Uncertainty principle
Part of a series of articles about 
Quantum mechanics 

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, x, and momentum, p.^{[1]} Such pairedvariables are known as complementary variables or canonically conjugate variables.
Introduced first in 1927 by German physicist Werner Heisenberg,^{[2]} the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. In the published 1927 paper, Heisenberg originally concluded that the uncertainty principle was ΔpΔq ≈ h using the full Planck constant.^{[2]}^{[3]}^{[4]}^{[5]} The formal inequality relating the standard deviation of position σ_{x} and the standard deviation of momentum σ_{p} was derived by Earle Hesse Kennard^{[6]} later that year and by Hermann Weyl^{[7]} in 1928:
where ħ is the reduced Planck constant, .
Historically, the uncertainty principle has been confused^{[8]}^{[9]} with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system,^{[10]}^{[11]} that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.^{[12]} It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wavelike systems,^{[13]} and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects.^{[14]} Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.^{[15]} Indeed the uncertainty principle has its roots in how we apply calculus to write the basic equations of mechanics. It must be emphasized that measurement does not mean only a process in which a physicistobserver takes part, but rather any interaction between classical and quantum objects regardless of any observer.^{[16]}
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^{[17]} or quantum optics^{[18]} systems. Applications dependent on the uncertainty principle for their operation include extremely lownoise technology such as that required in gravitational wave interferometers.^{[19]}
Introduction[edit]
It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic^{[20]} scales that humans experience. 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.
Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized at the same time.^{[21]} A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation p = ħk, where k is the wavenumber.
In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of noncommuting selfadjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.^{[22]}
Visualization[edit]
The uncertainty principle can be visualized using the position and momentumspace wavefunctions for one spinless particle with mass in one dimension.
The more localized the positionspace wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentumspace wavefunction is less localized so the possible momentum components the particle could have are more widespread. Conversely, the more localized the momentumspace wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the positionspace wavefunction, so the position coordinates the particle could occupy are more widespread. These wavefunctions are Fourier transforms of each other: mathematically, the uncertainty principle expresses the relationship between conjugate variables in the transform.
Wave mechanics interpretation[edit]
According to the de Broglie hypothesis, every object in the universe is associated with a wave. Thus every object, from an elementary particle to atoms, molecules and on up to planets and beyond are subject to the uncertainty principle.
The timeindependent wave function of a singlemoded plane wave of wavenumber k_{0} or momentum p_{0} 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 singlemode plane wave, is 1 if and 0 otherwise. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet.
On the other hand, consider a wave function that is a sum of many waves, which we may write as
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.
The precision of the position is improved, i.e. reduced σ_{x}, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σ_{p}. Another way of stating this is that σ_{x} and σ_{p} have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound.
We are interested in the variances of position and momentum, defined as
Without loss of generality, we will assume that the means vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form
The function can be interpreted as a vector in a function space. We can define an inner product for a pair of functions u(x) and v(x) in this vector space:
With this inner product defined, we note that the variance for position can be written as
We can repeat this for momentum by interpreting the function as a vector, but we can also take advantage of the fact that and are Fourier transforms of each other. We evaluate the inverse Fourier transform through integration by parts:
The Cauchy–Schwarz inequality asserts that
The modulus squared of any complex number z can be expressed as
All that remains is to evaluate these inner products.
Plugging this into the above inequalities, we get
with equality if and only if p and x are linearly dependent. Note that the only physics involved in this proof was that and are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for any pair of conjugate variables.
Matrix mechanics interpretation[edit]
(Ref ^{[16]})
In matrix mechanics, observables such as position and momentum are represented by selfadjoint operators. When considering pairs of observables, an important quantity is the commutator. For a pair of operators Â and , one defines their commutator as
The physical meaning of the noncommutativity 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 x_{0}. By definition, this means that Applying the commutator to yields
Suppose, for the sake of proof by contradiction, that is also a right eigenstate of momentum, with constant eigenvalue p_{0}. If this were true, then one could write
When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is not a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words, the momentum must be less precise. This precision may be quantified by the standard deviations,
As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle.
Heisenberg limit[edit]
In quantum metrology, and especially interferometry, the Heisenberg limit is the optimal rate at which the accuracy of a measurement can scale with the energy used in the measurement. Typically, this is the measurement of a phase (applied to one arm of a beamsplitter) and the energy is given by the number of photons used in an interferometer. Although some claim to have broken the Heisenberg limit, this reflects disagreement on the definition of the scaling resource.^{[23]} Suitably defined, the Heisenberg limit is a consequence of the basic principles of quantum mechanics and cannot be beaten, although the weak Heisenberg limit can be beaten.^{[24]}
Robertson–Schrödinger uncertainty relations[edit]
The most common general form of the uncertainty principle is the Robertson uncertainty relation.^{[25]}
For an arbitrary Hermitian operator we can associate a standard deviation
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,^{[26]}
where we have introduced the anticommutator,
The derivation shown here incorporates and builds off of those shown in Robertson,^{[25]} Schrödinger^{[26]} and standard textbooks such as Griffiths.^{[27]} For any Hermitian operator , based upon the definition of variance, we have
Similarly, for any other Hermitian operator in the same state
The product of the two deviations can thus be expressed as

(1)
In order to relate the two vectors and , we use the Cauchy–Schwarz inequality^{[28]} which is defined as

(2)
Since is in general a complex number, we use the fact that the modulus squared of any complex number is defined as , where is the complex conjugate of . The modulus squared can also be expressed as

(3)
we let and and substitute these into the equation above to get

(4)
The inner product is written out explicitly as
Similarly it can be shown that
Thus, we have
We now substitute the above two equations above back into Eq. (4) and get
Substituting the above into Equation (2) we get the Schrödinger uncertainty relation
This proof has an issue^{[29]} related to the domains of the operators involved. For the proof to make sense, the vector has to be in the domain of the unbounded operator , which is not always the case. In fact, the Robertson uncertainty relation is false if is an angle variable and is the derivative with respect to this variable. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero.^{[30]} (See the counterexample section below.) This issue can be overcome by using a variational method for the proof,^{[31]}^{[32]} or by working with an exponentiated version of the canonical commutation relations.^{[30]}
Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators and are selfadjoint operators. It suffices to assume that they are merely symmetric operators. (The distinction between these two notions is generally glossed over in the physics literature, where the term Hermitian is used for either or both classes of operators. See Chapter 9 of Hall's book^{[33]} for a detailed discussion of this important but technical distinction.)
Mixed states[edit]
The Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe mixed states.
The Maccone–Pati uncertainty relations[edit]
The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Maccone and Pati give nontrivial bounds on the sum of the variances for two incompatible observables.^{[34]} (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.^{[35]} due to Huang.) For two noncommuting observables and the first stronger uncertainty relation is given by
The second stronger uncertainty relation is given by
Improving the Robertson–Schrödinger uncertainty relation based on decompositions of the density matrix[edit]
The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components in any decomposition of the density matrix given as
With similar arguments, one can derive a relation with a convex roof on the righthand side^{[36]}
A simpler inequality follows without a convex roof^{[39]}
Phase space[edit]
In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real starsquare function. Given a Wigner function with star product ★ and a function f, the following is generally true:^{[40]}
Choosing , we arrive at
Since this positivity condition is true for all a, b, and c, it follows that all the eigenvalues of the matrix are nonnegative.
The nonnegative eigenvalues then imply a corresponding nonnegativity condition on the determinant,
Examples[edit]
Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below.
 Position  Linear Momentum uncertainty relation: for the position and linear momentum operators, the canonical commutation relation implies the Kennard inequality from above:
 Angular Momentum uncertainty relation: For two orthogonal components of the total angular momentum operator of an object: where i, j, k are distinct, and J_{i} denotes angular momentum along the x_{i} 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 j ≥ m, among others.
 Time – Energy uncertainty relation: In nonrelativistic mechanics, time is privileged as an independent variable. Nevertheless, in 1945, L. I. Mandelstam and I. E. Tamm derived a nonrelativistic time–energy uncertainty relation, as follows.^{[41]}^{[42]} From Heisenberg mechanics, using generalized Ehrenfest theorem for an operator B without explicit time dependance:
 We substitute value of in the weaker form of uncertainty condition:
 where A is substituted with H.
 We get that for a quantum system in a nonstationary state ψ and an observable B represented by a selfadjoint operator , the following formula holds:where σ_{E} is the standard deviation of the energy operator (Hamiltonian) in the state ψ, σ_{B} stands for the standard deviation of B. Although the second factor in the lefthand side has dimension of time, it is different from the time parameter that enters the Schrödinger equation. It is a lifetime of the state ψ with respect to the observable B: In other words, this is the time interval (Δt) after which the expectation value changes by one standard deviation and it is expressed as:An informal, heuristic meaning of the principle is the following: A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in spectroscopy, excited states have a finite lifetime. By the time–energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fastdecaying states have a broad linewidth, while slowdecaying states have a narrow linewidth.^{[43]} The same linewidth effect also makes it difficult to specify the rest mass of unstable, fastdecaying particles in particle physics. The faster the particle decays (the shorter its lifetime), the less certain is its mass (the larger the particle's width).
 For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter^{[44]}^{[45]}
A counterexample[edit]
Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable , which we may take to lie in the interval . Define "position" and "momentum" operators and by
Now let be any of the eigenstates of , which are given by . These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator is bounded, since ranges over a bounded interval. Thus, in the state , the uncertainty of is zero and the uncertainty of is finite, so that
For the usual position and momentum operators and on the real line, no such counterexamples can occur. As long as and are defined in the state , the Heisenberg uncertainty principle holds, even if fails to be in the domain of or of .^{[48]}
Examples[edit]
(Refs ^{[16]}^{[27]})
Quantum harmonic oscillator stationary states[edit]
Consider a onedimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the creation and annihilation operators:
Using the standard rules for creation and annihilation operators on the energy eigenstates,
In particular, the above Kennard bound^{[6]} is saturated for the ground state n=0, for which the probability density is just the normal distribution.
Quantum harmonic oscillators with Gaussian initial condition[edit]
In a quantum harmonic oscillator of characteristic angular frequency ω, place a state that is offset from the bottom of the potential by some displacement x_{0} as
From the relations
Coherent states[edit]
A coherent state is a right eigenstate of the annihilation operator,
In the picture where the coherent state is a massive particle in a quantum harmonic oscillator, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances,
Particle in a box[edit]
Consider a particle in a onedimensional box of length . The eigenfunctions in position and momentum space are
The product of the standard deviations is therefore
Constant momentum[edit]
Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p_{0} according to
Since and , this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is
Additional uncertainty relations[edit]
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^{[9]} — encompassing both systematic and statistical errors — holds:
Heisenberg's 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 the 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.^{[49]}^{[50]} Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors and . There is increasing experimental evidence^{[13]}^{[51]}^{[52]}^{[53]} 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^{[54]} 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
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.^{[32]}^{[56]}^{[57]}^{[58]} 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 manyworlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.^{[59]} This conjecture, also studied by Hirschman^{[60]} and proven in 1975 by Beckner^{[61]} and by Iwo BialynickiBirula and Jerzy Mycielski^{[62]} is that, for two normalized, dimensionless Fourier transform pairs f(a) and g(b) where
 and
the Shannon information entropies
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 h is Planck's constant.
Depending on one's choice of the x_{0} p_{0} product, the expression may be written in many ways. If x_{0} p_{0} is chosen to be h, then
If, instead, x_{0} p_{0} is chosen to be ħ, then
If x_{0} and p_{0} are chosen to be unity in whatever system of units are being used, then
The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities^{[64]}
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).
Entropic uncertainty of the normal distribution 

We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. The length scale can be set to whatever is convenient, so we assign
The probability distribution is the normal distribution
with Shannon entropy
A completely analogous calculation proceeds for the momentum distribution. Choosing a standard momentum of : The entropic uncertainty is therefore the limiting value 
A measurement apparatus will have a finite resolution set by the discretization of its possible outputs into bins, with the probability of lying within one of the bins given by the Born rule. We will consider the most common experimental situation, in which the bins are of uniform size. Let δx be a measure of the spatial resolution. We take the zeroth bin to be centered near the origin, with possibly some small constant offset c. The probability of lying within the jth interval of width δx is
To account for this discretization, we can define the Shannon entropy of the wave function for a given measurement apparatus as
Under the above definition, the entropic uncertainty relation is
Here we note that δx δp/h is a typical infinitesimal phase space volume used in the calculation of a partition function. The inequality is also strict and not saturated. Efforts to improve this bound are an active area of research.
Normal distribution example 

We demonstrate this method first on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations.
The probability of lying within one of these bins can be expressed in terms of the error function. The momentum probabilities are completely analogous. For simplicity, we will set the resolutions to
so that the probabilities reduce to
The Shannon entropy can be evaluated numerically. The entropic uncertainty is indeed larger than the limiting value. Note that despite being in the optimal case, the inequality is not saturated. 
Sinc function example 

An example of a unimodal distribution with infinite variance is the sinc function. If the wave function is the correctly normalized uniform distribution,
then its Fourier transform is the sinc function,
which yields infinite momentum variance despite having a centralized shape. The entropic uncertainty, on the other hand, is finite. Suppose for simplicity that the spatial resolution is just a twobin measurement, δx = a, and that the momentum resolution is δp = h/a.
Partitioning the uniform spatial distribution into two equal bins is straightforward. We set the offset c = 1/2 so that the two bins span the distribution. The bins for momentum must cover the entire real line. As done with the spatial distribution, we could apply an offset. It turns out, however, that the Shannon entropy is minimized when the zeroth bin for momentum is centered at the origin. (The reader is encouraged to try adding an offset.) The probability of lying within an arbitrary momentum bin can be expressed in terms of the sine integral. The Shannon entropy can be evaluated numerically. The entropic uncertainty is indeed larger than the limiting value. 
Uncertainty relation with three angular momentum components[edit]
For a particle of spin the following uncertainty relation holds
Harmonic analysis[edit]
In the context of harmonic analysis, a branch of mathematics, the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds,
Further mathematical uncertainty inequalities, including the above entropic uncertainty, hold between a function f and its Fourier transform ƒ̂:^{[66]}^{[67]}^{[68]}
Signal processing [edit]
In the context of signal processing, and in particular time–frequency analysis, uncertainty principles are referred to as the Gabor limit, after Dennis Gabor, or sometimes the Heisenberg–Gabor limit. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both time limited and band limited (a function and its Fourier transform cannot both have bounded domain)—see bandlimited versus timelimited. More accurately, the timebandwidth or durationbandwidth product satisfies
Stated alternatively, "One cannot simultaneously sharply localize a signal (function f) in both the time domain and frequency domain (ƒ̂, its Fourier transform)".
When applied to filters, the result implies that one cannot achieve high temporal resolution and frequency resolution at the same time; a concrete example are the resolution issues of the shorttime Fourier transform—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite tradeoff.
Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2dimensional) time–frequency plane. In practice, the Gabor limit limits the simultaneous time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.
As a result, in order to analyze signals where the transients are important, the wavelet transform is often used instead of the Fourier.
Discrete Fourier transform[edit]
Let be a sequence of N complex numbers and its discrete Fourier transform.
Denote by the number of nonzero elements in the time sequence and by the number of nonzero elements in the frequency sequence . Then,
This inequality is sharp, with equality achieved when x or X is a Dirac mass, or more generally when x is a nonzero multiple of a Dirac comb supported on a subgroup of the integers modulo N (in which case X is also a Dirac comb supported on a complementary subgroup, and vice versa).
More generally, if T and W are subsets of the integers modulo N, let denote the timelimiting operator and bandlimiting operators, respectively. Then
When N is a prime number, a stronger inequality holds:
Benedicks's theorem[edit]
Amrein–Berthier^{[72]} and Benedicks's theorem^{[73]} intuitively says that the set of points where f is nonzero and the set of points where ƒ̂ is nonzero cannot both be small.
Specifically, it is impossible for a function f in L^{2}(R) and its Fourier transform ƒ̂ to both be supported on sets of finite Lebesgue measure. A more quantitative version is^{[74]}^{[75]}
One expects that the factor Ce^{CSΣ} may be replaced by Ce^{C(SΣ)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:^{[76]} it is not possible for f and ƒ̂ to both be "very rapidly decreasing". Specifically, if f in is such that
then, if ab > 1, f = 0, while if ab = 1, then there is a polynomial P of degree ≤ N such that
This was later improved as follows: if is such that
This result was stated in Beurling's complete works without proof and proved in Hörmander^{[77]} (the case ) and Bonami, Demange, and Jaming^{[78]} 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.^{[79]}
A full description of the case ab < 1 as well as the following extension to Schwartz class distributions appears in ref.^{[80]}
Theorem — If a tempered distribution is such that
History[edit]
In 1925 Heisenberg published the first quantum mechanics theory, now known as the matrix mechanics. Working with Max Born and Pascual Jordan, he continued to develop the increasingly successful theory.^{[81]}^{: II:267 } A central aspect of this theory was noncommutativity: the theory implied that the relative order of position and momentum measurement was significant.
In March 1926, working in Bohr's institute, Heisenberg realized that the noncommutativity implies the uncertainty principle. Writing to Wolfgang Pauli in February 1927, he worked out the basic concepts.^{[82]}
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. His paper gave an analysis in terms of a microscope that Bohr showed was incorrect; Heisenberg included an addendum to the publication.
In his 1930 Chicago lecture^{[83]} he refined his principle:

(A1)
Later work broadened the concept. Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. Heisenberg wrote:
It can be expressed in its simplest form as follows: One can never know with perfect accuracy both of those two important factors which determine the movement of one of the smallest particles—its position and its velocity. It is impossible to determine accurately both the position and the direction and speed of a particle at the same instant.^{[84]}
Kennard^{[6]} in 1927 first proved the modern inequality:

(A2)
where ħ = h/2π, and σ_{x}, σ_{p} are the standard deviations of position and momentum. Heisenberg only proved relation (A2) for the special case of Gaussian states.^{[83]}
Energy time uncertainty principle[edit]
Other than the position–momentum uncertainty relation, the most important uncertainty relation is that between energy and time. A state with halflife will transition with an energy spread according to
The energy–time uncertainty relation is not, however, an obvious consequence of the general Robertson–Schrödinger relation. Since energy bears the same relation to time as momentum does to space in special relativity, it was clear to many early founders, Niels Bohr among them, that the following relation should hold:^{[86]}^{[83]}
One false formulation of the energy–time uncertainty principle says that measuring the energy of a quantum system to an accuracy requires a time interval .^{[85]} This formulation is similar to the one alluded to in Landau's joke, and was explicitly invalidated by Y. Aharonov and D. Bohm in 1961.^{[88]} The time in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on, whereas the position in the other version of the principle refers to where the particle has some probability to be and not where the observer might look.
Another common misconception is that the energy–time uncertainty principle says that the conservation of energy can be temporarily violated—energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time.^{[27]} Although this agrees with the spirit of relativistic quantum mechanics, it is based on the false axiom that the energy of the universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is violated when quantum field theory uses temporary electron–positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of all histories must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.
Terminology and translation[edit]
Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit",^{[2]} to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit". Later on, he always used "Unbestimmtheit". When the Englishlanguage version of Heisenberg's textbook, The Physical Principles of the Quantum Theory, was published in 1930, however, only the English word "uncertainty" was used, and it became the term in the English language.^{[89]}
Heisenberg's microscope[edit]
The principle is quite counterintuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violating the uncertainty principle is by utilizing the observer effect of an imaginary microscope as a measuring device.^{[83]}
He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.^{[90]}^{: 49–50 }
 Problem 1 – If the photon has a short wavelength, and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
 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 electron's 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.
The combination of these tradeoffs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant.^{[91]} Heisenberg did not care to formulate the uncertainty principle as an exact limit, and preferred to use it instead, as a heuristic quantitative statement, correct up to small numerical factors, which makes the radically new noncommutativity of quantum mechanics inevitable.
Critical reactions[edit]
The Copenhagen interpretation of quantum mechanics and Heisenberg's Uncertainty Principle were, in fact, seen as twin targets by detractors who believed in an underlying determinism and realism. According to the Copenhagen interpretation of quantum mechanics, there is no fundamental reality that the quantum state describes, just a prescription for calculating experimental results. There is no way to say what the state of a system fundamentally is, only what the result of observations might be.
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.
The ideal of the detached observer[edit]
Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):
"Like the moon has a definite position" Einstein said to me last winter, "whether or not we look at the moon, the same must also hold for the atomic objects, as there is no sharp distinction possible between these and macroscopic objects. Observation cannot create an element of reality like a position, there must be something contained in the complete description of physical reality which corresponds to the possibility of observing a position, already before the observation has been actually made." I hope, that I quoted Einstein correctly; it is always difficult to quote somebody out of memory with whom one does not agree. It is precisely this kind of postulate which I call the ideal of the detached observer.
— Letter from Pauli to Niels Bohr, February 15, 1955^{[92]}
Einstein's slit[edit]
The first of Einstein's thought experiments challenging the uncertainty principle went as follows:
Consider a particle passing through a slit of width d. The slit introduces an uncertainty in momentum of approximately h/d because the particle passes through the wall. But let us determine the momentum of the particle by measuring the recoil of the wall. In doing so, we find the momentum of the particle to arbitrary accuracy by conservation of momentum.
Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy Δp, the momentum of the wall must be known to this accuracy before the particle passes through. This introduces an uncertainty in the position of the wall and therefore the position of the slit equal to h/Δp, and if the wall's momentum is known precisely enough to measure the recoil, the slit's position is uncertain enough to disallow a position measurement.
A similar analysis with particles diffracting through multiple slits is given by Richard Feynman.^{[93]}
Einstein's box[edit]
Bohr was present when Einstein proposed the thought experiment which has become known as Einstein's box. Einstein argued that "Heisenberg's uncertainty equation implied that the uncertainty in time was related to the uncertainty in energy, the product of the two being related to Planck's constant."^{[94]} Consider, he said, an ideal box, lined with mirrors so that it can contain light indefinitely. The box could be weighed before a clockwork mechanism opened an ideal shutter at a chosen instant to allow one single photon to escape. "We now know, explained Einstein, precisely the time at which the photon left the box."^{[95]} "Now, weigh the box again. The change of mass tells the energy of the emitted light. In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."^{[94]}
Bohr spent a sleepless night considering this argument, and eventually realized that it was flawed. He pointed out that if the box were to be weighed, say by a spring and a pointer on a scale, "since the box must move vertically with a change in its weight, there will be uncertainty in its vertical velocity and therefore an uncertainty in its height above the table. ... Furthermore, the uncertainty about the elevation above the Earth's surface will result in an uncertainty in the rate of the clock,"^{[96]} because of Einstein's own theory of gravity's effect on time. "Through this chain of uncertainties, Bohr showed that Einstein's light box experiment could not simultaneously measure exactly both the energy of the photon and the time of its escape."^{[97]}
EPR paradox for entangled particles[edit]
Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolsky and Rosen published an analysis of widely separated entangled particles (EPR paradox).^{[98]} Measuring one particle, Einstein realized, would alter the probability distribution of the other, yet here the other particle could not possibly be disturbed. This example led Bohr to revise his understanding of the principle, concluding that the uncertainty was not caused by a direct interaction.^{[99]}
But Einstein came to much more farreaching conclusions from the same thought experiment. He believed the "natural basic assumption" that a complete description of reality would have to predict the results of experiments from "locally changing deterministic quantities" and therefore would have to include more information than the maximum possible allowed by the uncertainty principle.
In 1964, John Stewart Bell showed that this assumption can be falsified, since it would imply a certain inequality between the probabilities of different experiments. Experimental results confirm the predictions of quantum mechanics, ruling out Einstein's basic assumption that led him to the suggestion of his hidden variables. These hidden variables may be "hidden" because of an illusion that occurs during observations of objects that are too large or too small. This illusion can be likened to rotating fan blades that seem to pop in and out of existence at different locations and sometimes seem to be in the same place at the same time when observed. This same illusion manifests itself in the observation of subatomic particles. Both the fan blades and the subatomic particles are moving so fast that the illusion is seen by the observer. Therefore, it is possible that there would be predictability of the subatomic particles behavior and characteristics to a recording device capable of very high speed tracking....Ironically this fact is one of the best pieces of evidence supporting Karl Popper's philosophy of invalidation of a theory by falsificationexperiments. That is to say, here Einstein's "basic assumption" became falsified by experiments based on Bell's inequalities. For the objections of Karl Popper to the Heisenberg inequality itself, see below.
While it is possible to assume that quantum mechanical predictions are due to nonlocal, hidden variables, and in fact David Bohm invented such a formulation, this resolution is not satisfactory to the vast majority of physicists. The question of whether a random outcome is predetermined by a nonlocal theory can be philosophical, and it can be potentially intractable. If the hidden variables were not constrained, they could just be a list of random digits that are used to produce the measurement outcomes. To make it sensible, the assumption of nonlocal hidden variables is sometimes augmented by a second assumption—that the size of the observable universe puts a limit on the computations that these variables can do. A nonlocal theory of this sort predicts that a quantum computer would encounter fundamental obstacles when attempting to factor numbers of approximately 10,000 digits or more; a potentially achievable task in quantum mechanics.^{[100]}^{[full citation needed]}
Popper's criticism[edit]
Karl Popper approached the problem of indeterminacy as a logician and metaphysical realist.^{[101]} 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".^{[101]}^{[102]} 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 nondeterministic but lacks local hidden variables.
In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) in Naturwissenschaften,^{[103]} 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:
[Heisenberg's] formulae are, beyond all doubt, derivable statistical formulae of the quantum theory. But they have been habitually misinterpreted by those quantum theorists who said that these formulae can be interpreted as determining some upper limit to the precision of our measurements. [original emphasis]^{[104]}
Popper proposed an experiment to falsify the uncertainty relations, although he later withdrew his initial version after discussions with Weizsäcker, Heisenberg, and Einstein; this experiment may have influenced the formulation of the EPR experiment.^{[101]}^{[105]}
Manyworlds uncertainty[edit]
The manyworlds interpretation originally outlined by Hugh Everett III in 1957 is partly meant to reconcile the differences between Einstein's and Bohr's views by replacing Bohr's wave function collapse with an ensemble of deterministic and independent universes whose distribution is governed by wave functions and the Schrödinger equation. Thus, uncertainty in the manyworlds interpretation follows from each observer within any universe having no knowledge of what goes on in the other universes.
Free will[edit]
Some scientists including Arthur Compton^{[106]} and Martin Heisenberg^{[107]} have suggested that the uncertainty principle, or at least the general probabilistic nature of quantum mechanics, could be evidence for the twostage 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.^{[108]} Proponents of this theory commonly say that this decoherence is overcome by both screening and decoherencefree subspaces found in biological cells.^{[108]}
Thermodynamics[edit]
There is reason to believe that violating the uncertainty principle also strongly implies the violation of the second law of thermodynamics.^{[109]} See Gibbs paradox.
See also[edit]
 Canonical commutation relation
 Correspondence principle
 Discrete Fourier transform#Uncertainty principles
 Introduction to quantum mechanics
 Küpfmüller's uncertainty principle
 Observer effect (physics)
 Quantum indeterminacy
 Quantum tunnelling
 Physics and Beyond (book)
 Stronger uncertainty relations
 Weak measurement
References[edit]
 ^ ^{a} ^{b} ^{c} Sen, D. (2014). "The Uncertainty relations in quantum mechanics" (PDF). Current Science. 107 (2): 203–218.
 ^ ^{a} ^{b} ^{c} ^{d} Heisenberg, W. (1927) [19270301]. "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Zeitschrift für Physik (in German). 43 (3): 172–198. Bibcode:1927ZPhy...43..172H. doi:10.1007/BF01397280. ISSN 00443328. S2CID 122763326.Heisenberg, W (1983) [1927]. "The actual content of quantum theoretical kinematics and mechanics". No. NAS 1.15: 77379. 1983. 43 (3–4): 172. Bibcode:1983ZhPhy..43..172H.
English translation of Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik
 ^ Werner Heisenberg, Encounters with Einstein and Other Essays on People, Places and Particles, Published October 21st 1989 by Princeton University Press, p.53.
 ^ Dolling, Lisa M.; Gianelli, Arthur F.; Statile, Glenn N., eds. (2003). The Tests of Time. doi:10.1515/9781400889167. ISBN 9781400889167.
 ^ Kumar, Manjit. Quantum: Einstein, Bohr, and the great debate about the nature of reality / Manjit Kumar.—1st American ed., 2008. Chap.10, Note 37.
 ^ ^{a} ^{b} ^{c} Kennard, E. H. (1927), "Zur Quantenmechanik einfacher Bewegungstypen", Zeitschrift für Physik (in German), 44 (4–5): 326–352, Bibcode:1927ZPhy...44..326K, doi:10.1007/BF01391200, S2CID 121626384.
 ^ Weyl, H. (1928), Gruppentheorie und Quantenmechanik, Leipzig: Hirzel
 ^ Furuta, Aya (2012), "One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead", Scientific American
 ^ ^{a} ^{b} Ozawa, Masanao (2003), "Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement", Physical Review A, 67 (4): 42105, arXiv:quantph/0207121, Bibcode:2003PhRvA..67d2105O, doi:10.1103/PhysRevA.67.042105, S2CID 42012188
 ^ Wheeler, John Archibald (19780101), Marlow, A. R. (ed.), "The "Past" and the "DelayedChoice" DoubleSlit Experiment", Mathematical Foundations of Quantum Theory, Academic Press, pp. 9–48, doi:10.1016/b9780124732506.500066, ISBN 9780124732506, retrieved 20230719
 ^ Wheeler, John Archibald (1977), Lopes, José Leite; Paty, Michel (eds.), "Include the Observer in the Wave Function?", Quantum Mechanics, A Half Century Later: Papers of a Colloquium on Fifty Years of Quantum Mechanics, Held at the University Louis Pasteur, Strasbourg, May 2–4, 1974, Episteme, Dordrecht: Springer Netherlands, pp. 1–18, doi:10.1007/9789401011969_1, ISBN 9789401011969, retrieved 20230719
 ^ Werner Heisenberg, The Physical Principles of the Quantum Theory, p. 20
 ^ ^{a} ^{b} Rozema, L. A.; Darabi, A.; Mahler, D. H.; Hayat, A.; Soudagar, Y.; Steinberg, A. M. (2012). "Violation of Heisenberg's Measurement–Disturbance Relationship by Weak Measurements". Physical Review Letters. 109 (10): 100404. arXiv:1208.0034v2. Bibcode:2012PhRvL.109j0404R. doi:10.1103/PhysRevLett.109.100404. PMID 23005268. S2CID 37576344.
 ^ De Broglie, Louis (October 1923). "Waves and Quanta". Nature. 112 (2815): 540. Bibcode:1923Natur.112..540D. doi:10.1038/112540a0. ISSN 14764687. S2CID 186242764.
 ^ Indian Institute of Technology Madras, Professor V. Balakrishnan, Lecture 1 – Introduction to Quantum Physics; Heisenberg's uncertainty principle, National Programme of Technology Enhanced Learning on YouTube
 ^ ^{a} ^{b} ^{c} ^{d} Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1977). Quantum Mechanics: NonRelativistic Theory. Vol. 3 (3rd ed.). Pergamon Press. ISBN 9780080209401.
 ^ Elion, W. J.; Matters, M.; Geigenmüller, U.; Mooij, J. E. (1994), "Direct demonstration of Heisenberg's uncertainty principle in a superconductor", Nature, 371 (6498): 594–595, Bibcode:1994Natur.371..594E, doi:10.1038/371594a0, S2CID 4240085
 ^ Smithey, D. T.; M. Beck, J. Cooper, M. G. Raymer; Cooper, J.; Raymer, M. G. (1993), "Measurement of number–phase uncertainty relations of optical fields", Phys. Rev. A, 48 (4): 3159–3167, Bibcode:1993PhRvA..48.3159S, doi:10.1103/PhysRevA.48.3159, PMID 9909968
{{citation}}
: CS1 maint: multiple names: authors list (link)  ^ Caves, Carlton (1981), "Quantummechanical noise in an interferometer", Phys. Rev. D, 23 (8): 1693–1708, Bibcode:1981PhRvD..23.1693C, doi:10.1103/PhysRevD.23.1693
 ^ Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?". American Journal of Physics. 82 (9): 896–905. Bibcode:2014AmJPh..82..896J. doi:10.1119/1.4878358.
 ^ https://arxiv.org/pdf/0903.3712.pdf, Appendix B
 ^ Claude CohenTannoudji; Bernard Diu; Franck Laloë (1996), Quantum mechanics, WileyInterscience: Wiley, pp. 231–233, ISBN 9780471569527
 ^ Giovannetti, V.; Lloyd, S.; Maccone, L. (2011). "Advances in quantum metrology". Nature Photonics. 5 (4): 222. arXiv:1102.2318. Bibcode:2011NaPho...5..222G. doi:10.1038/nphoton.2011.35. S2CID 12591819.; arXiv
 ^ Luis, Alfredo (20170313). "Breaking the weak Heisenberg limit". Physical Review A. 95 (3): 032113. arXiv:1607.07668. Bibcode:2017PhRvA..95c2113L. doi:10.1103/PhysRevA.95.032113. ISSN 24699926. S2CID 55838380.
 ^ ^{a} ^{b} Robertson, H. P. (1929), "The Uncertainty Principle", Phys. Rev., 34 (1): 163–64, Bibcode:1929PhRv...34..163R, doi:10.1103/PhysRev.34.163
 ^ ^{a} ^{b} Schrödinger, E. (1930), "Zum Heisenbergschen Unschärfeprinzip", Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalischmathematische Klasse, 14: 296–303
 ^ ^{a} ^{b} ^{c} Griffiths, David (2005), Quantum Mechanics, New Jersey: Pearson
 ^ Riley, K. F.; M. P. Hobson and S. J. Bence (2006), Mathematical Methods for Physics and Engineering, Cambridge, p. 246
 ^ Davidson, E. R. (1965), "On Derivations of the Uncertainty Principle", J. Chem. Phys., 42 (4): 1461–1462, Bibcode:1965JChPh..42.1461D, doi:10.1063/1.1696139
 ^ ^{a} ^{b} ^{c} Hall, B. C. (2013), Quantum Theory for Mathematicians, Springer, p. 245, Bibcode:2013qtm..book.....H
 ^ Jackiw, Roman (1968), "Minimum Uncertainty Product, NumberPhase Uncertainty Product, and Coherent States", J. Math. Phys., 9 (3): 339–346, Bibcode:1968JMP.....9..339J, doi:10.1063/1.1664585
 ^ ^{a} ^{b} Carruthers, P.; Nieto, M. M. (1968), "Phase and Angle Variables in Quantum Mechanics", Rev. Mod. Phys., 40 (2): 411–440, Bibcode:1968RvMP...40..411C, doi:10.1103/RevModPhys.40.411
 ^ Hall, B. C. (2013), Quantum Theory for Mathematicians, Springer, Bibcode:2013qtm..book.....H
 ^ Maccone, Lorenzo; Pati, Arun K. (31 December 2014). "Stronger Uncertainty Relations for All Incompatible Observables". Physical Review Letters. 113 (26): 260401. arXiv:1407.0338. Bibcode:2014PhRvL.113z0401M. doi:10.1103/PhysRevLett.113.260401. PMID 25615288. S2CID 21334130.
 ^ Huang, Yichen (10 August 2012). "Variancebased uncertainty relations". Physical Review A. 86 (2): 024101. arXiv:1012.3105. Bibcode:2012PhRvA..86b4101H. doi:10.1103/PhysRevA.86.024101. S2CID 118507388.
 ^ ^{a} ^{b} ^{c} ^{d} Tóth, Géza; Fröwis, Florian (31 January 2022). "Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices". Physical Review Research. 4 (1): 013075. arXiv:2109.06893. Bibcode:2022PhRvR...4a3075T. doi:10.1103/PhysRevResearch.4.013075. S2CID 237513549.
 ^ Tóth, Géza; Petz, Dénes (20 March 2013). "Extremal properties of the variance and the quantum Fisher information". Physical Review A. 87 (3): 032324. arXiv:1109.2831. Bibcode:2013PhRvA..87c2324T. doi: