Uncertainty Principle
The uncertainty principle states that certain pairs of physical properties, most famously position and momentum, cannot both be known to arbitrary precision simultaneously; the product of their uncertainties is bounded below by a fixed constant related to Planck's constant.
Curious
In everyday experience, there's no obstacle in principle to measuring both where something is and how fast it's going at the same time — a radar gun measures a car's speed while you can also see exactly where it is. The uncertainty principle says that at the quantum scale, this isn't just practically hard — it's fundamentally impossible to do both with unlimited precision at once.
This isn't because our measuring instruments are clumsy or because measuring one property physically disturbs the other (though that intuition captures a related idea). It's a mathematical consequence of the wave nature of quantum particles: a wave that is very localised in space is necessarily spread out in terms of the wavelengths (and hence momenta) that make it up, and vice versa.
The effect is utterly negligible for everyday objects — a thrown baseball's position and momentum uncertainties are astronomically smaller than anything measurable — but becomes significant and unavoidable at the scale of electrons, atoms, and photons.
Exploring
The most common form, for position (x) and momentum (p), is:
| Symbol | Quantity | SI Unit |
|---|---|---|
| Δx | Standard deviation (uncertainty) in position | metre (m) |
| Δp | Standard deviation (uncertainty) in momentum | kg·m/s |
| ħ | Reduced Planck constant | J·s |
An analogous relation holds for energy and time, ΔE·Δt ≥ ħ/2, though its interpretation is more subtle since time is not an operator in standard quantum mechanics the way position and momentum are; it is typically understood as relating the lifetime of a state to the uncertainty in its energy.
Worked example: An electron is confined to a region of size Δx = 1 × 10⁻¹⁰ m (roughly the size of an atom). The minimum momentum uncertainty is Δp ≥ ħ/(2Δx) = (1.0546 × 10⁻³⁴)/(2 × 10⁻¹⁰) ≈ 5.27 × 10⁻²⁵ kg·m/s. Dividing by the electron mass (9.109 × 10⁻³¹ kg) gives a minimum velocity uncertainty of roughly 5.8 × 10⁵ m/s — showing why electrons confined to atomic dimensions must have substantial, irreducible kinetic energy, a key reason atoms don't collapse.
Deep Dive
Formal derivation (from the canonical commutation relation): In quantum mechanics, position and momentum are represented by operators x̂ and p̂ satisfying the canonical commutation relation:
The general Robertson uncertainty relation states that for any two observables A and B represented by Hermitian operators, ΔA·ΔB ≥ |⟨[Â,B̂]⟩|/2. Substituting the canonical commutator [x̂,p̂] = iħ gives directly:
This shows the uncertainty principle is not a separate postulate bolted onto quantum mechanics — it is a direct mathematical consequence of position and momentum being non-commuting operators, which is itself tied to the Fourier-transform relationship between position-space and momentum-space wave functions (a highly localised function in one representation is necessarily broad in the Fourier-conjugate representation).
Historical note and interpretation debate: Werner Heisenberg first presented the principle in 1927 using a thought experiment involving a hypothetical gamma-ray microscope, framing it partly in terms of measurement disturbance. Earle Kennard and Howard Robertson later gave the rigorous, disturbance-independent mathematical derivation from the commutator, which is now understood as the principle's more fundamental basis. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics."
Limits and generalisations: The uncertainty principle applies to any pair of non-commuting observables, not only position and momentum — for example, different components of angular momentum or spin do not commute and obey analogous relations. It places no restriction on simultaneously measuring commuting observables (such as position along two different, independent axes) to arbitrary joint precision.
Concept Relationship Map
Requires: wave function, commutator [PENDING]
Enables: zero-point energy explanations, quantum tunnelling context, atomic stability arguments
Commonly confused with: the observer effect / measurement disturbance (a related but distinct, and less fundamental, phenomenon)
Sources
- Heisenberg, W. (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik." Zeitschrift für Physik, 43(3–4), 172–198.
- Robertson, H.P. (1929). "The Uncertainty Principle." Physical Review, 34(1), 163–164.
- Griffiths, D.J., Schroeter, D.F. (2018). Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, Ch. 3.