The Heisenberg uncertainty calculator above applies the two standard uncertainty relations. Give it a spread in position and it returns the smallest spread in momentum that quantum mechanics permits; give it the lifetime of an excited state and it returns the smallest possible spread in that state's energy. It handles both directions of both pairs, in the units the quantities are actually quoted in.
Arb Digital builds free tools that get the physics right rather than just the arithmetic. The most misunderstood thing about this principle is what it is a statement about, and the section below on that is the reason this page exists. The short version: it is a property of the system, not a limitation of measurement.
What This Heisenberg Uncertainty Calculator Does
It solves two inequalities. The position–momentum relation says the product of the uncertainty in a particle's position and the uncertainty in its momentum can never fall below half the reduced Planck constant. The energy–time relation says the same about the spread in a state's energy and the time window over which it exists.
You choose which quantity you know, and the tool returns the minimum for its partner along with the derived quantities that go with it. In the position–momentum mode it reports the position spread, the momentum spread, the velocity spread for the mass you selected, and the kinetic energy that corresponds to that momentum, expressed in electronvolts. In the energy–time mode it reports both energies in joules and electronvolts, the time, and the natural linewidth in hertz that a spectroscopist would actually measure.
Every answer is a floor. Real states can and usually do have larger uncertainties than the minimum; only a Gaussian wave packet sits exactly on the bound. Nothing can sit below it.
How to Use It
- Pick the quantity you already know. Selecting energy or time automatically switches to the energy–time pair, so you never have to set the pair separately.
- Enter the value in its natural unit. Atomic distances in nanometres or ångströms, nuclear distances in femtometres, excited-state lifetimes in nanoseconds.
- Set the mass if you want a velocity or an energy. The mass plays no part in the uncertainty relation itself; it only converts momentum into velocity and kinetic energy.
- Leave the bound on ℏ/2 unless you have a reason not to. That is the rigorous result. The ℏ option is a rougher convention used for quick estimates and it doubles every answer.
- Read the answer as a minimum. If your measured spread is larger than the calculated floor, nothing is wrong. If it were smaller, something in the measurement would be.
The Formula: How the Uncertainty Bound Is Calculated
The position–momentum relation is Δx × Δp ≥ ℏ ÷ 2, and the energy–time relation is ΔE × Δt ≥ ℏ ÷ 2. The constant ℏ is the reduced Planck constant, equal to the Planck constant divided by 2π, with the CODATA value 1.054571817 × 10−34 J s. You will also see the relation written as Δx × Δp ≥ h ÷ 4π, which is the identical statement, because h ÷ 4π is the same number as ℏ ÷ 2.
Work the default. An electron confined to roughly an atomic diameter, Δx = 0.1 nm, has a minimum momentum spread of Δp = 1.0546 × 10−34 ÷ (2 × 10−10) = 5.273 × 10−25 kg·m/s. Divide by the electron mass and that is a velocity spread of about 579,000 metres per second — roughly a fifth of one per cent of the speed of light. The corresponding kinetic energy, Δp² ÷ 2m, is about 0.95 eV.
Those numbers are the reason atoms are the size they are, and they are derived in full in OpenStax University Physics, The Heisenberg Uncertainty Principle. The quantities in the relation are standard deviations of the distributions predicted by the wavefunction, not error bars on an instrument — a distinction the next section is entirely about.
It Is a Property of the System, Not a Limit of Measurement
This is the point almost every popular account gets wrong, and it is worth being blunt about. The uncertainty principle does not say that measuring a particle's position disturbs its momentum. It does not say your apparatus is too crude. It says that a quantum state which is sharply localised in position simply does not possess a definite momentum — there is no hidden true value that better equipment would reveal.
The reasoning is mathematical rather than experimental. A quantum state is described by a wavefunction, and momentum is related to that wavefunction by a Fourier transform. There is a theorem in Fourier analysis, entirely independent of physics, that a function and its transform cannot both be arbitrarily narrow: squeeze one and the other necessarily spreads. The same theorem governs why a very short pulse of sound cannot have a well-defined pitch. Quantum mechanics inherits that constraint, and ℏ/2 is simply where the constant lands once the physical units are inserted.
Heisenberg's own original explanation used the microscope thought experiment, in which a photon used to locate an electron kicks it and disturbs its momentum. That argument gives roughly the right magnitude, which is why it survives in textbooks, but it describes a different quantity: measurement disturbance, which is a real and separately studied effect. The Kennard inequality that this calculator implements is about the state itself, before anybody measures anything. A perfect, non-disturbing instrument would not change a single digit of the answers above.
Why This Is the Reason Atoms Have a Size
The classical picture of an electron orbiting a nucleus has a fatal problem: an accelerating charge radiates, so the electron should spiral in and the atom should collapse in a fraction of a nanosecond. It does not, and uncertainty is why.
Confining the electron closer to the nucleus reduces Δx, which forces Δp up, which raises the kinetic energy as the square of the momentum. Meanwhile the Coulomb potential energy falls only as one over the distance. Total energy is therefore a sum of a term rising like 1/r² and a term falling like 1/r, and that sum has a minimum at a finite radius. Put the numbers in and the minimum lands near 0.05 nm, which is the Bohr radius. The atom's size is set by the balance between electrostatic attraction and the energy cost of confinement.
The same argument explains a great deal else. Nuclei are femtometres across, and the same calculation with Δx = 1 fm gives a momentum uncertainty implying energies in the tens of MeV, which is exactly the scale of nuclear binding. It also explains why a free electron cannot be confined inside a nucleus, and why white dwarfs and neutron stars resist gravitational collapse: degeneracy pressure is confinement energy under another name. The de Broglie wavelength calculator approaches the same wave-particle relationship from the wavelength side.
Energy, Time and the Width of a Spectral Line
The energy–time relation is subtler than its partner, because time is not an observable in quantum mechanics the way position is. It has no operator, so Δt is not the standard deviation of a measurement of time. It is better read as the characteristic timescale over which the system changes appreciably — most usefully, the lifetime of an unstable state.
Read that way it makes a hard, measurable prediction. An excited atomic state that lives for 10 nanoseconds cannot have a perfectly sharp energy; its energy is spread by at least ℏ/(2 × 10−8) joules, which is a few tens of nanoelectronvolts. Divide by the Planck constant and that becomes a frequency spread of around 8 MHz. This is the natural linewidth, and it is the irreducible floor on how narrow a spectral line can be. Spectroscopists measure it routinely and it agrees.
It also explains why short-lived particles have a mass that is quoted with a width. A resonance living 10−23 seconds has an energy spread of tens of MeV, which is why particle physicists talk about a resonance's width in energy units rather than its lifetime in seconds — the two are the same statement. The linewidth output in the grid does that conversion directly.
Why You Never Notice It in Everyday Life
Run a macroscopic object through the same formula and the reason becomes obvious. Take a one-gram bead whose position you know to within a micrometre. The minimum momentum uncertainty is around 5 × 10−29 kg·m/s, which for a gram is a velocity uncertainty of roughly 5 × 10−26 metres per second. At that speed the bead would take longer than the age of the universe to drift across an atom.
The principle is not switched off for large objects; it is simply irrelevant, because ℏ is about 10−34 in SI units and everything in daily experience is enormous by comparison. The effect matters exactly when the product of the mass and the length scale approaches ℏ, which happens for electrons in atoms, nucleons in nuclei, cold atoms in traps, and increasingly for engineered mechanical resonators that experimentalists have now cooled to their quantum ground state.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Calling it a measurement limitation — it describes what a quantum state is. A perfect, non-disturbing instrument would not lower the bound at all.
- Mixing up ℏ and h — the reduced constant is h divided by 2π. Writing ℏ/2 when you mean h/2 is wrong by a factor of about 6.3.
- Using the bound as an equality — it is a floor. Only a Gaussian wave packet reaches it exactly; every other state sits above it.
- Treating Δt as a measured duration — time has no operator in quantum mechanics. In the energy–time relation it means the timescale over which the system changes, usually a state lifetime.
- Forgetting the mass is only a conversion — it turns momentum into velocity and energy. The position–momentum bound itself contains no mass, so a heavier particle does not obey a different inequality.
Related Free Tools From Arb Digital
The wave nature that produces this bound is quantified by the de Broglie wavelength calculator, and the energy of the photons involved in any spectroscopic measurement comes from the photon energy calculator. Converting a linewidth in hertz to a wavelength is the job of the wavelength calculator. On the classical side, the momentum calculator and the kinetic energy calculator handle the same two quantities without the quantum bound, and the mass energy equivalence calculator covers the rest energies that particle widths are quoted against. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
The product of the uncertainty in position and the uncertainty in momentum is at least the reduced Planck constant divided by two. The same bound applies to the product of an energy spread and the timescale over which a state exists.
No. It is a property of the quantum state itself. A state sharply localised in position does not possess a definite momentum at all, so there is no hidden true value that a better instrument could uncover. Measurement disturbance is a real but separate effect.
It falls out of the mathematics of Fourier transforms, which say a function and its transform cannot both be arbitrarily narrow. The factor of one half is where that general theorem lands once the physical units of quantum mechanics are inserted.
Yes, but the numbers are unnoticeably small. A one-gram bead located to within a micrometre has a velocity uncertainty of about ten to the minus twenty-six metres per second, far below anything that could ever be observed.
Time is not an observable with an operator, so the time term is not a measurement error. It is the timescale over which the system changes, most usefully the lifetime of an unstable state, and it sets the minimum spread in that state's energy.
It is the irreducible frequency width of a spectral line caused by the finite lifetime of the excited state. Divide the minimum energy spread by the Planck constant and you get it directly; a ten nanosecond lifetime gives roughly eight megahertz.
Squeezing an electron closer to the nucleus forces its momentum uncertainty up, and kinetic energy rises as the square of momentum while the attractive potential energy falls only as one over distance. The total energy therefore has a minimum at a finite radius rather than at zero.
This tool is provided for educational use. It computes the minimum uncertainty permitted by the standard inequalities and does not model any particular experimental apparatus or state preparation.
