Uncertainty Principle Calculator

Calculator

Pick a relation, set units, then compute a minimum or verify a product against the bound.

Relation

Solve mode

Constant form

Use ħ/2

Use h/(4π)

Both forms are equivalent when ħ = h/(2π).

State factor k (≥ 1)

k = 1 corresponds to the tight (saturating) case.

Δx (position uncertainty)

Δp (momentum uncertainty)

Required for “Check” and “Compute minimum for Input 1”.

Enable batch check (paste pairs: value1,value2 per line)

Batch uses the same units you select above.

Quick notes

Inputs are uncertainties (standard deviations) in typical treatments.
Units are converted to SI for the bound comparison.
Results show a minimum assuming the tight case (k = 1).
Energy–time is a useful estimate, not an operator pair.

Example data

These sample rows illustrate typical scales. Try copying them into the batch box.

Relation	Input 1	Input 2	What to expect
Δx · Δp	Δx = 1 nm	Compute Δp min	Minimum momentum uncertainty is on the order of 10^-26 kg·m/s.
ΔE · Δt	Δt = 1 fs	Compute ΔE min	Minimum energy uncertainty is on the order of 10^-19 J (≈ eV scale).
Δθ · ΔL	Δθ = 1°	Compute ΔL min	Minimum angular momentum uncertainty is a small fraction of ħ.

Formula used

This tool applies the Heisenberg-type bound:

Δx · Δp ≥ ħ/2 (position–momentum)
ΔE · Δt ≥ ħ/2 (energy–time estimate)
Δθ · ΔL ≥ ħ/2 (angle–angular momentum)

You can also use the equivalent form h/(4π), since ħ = h/(2π).

The optional factor k ≥ 1 models states that do not saturate the inequality.

How to use this calculator

Select the relation you want to study.
Choose a solve mode: minimum or product check.
Enter the known uncertainty and pick the unit.
Set k if you expect a looser-than-minimum state.
Press Submit to see results above the form.
Use Download CSV or PDF to save your output.
Enable batch mode to verify multiple measurement pairs.

FAQs

1) What does Δ represent in this tool?

Δ is an uncertainty measure, often treated as a standard deviation. The calculator assumes positive uncertainties and uses SI conversions to evaluate the bound consistently.

2) Why can I choose ħ/2 or h/(4π)?

They are the same bound written differently. Since ħ equals h divided by 2π, substituting gives h/(4π). Pick whichever form you prefer.

3) What is the state factor k?

k lets you model states where the product is larger than the minimum. k = 1 is the tight case, while k > 1 represents additional spread from the system or preparation.

4) Is the energy–time relation always strict?

Energy–time is commonly used as an estimate for lifetimes and bandwidths. Time is not an operator like position, so interpretations vary. This tool treats it as a practical bound-style estimate.

5) What units should I use for momentum?

Use kg·m/s for SI momentum. If you use eV/c, the calculator converts it to SI using 1 eV = 1.602×10^-19 J and c = 299,792,458 m/s.

6) Why does the minimum change when I change units?

The physics does not change, but the displayed number depends on the chosen unit scale. Internally, the tool converts to SI, computes the result, then converts back to your selected unit.

7) What does “Check product vs limit” mean?

You enter both uncertainties and the tool computes their product, then compares it to the bound. It also reports the ratio product/limit to show how close you are.

8) Can I export multiple rows at once?

Yes. Enable batch mode and paste lines like “value1,value2”. The tool evaluates each pair using your selected units and includes the full batch table in the CSV export.