Physics 1927

On the Physical Content of Quantum-Theoretical Kinematics and Mechanics

Werner Heisenberg

You cannot pin down where a particle is and how fast it goes at once — sharpen one and the other blurs.

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In depth · the introduction

The closer you look at where a particle is, the less you can say about where it's headed — and nature, not your equipment, draws the line.

The big idea

In the everyday world you can know a car's position and its speed at the same time, as precisely as you like. Heisenberg showed that the tiny world doesn't work that way. For a particle like an electron, position and momentum (its mass times velocity) come as a package deal: the sharper you make one, the fuzzier the other becomes. There is a floor — set by Planck's constant — below which the product of the two blurs simply cannot go.

The radical part isn't that measuring is hard. It's that the sharp values aren't there to be found. A particle doesn't secretly have an exact position and an exact speed that we're just too clumsy to read at once. According to quantum mechanics, having both at once is not a thing the world offers.

How it came about

By early 1927 quantum mechanics existed but nobody was sure what it meant. The 25-year-old Heisenberg was working as Niels Bohr's assistant in Copenhagen. When Bohr left for a skiing holiday in Norway in February, Heisenberg stayed behind, pacing the institute at night, wrestling with a simple question: what does it even mean to say where an electron is?

His answer came as a thought experiment — an imaginary microscope that uses light to spot an electron — and a relation he first sent to his friend Wolfgang Pauli in a long letter. He wrote it up and submitted it in March. When Bohr came back and read the manuscript, he didn't celebrate; he argued. Bohr thought Heisenberg's reasoning leaned too hard on picturing the electron as a little ball getting knocked about. The two quarrelled hard, then merged their views into what became the standard “Copenhagen” reading of quantum theory.

Why it mattered

This was the moment physics gave up a 300-year-old dream: that, in principle, everything about a system could be known exactly. Heisenberg's relation said no — and meant it not as a temporary limit but as a law. It is half the reason quantum mechanics feels so strange, and it forced a generation of physicists to rethink what a scientific theory is even allowed to claim.

It is also intensely practical. The same principle explains why atoms are stable, why some materials behave as they do near absolute zero, and it sets the ultimate precision of the finest clocks and sensors humans have built.

A way to picture it

Think of a photograph of a moving cyclist. Use a very fast shutter and you freeze them razor-sharp — but the frozen frame tells you nothing about how fast they were going. Use a slow shutter and they blur into a streak — now you can see the motion, but their exact location is smeared out. Sharpness of place and clearness of motion trade against each other.

A quantum particle is like that photo, except the trade-off isn't a limit of the camera — it's a limit of reality. There is no perfect shutter speed that captures both the exact place and the exact motion at once. Nature only sells the package.

Where it sits

Planck (1900) and Einstein (1905) found that light comes in lumps; Bohr (1913) gave the atom fixed energy levels; Schrödinger (1926) wrote the wave equation that explained them. Heisenberg's uncertainty relation completed the picture by saying what the new theory does not let you know. With Born's probability rule and Bohr's complementarity it formed the Copenhagen interpretation. Its echo runs on to the present: the squeezed light inside the LIGO gravitational-wave detector, and the no-cloning security of quantum cryptography, are uncertainty in modern dress.

The original document

Original source text

W. Heisenberg · Zeitschrift für Physik 43 (1927): 172–198 · submitted March 1927, from Bohr's institute in Copenhagen

The opening: what do “position” and “velocity” even mean?

Heisenberg begins not with an experiment but with words. The trouble, he argues, is that we keep importing classical pictures — a particle with a definite place and a definite speed — into a theory that does not contain them. A word like “position of the electron” has meaning only once you say how you would measure it; and every way of measuring it disturbs what it measures.

The γ-ray microscope

To see an electron's position you must hit it with light; to see it sharply you must use light of very short wavelength. But a short-wavelength photon carries large momentum, and bouncing off the electron it kicks it — the Compton recoil. The sharper your view of where the electron is, the harder the unavoidable kick, and the less you can say about where it is going.

At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum.

From this Heisenberg distils the relation in its original, order-of-magnitude form. Writing q₁ for the imprecision in position and p₁ for the imprecision in momentum, the product cannot be made smaller than roughly Planck's constant:

p₁ q₁ ∼ h. … the more precisely the position is determined, the less precisely the momentum is known, and conversely.

Not a limit of instruments, but of meaning

Crucially, Heisenberg ties the relation to the formalism: it is the direct expression of the non-commuting algebra of his own matrix mechanics, in which position and momentum obey pq − qp = h/2πi. The imprecision is therefore not an experimental nuisance to be engineered away; it is built into the mathematics of the theory. He closes by arguing that quantum mechanics, read this way, is complete and consistent — there are no hidden, sharper values waiting underneath.

[ … ]

The bound here is heuristic — an order of magnitude. Within months Earle Kennard recast it for the standard deviations of position and momentum and proved the exact inequality Δx·Δp ≥ ħ/2 for any state; H. P. Robertson generalised it to any pair of observables in 1929. The sharp form so often written under Heisenberg's name is, strictly, Kennard's.

Copenhagen · 1927