What Is the Uncertainty Principle?

In February 1927 Niels Bohr left Copenhagen for a skiing holiday in Norway. His 25-year-old assistant, Werner Heisenberg, stayed behind and wrote the paper that introduced the uncertainty principle. It turns 100 next year, and it is probably the most misquoted idea in physics. Heisenberg never called it a principle. Someone else proved the exact inequality found in every textbook. And the explanation nearly everyone gives, that measuring a particle disturbs it, describes a separate claim whose correct form physicists were still arguing over in the 2010s. Here is what the uncertainty principle really says, why it holds, and what it does.

What the principle actually says

The textbook statement is one line:

Δx Δp ≥ ħ/2

Here Δx is the spread in a particle’s position and Δp the spread in its momentum. ħ is Planck’s constant divided by 2π, about 1.05 × 10⁻³⁴ joule-seconds.

The word “spread” carries the whole meaning. Prepare many particles in exactly the same way, measure position on some of them and momentum on others, and look at how scattered the results are. Those scatters are Δx and Δp. The inequality says that no way of preparing the particles can make both scatters small at once. Squeeze one, and the other must grow.

Notice what the statement is not about. It says nothing about clumsy instruments, and nothing about a single measurement disturbing a single particle. It is a property of the state itself. In standard quantum mechanics, there is simply no state in which position and momentum both have sharp values.

It is a property of waves

The reason is that quantum mechanics describes particles with waves, and waves obey this trade-off on their own. Louis de Broglie tied a particle’s momentum to its wavelength: a definite momentum means a single, pure wavelength. But a pure wave repeats forever, so the particle could be anywhere.

To confine a particle to a small region, you must build a wave packet that is large there and zero elsewhere. The only way to make one is to add together waves of many different wavelengths, which is to say many different momenta. The tighter the packet, the wider the mix it needs. That is a theorem about waves, and the uncertainty principle is what it looks like when the waves describe matter.

Squeeze the position, and the momentum spreads

Why nothing you can see is fuzzy

If the principle applies to everything, why does a football not blur? Because ħ is extraordinarily small, and the trade-off only bites when the mass is tiny too.

Take a one-gram ball and pin its position down to a micrometre. The smallest possible spread in its velocity is about 5 × 10⁻²⁶ metres per second. Let that uncertainty run for the entire age of the universe, and it adds up to about 23 nanometres, roughly a hundred atoms across. No measurement of a ball will ever notice.

Now take an electron confined to the size of an atom, about a tenth of a nanometre. Its velocity spread must be at least 580 kilometres per second. That is a sizeable fraction of the speed an electron actually has inside hydrogen. At the scale of atoms the principle is not a footnote. It shapes everything.

Heisenberg’s microscope

Heisenberg’s own 1927 argument was a thought experiment. To see where an electron is, shine light on it and catch the scattered light in a microscope. To see it sharply you need short wavelengths, so imagine gamma rays.

The catch is that light carries momentum. A photon bouncing off the electron gives it a kick, and because the photon could have entered the lens anywhere across its width, the size and direction of that kick are uncertain. A wider lens or a shorter wavelength sharpens the position but makes the possible kick bigger. Work through the optics and the product of the two uncertainties comes out at around Planck’s constant, whatever you choose.

Heisenberg’s microscope

The exact version came from someone else

Heisenberg’s paper gave only a rough estimate, the product being of order h. When Bohr returned from Norway, he objected to how Heisenberg had analysed the microscope, and the published paper carries a postscript acknowledging points Bohr had raised.

The American physicist Earle Kennard proved the precise inequality, Δx Δp ≥ ħ/2 for every possible state, later in 1927. In 1929 Howard Percy Robertson generalised it to any pair of measurable quantities A and B:

ΔA ΔB ≥ ½ |⟨[A, B]⟩|

The bracket [A, B] measures whether the order of two operations matters. For position and momentum it does, and the answer is iħ, which gives back Kennard’s result. Wherever order matters, a trade-off follows, including between spin measured along different directions.

The name did not come from Heisenberg either. He spoke of inaccuracy relations or indeterminacy relations. Arthur Eddington popularised the word “principle” in 1928.

Disturbance is a different question

Kennard’s inequality is about how a state is prepared. Heisenberg’s microscope is about something else: how much a measurement’s error in one quantity forces a disturbance to the other. The two ideas were treated as the same for decades. They are not. The double-slit experiment makes a related point: which-path information, not a kick, is what erases interference.

In 2003 Masanao Ozawa showed that the naive measurement version, error times disturbance at least ħ/2, is not true in general, and proposed a corrected inequality with extra terms. In 2012 two experiments reported beating the naive form: one with neutron spins in Vienna, one with weakly measured photons in Toronto. Some headlines announced that Heisenberg had been proved wrong.

In 2013 Paul Busch, Pekka Lahti and Reinhard Werner replied with a proof that a relation of exactly Heisenberg’s form does hold, provided error and disturbance are defined differently. One of the 2012 teams later acknowledged that the disagreement turns on those definitions. Both results are mathematically sound. They answer subtly different questions.

Why atoms do not collapse

Classical physics has an embarrassing prediction: an electron orbiting a nucleus should radiate its energy away and spiral inward in a fraction of a second. The uncertainty principle gives the quickest explanation of why it does not.

Squeeze an electron into a region of size r and its momentum spread grows to roughly ħ/r, which costs kinetic energy. Meanwhile the pull of the nucleus rewards getting closer. The total energy is roughly:

E(r) ≈ ħ²/2mr² − e²/4πε₀r

Too close and the first term explodes; too far and the second loses its grip. The best compromise sits at about 0.053 nanometres with an energy of about −13.6 electronvolts, which are the real size and binding energy of hydrogen. The exact match is partly luck in choosing ħ/r, and a looser choice gives a different factor. The order of magnitude is not luck.

The same logic explains zero-point energy. A particle trapped in a box can never sit perfectly still, because stillness would need zero momentum spread in a finite space. The full calculation lives in the Schrödinger equation.

Energy and time: a different kind of relation

You will often see a second relation, between energy and time. It looks similar but works differently. In quantum mechanics position and momentum are both measurable quantities with operators, but time is not: it is the clock the theory runs on.

The careful reading, due to Leonid Mandelstam and Igor Tamm in 1945, is that Δt is how long a system takes to change noticeably. A state that lasts only briefly cannot have a sharply defined energy. That is why short-lived particles show up with a spread of masses rather than a single value. The popular story of particles briefly “borrowing” energy from nothing is a loose metaphor for this, not a statement of the rule. For what empty space really contains, see our piece on the quantum vacuum.

Bargaining with Heisenberg

The principle sets a budget, not a wall, and the most sensitive instruments ever built spend that budget deliberately.

The gravitational-wave detector LIGO measures mirror movements far smaller than a proton. Even its laser light carries quantum noise, split between the light’s phase and its amplitude, with a trade-off between them. Since 2019 LIGO has injected “squeezed” light, with less uncertainty in phase and more in amplitude. Quieter phase sharpens high-frequency signals.

The price is paid elsewhere. The extra amplitude noise pushes on the detector’s 40-kilogram mirrors and blurs low-frequency signals. In 2023 LIGO began rotating the squeezing with frequency, using a long filter cavity, so each frequency gets the trade that suits it best. The result, reported in Physical Review X, improved sensitivity across the band. Heisenberg’s limit was never broken. It was negotiated.

Established, contested, unproven

Established. No quantum state has sharp position and momentum at once, and Δx Δp ≥ ħ/2 holds for every state, as Kennard proved. The trade-off follows from the wave nature of matter and generalises to any pair of quantities whose order matters. It explains the size of atoms and zero-point energy, and it is engineered around daily at LIGO.

Contested. How to state the measurement version, error against disturbance. Ozawa’s relation and the Busch, Lahti and Werner relation are both proven, under different definitions, and there is no consensus on which definitions capture Heisenberg’s intuition best. What the spreads mean is also interpretation-dependent: in pilot-wave theory particles do have definite positions at all times.

Unproven. Whether the familiar trade-offs survive unchanged in a theory of quantum gravity. Several approaches propose a minimum measurable length, which would modify the principle at the smallest scales. Experiments so far only set loose limits on such effects, and a decisive test may be a long way off.