Heisenberg Uncertainty Calculator
Heisenberg's uncertainty principle, formulated in 1927, is one of the cornerstones of quantum mechanics. It states that the uncertainties in position (delta_x) and momentum (delta_p) of a particle are related by delta_x * delta_p >= hbar/2, where hbar = 1.054571817 x 10^-34 J s. This is not about imprecise instruments: it reflects the fundamental wave nature of matter. Specifying position precisely (small delta_x) forces momentum to be spread over a wide range (large delta_p), and vice versa. This calculator computes the minimum uncertainty in one variable given the known uncertainty in the other, or computes the uncertainty product to check whether a given state satisfies the principle.
Heisenberg uncertainty principle formula
delta_x * delta_p >= hbar / 2
hbar = h / (2 * pi) = 1.054571817e-34 J s
Minimum: delta_p_min = hbar / (2 * delta_x)
The minimum uncertainty state corresponds to a Gaussian wavepacket (coherent state). For an electron localized to 1 nm, the minimum momentum uncertainty is about 5.27 x 10^-26 kg m/s, corresponding to a velocity uncertainty of about 5.8 x 10^4 m/s.
Physical examples
- Electron in a hydrogen atom (Bohr radius = 5.29 x 10^-11 m): minimum momentum uncertainty ~ 9.97 x 10^-25 kg m/s, consistent with the observed electron velocity.
- A proton confined to a nucleus (radius ~ 10^-15 m): momentum uncertainty ~ 5.27 x 10^-20 kg m/s, giving kinetic energy in the MeV range, confirming nuclear energy scales.
- A 1 kg ball localized to 10^-10 m: minimum momentum uncertainty ~ 5.27 x 10^-25 kg m/s, completely negligible at macroscopic scale.
Heisenberg uncertainty: frequently asked questions
What is the Heisenberg uncertainty principle?
The Heisenberg uncertainty principle states that the product of the uncertainties in position and momentum of a quantum particle must satisfy delta_x * delta_p >= hbar/2, where hbar = h/(2 pi). This is not a limitation of measurement technology but a fundamental feature of quantum mechanics: quantum states simply do not have definite position and momentum simultaneously.
What is hbar?
hbar (h-bar) is the reduced Planck constant, equal to h / (2 pi) = 1.054571817 x 10^-34 J s. It appears naturally in quantum mechanics because many quantum phenomena involve angular frequencies (2 pi times ordinary frequency) rather than ordinary frequencies.
What does the minimum uncertainty mean?
The minimum uncertainty state (equality in the inequality) corresponds to a Gaussian wave packet, called a coherent state or minimum uncertainty state. Real particles can have larger uncertainties than the minimum, but never smaller. The calculator gives the minimum possible uncertainty in one variable given the other.
Is the uncertainty principle related to the observer effect?
The observer effect (that measuring a system disturbs it) is related but distinct. The Heisenberg uncertainty principle is a property of quantum states themselves, independent of how we measure. Even in principle, no matter how perfect the measurement device, both quantities cannot be known simultaneously to arbitrary precision.
Does the uncertainty principle apply to energy and time?
Yes. An analogous relation holds: delta_E * delta_t >= hbar/2. This has real consequences: a quantum state with a short lifetime delta_t has an energy uncertainty delta_E >= hbar/(2 delta_t), which is why atomic spectral lines have a natural linewidth rather than being infinitely sharp.
Official sources
- NIST CODATA 2018: Reduced Planck Constant.
- OpenStax University Physics Vol. 3: The Heisenberg Uncertainty Principle.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.