Uncertainty principle

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In quantum mechanics, the uncertainty principle, also known as Heisenberg's uncertainty principle, is any of a variety of mathematical inequalities[1] asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known.

Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.[2] The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard[3] later that year and by Hermann Weyl[4] in 1928:

(ħ is the reduced Planck constant, h / 2π).

Historically, the uncertainty principle has been confused[5][6] with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.[7] It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.[9] It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.[10] [note 1]

Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number–phase uncertainty relations in superconducting[12] or quantum optics[13] systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers.[14]



The uncertainty principle is not readily apparent on the macroscopic scales of everyday experience.[15] So it is helpful to demonstrate how it applies to more easily understood physical situations. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in a way that generalizes more easily.[citation needed]



Wave mechanics interpretation[edit]

(Ref [10])

Main article: Wave packet
Main article: Schrödinger equation

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function . The time-independent wave function of a single-moded plane wave of wavenumber k0 or momentum p0 is

The Born rule states that this should be interpreted as a probability density amplitude function in the sense that the probability of finding the particle between a and b is

In the case of the single-moded plane wave, is a uniform distribution. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. Consider a wave function that is a sum of many waves, however, we may write this as


with representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that is the Fourier transform of and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.

One way to quantify the precision of the position and momentum is the standard deviation σ. Since is a probability density function for position, we calculate its standard deviation.


Matrix mechanics interpretation[edit]

(Ref [10])

Main article: Matrix mechanics



The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. Let be a right eigenstate of position with a constant eigenvalue x0. By definition, this means that Applying the commutator to yields

where Î is the identity operator.

Suppose, for the sake of proof by contradiction, that is also a right eigenstate of momentum, with constant eigenvalue p0. If this were true, then one could write

On the other hand, the above canonical commutation relation requires that




Robertson–Schrödinger uncertainty relations[edit]

The most common general form of the uncertainty principle is the Robertson uncertainty relation.[17]

For an arbitrary Hermitian operator we can associate a standard deviation

where the brackets indicate an expectation value. For a pair of operators Â and , we may define their commutator as

In this notation, the Robertson uncertainty relation is given by

The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the Schrödinger uncertainty relation,[18]

where we have introduced the anticommutator,


  • For position and linear momentum, the canonical commutation relation implies the Kennard inequality from above:
where i, j, k are distinct and Ji denotes angular momentum along the xi axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for , a choice , in angular momentum multiplets, ψ = |j, m 〉, bounds the Casimir invariant (angular momentum squared, ) from below and thus yields useful constraints such as j (j + 1) ≥ m (m + 1), and hence jm, among others.
  • In non-relativistic mechanics, time is privileged as an independent variable. Nevertheless, in 1945, L. I. Mandelshtam and I. E. Tamm derived a non-relativistic time–energy uncertainty relation, as follows.[26][27] For a quantum system in a non-stationary state ψ and an observable B represented by a self-adjoint operator , the following formula holds:


(Refs [10][19])

Quantum harmonic oscillator stationary states[edit]





the variances may be computed directly,

The product of these standard deviations is then

In particular, the above Kennard bound[3] is saturated for the ground state n=0, for which the probability density is just the normal distribution.

Quantum harmonic oscillator with Gaussian initial condition[edit]






where we have used the notation to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

From the relations


we can conclude


Coherent states[edit]

Main article: Coherent state

A coherent state is a right eigenstate of the annihilation operator,


which may be represented in terms of Fock states as



Therefore, every coherent state saturates the Kennard bound


with position and momentum each contributing an amount in a "balanced" way. Moreover, every squeezed coherent state also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general.

Particle in a box[edit]

Main article: Particle in a box

Consider a particle in a one-dimensional box of length . The eigenfunctions in position and momentum space are



where and we have used the de Broglie relation . The variances of and can be calculated explicitly:


The product of the standard deviations is therefore

For all , the quantity is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when , in which case


Constant momentum[edit]

Main article: Wave packet


where we have introduced a reference scale , with describing the width of the distribution−−cf. nondimensionalization. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are


such that the uncertainty product can only increase with time as

Additional uncertainty relations[edit]

Mixed states[edit]

The Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe mixed states.[31]

Phase space[edit]

In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a Wigner function with star product ★ and a function f, the following is generally true:[32]

Choosing , we arrive at


or, explicitly, after algebraic manipulation,

Systematic and statistical errors[edit]

The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation . Heisenberg's original version, however, was dealing with the systematic error, a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect.

If we let represent the error (i.e., inaccuracy) of a measurement of an observable A and the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A, then the inequality proposed by Ozawa[6] — encompassing both systematic and statistical errors — holds:

Heisenberg uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the systematic error. Using the notation above to describe the error/disturbance effect of sequential measurements (first A, then B), it could be written as

The formal derivation of Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.[33][34] Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors and . There is increasing experimental evidence[8][35] [36][37] that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality.

Using the same formalism,[1] it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements (A and B at the same time):

The two simultaneous measurements on A and B are necessarily[38] unsharp or weak.

It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson[1]

and Ozawa relations we obtain

The four terms can be written as:


as the inaccuracy in the measured values of the variable A and

as the resulting fluctuation in the conjugate variable B, Fujikawa[39] established an uncertainty relation similar to the Heisenberg original one, but valid both for systematic and statistical errors:

Quantum entropic uncertainty principle[edit]

For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.[24][40][41][42] Other examples include highly bimodal distributions, or unimodal distributions with divergent variance.

A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.[43] This conjecture, also studied by Hirschman[44] and proven in 1975 by Beckner[45] and by Iwo Bialynicki-Birula and Jerzy Mycielski[46] is that, for two normalized, dimensionless Fourier transform pairs f(a) and g(b) where


the Shannon information entropies


are subject to the following constraint,

where the logarithms may be in any base.

The probability distribution functions associated with the position wave function ψ(x) and the momentum wave function φ(x) have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by

where x0 and p0 are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the Fourier transform relation between the position wave function ψ(x) and the momentum wavefuction φ(p), the above constraint can be written for the corresponding entropies as

where h is Planck's constant.

Depending on one's choice of the x0 p0 product, the expression may be written in many ways. If x0 p0 is chosen to be h, then

If, instead, x0 p0 is chosen to be ħ, then

If x0 and p0 are chosen to be unity in whatever system of units are being used, then

where h is interpreted as a dimensionless number equal to the value of Planck's constant in the chosen system of units.

The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities[47]


In other words, the Heisenberg uncertainty principle, is a consequence of the quantum entropic uncertainty principle, but not vice versa. A few remarks on these inequalities. First, the choice of base e is a matter of popular convention in physics. The logarithm can alternatively be in any base, provided that it be consistent on both sides of the inequality. Second, recall the Shannon entropy has been used, not the quantum von Neumann entropy. Finally, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is the maximum entropy probability distribution among those with fixed variance (cf. here for proof).

Harmonic analysis[edit]


Further mathematical uncertainty inequalities, including the above entropic uncertainty, hold between a function f and its Fourier transform ƒ̂:[48][49][50]

Signal processing [edit]





Benedicks's theorem[edit]

Amrein-Berthier[51] and Benedicks's theorem[52] intuitively says that the set of points where f is non-zero and the set of points where ƒ̂ is nonzero cannot both be small.


One expects that the factor CeC|S||Σ| may be replaced by CeC(|S||Σ|)1/d, which is only known if either S or Σ is convex.

Hardy's uncertainty principle[edit]

The mathematician G. H. Hardy formulated the following uncertainty principle:[55] it is not possible for f and ƒ̂ to both be "very rapidly decreasing." Specifically, if f in L2(R) is such that


( an integer),





This result was stated in Beurling's complete works without proof and proved in Hörmander[56] (the case ) and Bonami, Demange, and Jaming[57] for the general case. Note that Hörmander–Beurling's version implies the case ab > 1 in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in ref.[58]

A full description of the case ab < 1 as well as the following extension to Schwartz class distributions appears in ref.[59]

Theorem. If a tempered distribution is such that






Werner Heisenberg and Niels Bohr





In his celebrated 1927 paper, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement,[2] but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture[63] he refined his principle:






Kennard[3] in 1927 first proved the modern inequality:







Terminology and translation[edit]


Heisenberg's microscope[edit]



He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.[65]:49–50

Problem 2 – If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon affects the electrons beamline momentum and hence, the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.


Critical reactions[edit]


Albert Einstein believed that randomness is a reflection of our ignorance of some fundamental property of reality, while Niels Bohr believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. Einstein and Bohr debated the uncertainty principle for many years. Some experiments within the first decade of the twenty-first century have cast doubt on observer effect aspects of the uncertainty principle.[67][68]

Einstein's slit[edit]




A similar analysis with particles diffracting through multiple slits is given by Richard Feynman.[69]

Einstein's box[edit]



EPR paradox for entangled particles[edit]





Popper's criticism[edit]

Main article: Popper's experiment

Karl Popper approached the problem of indeterminacy as a logician and metaphysical realist.[76] He disagreed with the application of the uncertainty relations to individual particles rather than to ensembles of identically prepared particles, referring to them as "statistical scatter relations".[76][77] In this statistical interpretation, a particular measurement may be made to arbitrary precision without invalidating the quantum theory. This directly contrasts with the Copenhagen interpretation of quantum mechanics, which is non-deterministic but lacks local hidden variables.

In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) in Naturwissenschaften,[78] and in the same year Logik der Forschung (translated and updated by the author as The Logic of Scientific Discovery in 1959), outlining his arguments for the statistical interpretation. In 1982, he further developed his theory in Quantum theory and the schism in Physics, writing:



Many-worlds uncertainty[edit]


Free will[edit]

Some scientists including Arthur Compton[81] and Martin Heisenberg[82] have suggested that the uncertainty principle, or at least the general probabilistic nature of quantum mechanics, could be evidence for the two-stage model of free will. One critique, however, is that apart from the basic role of quantum mechanics as a foundation for chemistry, nontrivial biological mechanisms requiring quantum mechanics are unlikely, due to the rapid decoherence time of quantum systems at room temperature.[83] The standard view, however, is that this decoherence is overcome by both screening and decoherence-free subspaces found in biological cells.[83]

See also[edit]


  1. ^ N.B. on precision: If and are the precisions of position and momentum obtained in an individual measurement and , their standard deviations in an ensemble of individual measurements on similarly prepared systems, then "There are, in principle, no restrictions on the precisions of individual measurements and , but the standard deviations will always satisfy ".[11]


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