859085e-35.
- Minimum Δp=1.054571817e-34÷(2×1e-9)=5.272859085e-26.
- Minimum Δv=5.272859085e-26÷9.1093837139e-31=57883.817946478084.
Problem → model → reason → result
What problem does this model solve?
Calculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
Why does the model apply?
Position and momentum spreads cannot both be made arbitrarily small; their standard deviations obey a lower product bound.
What assumptions does it make?
Values use the displayed units and constants, states follow the stated normalization convention, and each calculation uses the named ideal non-relativistic quantum model.
Formula
ΔxΔp≥ℏ/2; Δvmin=ℏ/(2mΔx)
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
For an electron localized to 1 nm, the minimum momentum uncertainty is about 5.27×10⁻²⁶ kg·m/s.
Common mistake
This is a bound on statistical spreads, not ordinary instrument error or a claim that the particle has no position.
When does this model not apply?
These introductory ideal models do not replace a full Hamiltonian, boundary conditions, uncertainty analysis, relativistic treatment or experimental interpretation for a real quantum system.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Uncertainty principle do?
Calculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
How does the Uncertainty principle work?
The calculator applies ΔxΔp≥ℏ/2; Δvmin=ℏ/(2mΔx). Position and momentum spreads cannot both be made arbitrarily small; their standard deviations obey a lower product bound.
What can I learn from the Uncertainty principle?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.