Heisenberg's Uncertainty Principle Calculator
Enter any two known values to compute the remaining uncertainty. Choose between the position-momentum pair or the energy-time pair, pick a particle mode (electron, proton, neutron, or custom mass), and the calculator applies the minimum-uncertainty equality, sigma_x * sigma_p = hbar / 2, or Delta_E * Delta_t = hbar / 2. The "Show your work" panel walks through each arithmetic step with your exact numbers.
Formula
Worked example
An electron is confined to a region of 1 Angstrom (1e-10 m). The minimum momentum uncertainty is hbar/2 / sigma_x = 5.273e-35 / 1e-10 = 5.273e-25 kg m/s. Dividing by the electron mass (9.109e-31 kg) gives a velocity uncertainty of about 5.79e5 m/s, roughly 0.19% of light speed.
What is Heisenberg's uncertainty principle?
Heisenberg's uncertainty principle, formulated by Werner Heisenberg in 1927, states that the product of the standard deviations in position and momentum of any quantum particle cannot be smaller than half the reduced Planck constant: sigma_x * sigma_p >= hbar/2. A companion relation applies to energy and time: Delta_E * Delta_t >= hbar/2. These are not statements about imperfect instruments. They reflect the wave nature of quantum particles: a wave localised precisely in space must, by Fourier analysis, be made up of many different wavelengths, each corresponding to a different momentum. The more tightly you localise the wave, the wider the spread of constituent wavelengths must be.
Position-momentum uncertainty in practice
For everyday objects like baseballs and dust grains, the uncertainty bound is so tiny it is physically meaningless. For a 145-gram baseball known to within 1 mm, the minimum velocity uncertainty is about 3.7e-31 m/s, far below any conceivable measurement error. But for an electron in a hydrogen atom, confined to a radius of about 53 picometres, the momentum uncertainty implies a speed spread of roughly one million metres per second. This zero-point motion is why atomic electrons do not spiral into the nucleus: collapsing to a point would require infinite momentum uncertainty, which costs enormous kinetic energy. The principle also underpins scanning tunnelling microscopy, electron diffraction, and the design of quantum dots and semiconductor devices.
Energy-time uncertainty and natural linewidth
The energy-time form of the uncertainty relation governs how sharply defined the energy of a quantum state can be. If an excited atom has a mean lifetime of Delta_t before decaying, its energy is uncertain by at least hbar / (2 * Delta_t). This spread in energy corresponds to a spread in the frequency of emitted photons, observed as the natural linewidth of a spectral line. A state with a 10-nanosecond lifetime has a linewidth of about 33 MHz. Very short-lived particles such as the W and Z bosons have lifetimes around 3e-25 s, giving decay widths of about 2 GeV, which is directly measurable in particle colliders. The energy-time relation is also central to nuclear magnetic resonance, laser physics, and the precision of atomic clocks.
How to use this calculator
Select 'Position and momentum' or 'Energy and time' from the 'Uncertainty pair' dropdown. For position-momentum, choose whether you know sigma_x, sigma_p, or sigma_v (velocity uncertainty requires a mass). Pick a preset particle (electron, proton, neutron) or enter a custom mass. For energy-time, choose whether you know Delta_t or Delta_E. Enter the known value and all unknowns are computed to the minimum-uncertainty limit (equality). The 'Show your work' panel displays each arithmetic step with your exact numbers, and the reference table below gives intuition for how the principle behaves across scales from nuclear to macroscopic.
Typical uncertainty values for real quantum systems
| System | sigma_x (m) | sigma_p (kg m/s) | sigma_v (m/s) | Significance |
|---|---|---|---|---|
| Electron in hydrogen atom | 5.3 x 10^-11 | 9.9 x 10^-25 | ~1.1 x 10^6 | ~0.4% of light speed - quantum zero-point motion |
| Proton in nucleus | 1.0 x 10^-15 | 5.3 x 10^-20 | ~3.2 x 10^7 | ~10% of c - confirms nuclear quantum confinement |
| Dust grain (1 microgram, 1 nm) | 1.0 x 10^-9 | 5.3 x 10^-26 | ~5.3 x 10^-17 | Negligible - quantum effects vanish at macroscopic scales |
| Baseball (145 g, 1 mm) | 1.0 x 10^-3 | 5.3 x 10^-32 | ~3.7 x 10^-31 | Completely unmeasurable - explains why baseballs obey classical physics |
| Excited atom (lifetime 10 ns) | N/A | N/A | N/A | Delta_E >= 5.3 x 10^-27 J (natural linewidth ~33 MHz) |
Illustrative minimum uncertainties computed from the Heisenberg equality at atomic and nuclear scales.
Frequently asked questions
Is the uncertainty principle caused by measurement disturbance?
No. A common misconception is that measuring one quantity 'disturbs' the other. The principle is deeper: quantum particles are described by wavefunctions, and a wavefunction sharply localised in position is, by the mathematics of Fourier transforms, necessarily spread over many momentum values. The uncertainty is intrinsic to the quantum state, not a consequence of clumsy measurement. That said, there is a separate and real effect called measurement back-action where a measuring apparatus does disturb the system, but this is distinct from the Heisenberg limit.
Why does the principle use hbar/2, not h or hbar?
The precise lower bound depends on the definition used. The Robertson-Schrodinger inequality, which is the rigorous quantum-mechanical derivation, gives sigma_x * sigma_p >= hbar/2 for standard deviations. Some textbooks and older sources write Delta_x * Delta_p >= hbar or h/(4*pi), which is the same as hbar/2. Others write >= h/(2*pi) = hbar, which is looser. This calculator uses the strict minimum hbar/2, the smallest possible product of the two standard deviations.
Does the uncertainty principle apply to macroscopic objects?
Mathematically yes, but the effect is too small to observe or matter. For a 1-gram object known to within 1 micrometre, the implied momentum uncertainty is about 5.3e-29 kg m/s, which corresponds to a velocity uncertainty of 5.3e-26 m/s. No instrument in existence can measure velocity to that precision, so quantum uncertainty is completely buried under classical measurement noise and thermal fluctuations for any macroscopic object.
What is the reduced Planck constant (hbar)?
The reduced Planck constant is hbar = h / (2*pi), where h = 6.626e-34 J*s is Planck's constant. Its value is approximately 1.0546e-34 J*s. It appears naturally in quantum mechanics because angular frequency (radians per second) is more fundamental than ordinary frequency (cycles per second) in the wave equations of quantum theory. The factor of 1/2 in the uncertainty bound hbar/2 comes from the Robertson inequality applied to position and momentum operators.
How does the energy-time uncertainty relate to particle decay widths?
Unstable particles and excited states have a characteristic lifetime tau. The uncertainty in their rest-mass energy is at least hbar / (2*tau), called the decay width Gamma. For the W boson (tau ~ 3e-25 s), Gamma is about 2.1 GeV, directly measured at colliders. For a typical atomic excited state (tau ~ 10 ns), the linewidth is just tens of megahertz. This is why sharper spectral lines always come from longer-lived states, and it sets a fundamental limit on the precision of any spectroscopic measurement.
Can I use this calculator for relativistic particles?
The position-momentum inequality sigma_x * sigma_p >= hbar/2 is valid even relativistically, because it involves relativistic momentum p = gamma*m*v. However, the velocity-uncertainty conversion (sigma_v = sigma_p / m) is only accurate for non-relativistic speeds (v much less than c). For particles moving near light speed, use the relativistic relation between momentum and velocity: p = m*v / sqrt(1 - v^2/c^2).