Uncertainty principle

• 1. Introduction

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  The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a foundational idea in quantum mechanics. The German physicist Werner Heisenberg first put it forward, and he later summed it up in one blunt sentence. He wrote: "It is impossible to determine accurately both the position and the direction and speed of a particle at the same instant." That sounds like a complaint about clumsy instruments. It is not. The principle says that certain pairs of properties, such as position and momentum, come with a built-in trade-off. Sharpen one and the other blurs. Nobody notices this in everyday life, because it is indiscernible at the scales humans experience. Yet it reaches into the lifetimes of unstable particles, the behavior of sound waves, and long arguments among leading physicists. Where does the trade-off come from? Why did Heisenberg's first explanation need an addendum? Who tried to break the principle, and who rejected it altogether?

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• 2. Why Waves Trade Off

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  Think of a pure tone in a sound wave. Its frequency is a single sharp spike, yet in time the sound is a sine wave that is not localized anywhere. Mathematicians call the two descriptions Fourier transforms of one another. A nonzero function and its Fourier transform cannot both be sharply localized at the same time.

  The de Broglie hypothesis carries that logic into matter. It says every object in the universe is associated with a wave, from elementary particles, atoms and molecules up to planets. A particle's position then takes the form of a matter wave, and its momentum is the Fourier conjugate of that wave. A single plane wave has one exact momentum, but the particle could be essentially anywhere along it. Adding many plane waves together squeezes the wave packet into a smaller region, which sharpens the position. The cost is that momentum becomes a mixture of waves with many different momenta. The exact floor on this trade-off is called the Kennard bound.

  Matrix mechanics reaches the same destination by a more abstract road. It represents position and momentum with self-adjoint operators, and the order in which they act matters. Their commutator, the canonical commutation relation, forbids any quantum state from being both a position eigenstate and a momentum eigenstate. Measuring a particle's position leaves it in a position eigenstate. That state is a sum of many momentum eigenstates, so the momentum must be less precise. The wave picture is more visually intuitive, while the matrix picture generalizes more easily.

  Angular momentum obeys a version of the same rule. Unless all three components vanish together, only one component of a system's angular momentum can be defined with arbitrary precision. That component is normally the one parallel to an external magnetic or electric field.

  The ground state of a quantum harmonic oscillator sits exactly on the Kennard bound, with a probability density shaped like a normal distribution. So does every coherent state, and so does every squeezed coherent state.

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• 3. Copenhagen Years

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  In 1925 Heisenberg published the Umdeutung paper, a German word meaning reinterpretation. It showed that the central feature of quantum theory was non-commutativity, so the relative order of a position measurement and a momentum measurement was significant. Working with Max Born and Pascual Jordan, he kept developing matrix mechanics, which became the first modern formulation of quantum mechanics.

  In March 1926, working in Bohr's institute, Heisenberg realized that non-commutativity implied the uncertainty principle. In February 1927 he was writing to Wolfgang Pauli and working out the basic concepts.

  His celebrated 1927 paper on the perceptual meaning of quantum kinematics and mechanics named a minimum amount of unavoidable momentum disturbance caused by any position measurement. It gave no precise definition of the uncertainties in position and momentum. Heisenberg offered plausible estimates case by case instead. The paper's analysis relied on a microscope argument that Bohr showed was incorrect, so Heisenberg added an addendum to the publication.

  The precise inequality came from others. Earle Hesse Kennard proved the modern version in 1927, and Hermann Weyl derived it in 1928. Heisenberg had proved the relation only for the special case of Gaussian states. In 1929 Howard Percy Robertson generalized the inequality to all observables. In 1930 Erwin Schrödinger extended it to allow nonzero covariance between the operators, and the result is called the Robertson-Schrödinger inequality. Heisenberg refined his own statement in a 1930 lecture in Chicago.

  Even the name took years to settle. In the main body of his paper, Heisenberg used the German word Ungenauigkeit, meaning imprecision. Only in an endnote did he switch to Unsicherheit, meaning uncertainty. Later he always used Unbestimmtheit, meaning indefiniteness. The English edition of his textbook, The Physical Principles of the Quantum Theory, appeared in 1930 and used only the word uncertainty. That English word became the standard term.

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• 4. The Microscope Problem

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  Heisenberg's imaginary microscope puts an experimenter in front of an electron and hands them a photon to fire at it. Two trade-offs follow. A photon with a short wavelength has a large momentum and fixes the position accurately. But it scatters in a random direction and hands the electron a large, uncertain kick. A photon with a long wavelength barely disturbs the electron's momentum, yet it reveals the position only vaguely.

  The aperture creates a second trade-off. With a large aperture, the electron's location is well resolved. But conservation of momentum means the transverse momentum of the incoming photon affects the electron's momentum along the beamline, so that momentum is resolved poorly. A small aperture reverses the situation.

  Together these trade-offs mean that no choice of wavelength or aperture beats a lower limit on the product of the two uncertainties. That limit equals the Planck constant, up to a small numerical factor. Heisenberg did not care to turn this into an exact limit. He treated it as a heuristic quantitative statement, correct up to small factors, which made the radically new noncommutativity of quantum mechanics inevitable. Early students of quantum theory needed the picture because the principle felt so counter-intuitive. They had to be reassured that naive measurements meant to violate it were bound to be unworkable.

  The microscope also planted a lasting confusion with the observer effect, which says that measuring certain systems cannot avoid changing them. Heisenberg used an observer effect at the quantum level as a physical explanation of the uncertainty. It has since become clearer that the principle is a basic property of quantum systems. It is not a statement about whether present technology is good enough.

  There is a further distinction. Heisenberg's original version dealt with systematic error, meaning a disturbance of the system produced by the measuring apparatus. Modern inequalities such as Kennard's concern the statistical spread of observables. Masanao Ozawa proposed an inequality that covers both kinds of error, and Heisenberg's original wording mentions only its first term. A formal derivation of the Heisenberg error-disturbance relation is possible but far from intuitive. It was formulated in a mathematically consistent way only in recent years. Experimental evidence is increasing that total quantum uncertainty needs all three terms of the Ozawa inequality.

  Simultaneous measurements of two quantities are a different case, and they are necessarily unsharp or weak. Kazuo Fujikawa then built a relation that stays close to Heisenberg's original form but holds for both systematic and statistical errors.

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• 5. Einstein Against Bohr

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  The Copenhagen interpretation of quantum mechanics and Heisenberg's principle were at first seen by detractors as twin targets. In the Copenhagen view, the quantum state describes no fundamental reality. It is only a prescription for calculating experimental results. Albert Einstein believed that randomness reflects our ignorance of some fundamental property of reality. Niels Bohr believed the probability distributions are fundamental and irreducible, and that they depend on which measurements we choose to perform. The two men debated the principle for many years.

  Wolfgang Pauli called Einstein's underlying objection "the ideal of the detached observer." In Pauli's account, Einstein told him: "Like the moon has a definite position, whether or not we look at the moon, the same must also hold for the atomic objects." Pauli added that he hoped he had quoted Einstein correctly, since it is difficult to quote from memory someone you disagree with.

  Einstein's first thought experiment sent a particle through a slit in a wall. The slit blurs the particle's momentum, he granted. But he proposed measuring the recoil of the wall instead, which would give the particle's momentum to arbitrary accuracy by conservation of momentum. Bohr replied that the wall is quantum mechanical too. To measure the recoil that finely, the wall's momentum must be known that finely beforehand. That makes the position of the slit too uncertain to allow a position measurement. Richard Feynman gave a similar analysis for particles diffracting through multiple slits.

  Bohr was present when Einstein proposed his box. An ideal box is lined with mirrors so that it can hold light indefinitely. Weigh it, let a clockwork mechanism open a shutter at a chosen instant so that one photon escapes, then weigh it again. The change in mass gives the energy of the light, and the clock gives the moment of release, both with any desired precision. Bohr spent a sleepless night on the argument and finally found the flaw. Weighing the box with a spring and a pointer lets it move vertically, which creates an uncertainty in its velocity and so in its height above the table. That height uncertainty changes the rate of the clock, because of the effect of gravity on time in Einstein's own theory.

  In 1935 Einstein, Boris Podolsky and Nathan Rosen published an analysis of spatially separated entangled particles, now called the EPR paradox. Measure the position of one particle and the momentum of the other, they argued, and you could deduce both quantities for both particles to any precision. Avoiding that conclusion means a measurement on one particle must instantly modify the probability distribution of the other, possibly violating locality. In 1964 John Stewart Bell showed the assumption could be falsified, because it would imply an inequality between the probabilities of different experiments. Experimental results have since confirmed the predictions of quantum mechanics, ruling out the local hidden variables that EPR's reasoning assumed.

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• 6. Popper And The Doubters

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  In 1934 the philosopher of science Karl Popper published a paper titled Critique of the Uncertainty Relations in the journal Naturwissenschaften. In the same year he published Logik der Forschung, which he later translated and updated as The Logic of Scientific Discovery in 1959. Popper approached the problem of indeterminacy as a logician and a metaphysical realist.

  His objection was to applying the uncertainty relations to individual particles. He argued they apply to ensembles of identically prepared particles, and he called them statistical scatter relations. In that statistical interpretation, a single measurement may be made to arbitrary precision without harming quantum theory. In a 1982 book, Quantum theory and the schism in Physics, he called Heisenberg's formulae "beyond all doubt, derivable statistical formulae of the quantum theory." He said quantum theorists had habitually misread them as an upper limit to the precision of measurements.

  Popper also proposed an experiment to falsify the uncertainty relations. He withdrew his first version after discussions with Carl Friedrich von Weizsäcker, Heisenberg and Einstein. He had sent his paper to Einstein, and it may have influenced the formulation of the EPR paradox.

  Edwin C. Kemble made a different objection in 1937. Uncertainty relations tie quantum particles, such as electrons, to the classical concepts of position and momentum. That presumes the particles have those properties, which Kemble said cannot be experimentally verified, and assuming them gives rise to many contradictions. Rudolf Haag notes that position in quantum mechanics is an attribute of an interaction, say between an electron and a detector, rather than an intrinsic property. From this view the principle is not fundamental. It is a concept "carried over from the language of our ancestors," as Kemble put it.

  The principle has also wandered into debates about free will. Arthur Compton and Martin Heisenberg have suggested that it, or at least the probabilistic nature of quantum mechanics, could be evidence for the two-stage model of free will. One critique says that, apart from the role of quantum mechanics as a foundation for chemistry, nontrivial biological mechanisms needing quantum mechanics are unlikely. Quantum systems lose their coherence very quickly at room temperature. Proponents reply that screening and decoherence-free subspaces in biological cells overcome this.

  There is also reason to believe that violating the uncertainty principle would strongly imply violating the second law of thermodynamics, a link connected to the Gibbs paradox.

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• 7. Energy And Time

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  A Delta particle, a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10 to the power of minus 23 seconds. Its measured mass, equivalent to an energy of 1232 megaelectronvolts per speed of light squared, varies by plus or minus 120 of the same units. That spread is intrinsic to the particle and not caused by measurement errors. It is the energy-time form of the uncertainty relation at work.

  The informal logic runs like this. A state that exists only briefly cannot have a definite energy. A definite energy requires a precisely defined frequency, and that requires the state to last through many cycles. In spectroscopy, excited states have finite lifetimes, so each decay releases a slightly different energy. The average energy of the outgoing photon peaks at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, and slow-decaying states have a narrow one. The same effect makes it hard to specify the rest mass of fast-decaying unstable particles in particle physics. The shorter the lifetime, the less certain the mass.

  This relation has a long and controversial history, because time is awkward in quantum mechanics. There is no quantum theory of time measurement, and relativity is both fundamental to time and difficult to include. Position and momentum belong to a single particle, but time is a property of a whole system, with no operator for it. Three notions of time can be told apart. External time is the laboratory time the experimenter sees. Intrinsic time is inferred from changes in dynamic variables, like the hands of a clock. Observable time concerns the measurement of time-separated events, and it remains a challenge.

  Yakir Aharonov and David Bohm showed that in some quantum systems energy can be measured accurately within an arbitrarily short time. So energy-time principles built on external time are not universal. In 1945 Leonid Mandelstam and Igor Tamm used intrinsic time to derive a non-relativistic relation. Their characteristic time is how long it takes the expectation value of an observable to change by one standard deviation. It has the dimension of time, but it differs from the time parameter in the Schrödinger equation.

  Some formulations of quantum field theory use temporary electron-positron pairs called virtual particles. The mass-energy and lifetime of these particles are related by the energy-time uncertainty relation. Because energy is not known precisely enough to confine a system to a single simple history, calculations must include the influence of all histories. That includes histories with much more or much less energy than the mean. The relation does not mean energy can be borrowed from the universe and returned within a short time, and it does not temporarily violate conservation of energy.

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• 8. Beyond Quantum Mechanics

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  Dennis Gabor gave his name to the principle's signal-processing form, the Gabor limit, sometimes called the Heisenberg-Gabor limit. A function cannot be both time limited and band limited. For filters, this means high temporal resolution and high frequency resolution cannot be had at once. In the short-time Fourier transform, a wide window gives good frequency resolution at the cost of temporal resolution, while a narrow window does the opposite. To analyze signals where transients matter, the wavelet transform is often used instead of the Fourier transform.

  The discrete Fourier transform has its own version. Terence Tao discovered that when the length of the sequence is a prime number, a stronger and still sharp inequality holds. Benedicks's theorem says a function and its Fourier transform cannot both be supported on sets of finite Lebesgue measure. The mathematician G. H. Hardy showed that the function and its transform cannot both be very rapidly decreasing. Later improvements came from Beurling, Hörmander, and Bonami, Demange and Jaming.

  Hugh Everett III was formulating the many-worlds interpretation in 1957 when he conjectured a stronger, entropy-based version of the principle. I. I. Hirschman also studied it, and in 1975 W. Beckner proved it, as did Iwo Bialynicki-Birula and Jerzy Mycielski. Standard deviation is not a natural measure for every distribution, such as angles, highly bimodal distributions, or distributions with divergent variance. Entropy handles them. The sinc function is an example with infinite momentum variance but finite entropic uncertainty. The Heisenberg principle follows from the entropic one, but not the reverse. The normal distribution is the only one that saturates the entropic bound, and efforts to improve the bound remain an active area of research.

  The standard Robertson inequality does not always apply. For a quantum particle on a ring, the position operator is defined through an angle, and the states of definite angular momentum can be normalized. In such a state the uncertainty of angular momentum is zero while the uncertainty of the angle is finite. Multiplying by the angle breaks the periodic boundary conditions, so the Robertson relation fails. For the usual position and momentum on the real line, no such counterexample can occur.

  Lorenzo Maccone and Arun K. Pati proved stronger relations that bound the sum of the variances of two incompatible observables. The older Robertson-Schrödinger relation can become trivial when the state is an eigenstate of one observable. The Maccone-Pati bounds stay nonzero unless the state is an eigenstate of both.

  In quantum metrology, especially interferometry, the Heisenberg limit is the optimal rate at which measurement accuracy can scale with the energy used, such as the number of photons. Some claim to have broken it, but that reflects disagreement over how to define the scaling resource. Suitably defined, it cannot be beaten, although the weak Heisenberg limit can.

  Typical quantum experiments routinely observe aspects of the principle. All forms of spectroscopy, including particle physics, use it to relate measured energy line-width to the lifetime of quantum states. Some experiments test a particular form on purpose, such as number-phase uncertainty relations in superconducting or quantum optics systems. Extremely low-noise technology, such as gravitational wave interferometers, depends on the principle to operate.

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