Heisenberg's Uncertainty Principle Calculator

Your details

Position-momentum is the most common form. Energy-time governs the natural linewidth of excited states and unstable particles.
Select which quantity you already know. The calculator solves for the remaining one.
The standard deviation in the measured position of the particle, in metres. 1e-10 m = 1 Angstrom, roughly the size of an atom.
m
The standard deviation in the measured momentum of the particle, in kg*m/s.
kg m/s
The standard deviation in velocity, in m/s. Requires a particle mass to convert to momentum uncertainty.
m/s
Preset particle masses for common quantum-mechanics problems. Choose 'Custom mass' to enter your own.
Minimum uncertainty
sigma_p = 5.273 x 10^-25 kg m/s

The minimum allowed uncertainty for the unknown variable, from the Heisenberg equality.

Position uncertainty (sigma_x)1.000 x 10^-10m
Momentum uncertainty (sigma_p)5.273 x 10^-25kg m/s
Velocity uncertainty (sigma_v)5.788 x 10^5m/s
hbar / 2 (minimum product)5.273 x 10^-35

Minimum uncertainty for an electron: sigma_p = 5.273 x 10^-25 kg m/s

  • This result is a hard lower bound: no experiment can do better, regardless of how refined the instrument.
  • The smaller sigma_x (the more precisely you pin down position), the larger sigma_p must be, and vice versa.
  • This trade-off is not due to clumsy measurement. It is a fundamental property of quantum waves.
  • For an electron confined to atomic scales (~0.1 nm), the implied velocity uncertainty is hundreds of km/s, which is why electrons in atoms cannot be at rest.

Next stepSwitch to 'Velocity uncertainty' mode if you want to see what this momentum spread implies for the speed of the particle.

 = Powered by OnlyCalculators