Heisenberg Uncertainty Calculator
Calculate position-momentum or energy-time uncertainty limits using the Heisenberg uncertainty principle.
What Is the Heisenberg Uncertainty Principle?
The Heisenberg uncertainty principle states that certain pairs of physical properties cannot both be known with arbitrary precision at the same time. For position and momentum, the product of their uncertainties is bounded below by half of the reduced Planck constant.
Position-Momentum Relation
$$\Delta x \Delta p \geq \frac{\hbar}{2}$$\( \Delta x \) is the spread in position and \( \Delta p \) is the spread in momentum. If you localize a particle tightly in space, its momentum uncertainty must grow. A 0.1 nm position window implies a minimum momentum uncertainty on the order of \( 10^{-24} \) kg·m/s.
Energy-Time Relation
$$\Delta E \Delta t \geq \frac{\hbar}{2}$$This form limits how precisely energy and lifetime (or measurement duration) can be defined together. Short-lived excited states have broader energy spreads, which is why spectral lines have finite width.
Explore related tools: Avogadro Number Calculator and Kinematics Calculator.
Frequently Asked Questions
What is ℏ in the formula?
ℏ (h-bar) is the reduced Planck constant, equal to h divided by 2π. Its value is about 1.055 × 10⁻³⁴ J·s, so ℏ/2 is about 5.27 × 10⁻³⁵ J·s.
Does the uncertainty principle mean measurement error?
Not exactly. It is a fundamental limit on how sharply position and momentum can be defined for a quantum state, independent of instrument quality.
When should I use the energy-time form?
Use it for transitions, resonance widths, and short-lived states where energy spread and lifetime are linked.
What units does this calculator use?
Position is in nanometers, momentum in ×10⁻²⁵ kg·m/s, energy in electron volts, and time in femtoseconds. Results are shown in joule-seconds.
What if my product is below ℏ/2?
That combination of uncertainties is not allowed for a single quantum state. Increase one uncertainty or reduce the other so the product meets the bound.
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