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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 paired-variables are known as complementary variables or canonically conjugate variables.

First introduced in 1927 by German physicist Werner Heisenberg,^{[2]}^{[3]}^{[4]}^{[5]} the formal inequality relating the standard deviation of position *σ _{x}* and the standard deviation of momentum

where is the reduced Planck constant.

The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in many kinds of fundamental physical measurements.

It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic^{[8]} 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.^{[9]} 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 non-commuting self-adjoint 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.^{[10]}

The uncertainty principle can be visualized using the position- and momentum-space wavefunctions for one spinless particle with mass in one dimension.

The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so the possible momentum components the particle could have are more widespread. Conversely, the more localized the momentum-space 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 position-space 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.

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 time-independent wave function of a single-moded 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 single-mode 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
where *A*_{n} represents the relative contribution of the mode *p*_{n} to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit, where the wave function is an integral over all possible modes
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.^{[11]}

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:
where the asterisk denotes the complex conjugate.

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: where in the integration by parts, the cancelled term vanishes because the wave function vanishes at infinity, and the final two integrations re-assert the Fourier transforms. Often the term is called the momentum operator in position space. Applying Parseval's theorem, we see that the variance for momentum can be written as

The Cauchy–Schwarz inequality asserts that

The modulus squared of any complex number *z* can be expressed as
we let and and substitute these into the equation above to get

All that remains is to evaluate these inner products.

Plugging this into the above inequalities, we get or taking the square root

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.

(Ref ^{[11]})

In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity is the *commutator*. For a pair of operators Â and , one defines their commutator as
In the case of position and momentum, the commutator is the canonical commutation relation

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 *x*_{0}. 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 *p*_{0}. If this were true, then one could write
On the other hand, the above canonical commutation relation requires that
This implies that no quantum state can simultaneously be both a position and a momentum eigenstate.

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.

(Refs ^{[11]})

Consider a one-dimensional 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, the variances may be computed directly, The product of these standard deviations is then

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.

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
where Ω describes the width of the initial state but need not be the same as *ω*. Through integration over the propagator, we can solve for the full time-dependent solution. After many cancelations, the probability densities reduce to
where we have used the notation to denote a normal distribution of mean *μ* and variance *σ*^{2}. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

From the relations
we can conclude the following (the right most equality holds only when Ω = *ω*):

A coherent state is a right eigenstate of the annihilation operator, which may be represented in terms of Fock states as

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, 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.

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

Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum *p*_{0} according to
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

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 such that the uncertainty product can only increase with time as

An energy–time uncertainty relation like
has a long, controversial history; the meaning of and varies and different formulations have different arenas of validity.^{[12]} However, one well-known application is both well established^{[13]}^{[14]} and experimentally verified:^{[15]}^{[16]} the connection between the life-time of a resonance state, and its energy width :
In particle-physics, widths from experimental fits to the Breit–Wigner energy distribution are used to characterize the lifetime of quasi-stable or decaying states.^{[17]}

An informal, heuristic meaning of the principle is the following:^{[18]}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*. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.^{[19]} The same linewidth effect also makes it difficult to specify the rest mass of unstable, fast-decaying 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).

The concept of "time" in quantum mechanics offers many challenges.^{[20]} There is no quantum theory of time measurement; relativity is both fundamental to time and difficult to include in quantum mechanics.^{[12]} While position and momentum are associated with a single particle, time is a system property: it has no operator needed for the Robertson–Schrödinger relation.^{[1]} The mathematical treatment of stable and unstable quantum systems differ.^{[21]} These factors combine to make energy–time uncertainty principles controversial.

Three notions of "time" can be distinguished:^{[12]} external, intrinsic, and observable. External or laboratory time is seen by the experimenter; intrinsic time is inferred by changes in dynamic variables, like the hands of a clock or the motion of a free particle; observable time concerns time as an observable, the measurement of time-separated events.

An external-time energy–time uncertainty principle might say that measuring the energy of a quantum system to an accuracy requires a time interval .^{[14]} However, Yakir Aharonov and David Bohm^{[22]}^{[12]} have shown that, in some quantum systems, energy can be measured accurately within an arbitrarily short time: external-time uncertainty principles are not universal.

Intrinsic time is the basis for several formulations of energy–time uncertainty relations, including the Mandelstam–Tamm relation discussed in the next section. A physical system with an intrinsic time closely matching the external laboratory time is called a "clock".^{[20]}^{: 31 }

Observable time, measuring time between two events, remains a challenge for quantum theories; some progress has been made using positive operator-valued measure concepts.^{[12]}

In 1945, Leonid Mandelstam and Igor Tamm derived a non-relativistic *time–energy uncertainty relation* as follows.^{[23]}^{[12]} From Heisenberg mechanics, the generalized Ehrenfest theorem for an observable *B* without explicit time dependence, represented by a self-adjoint operator relates time dependence of the average value of to the average of its commutator with the Hamiltonian:

The value of is then substituted in the Robertson uncertainty relation for the energy operator and :
giving
(whenever the denonminator is nonzero).
While this is a universal result, it depends upon the observable chosen and that the deviations and are computed for a particular state.
Identifying and the characteristic time
gives an energy–time relationship
Although has the dimension of time, it is different from the time parameter *t* that enters the Schrödinger equation. This can be interpreted as time for which the expectation value of the observable, changes by an amount equal to one standard deviation.^{[24]}
Examples:

- The time a free quantum particle passes a point in space is more uncertain as the energy of the state is more precisely controlled: Since the time spread is related to the particle position spread and the energy spread is related to the momentum spread, this relation is directly related to position–momentum uncertainty.
^{[25]}^{: 144 } - A Delta particle, a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10
^{−23}s, so its measured mass equivalent to energy, 1232 MeV/*c*^{2}, varies by ±120 MeV/*c*^{2}; this variation is intrinsic and not caused by measurement errors.^{[25]}^{: 144 } - Two energy states with energies superimposed to create a composite state

- The probability amplitude of this state has a time-dependent interference term:
- The oscillation period varies inversely with the energy difference: .
^{[25]}^{: 144 }

Each example has a different meaning for the time uncertainty, according to the observable and state used.

Some formulations of quantum field theory uses temporary electron–positron pairs in its calculations called virtual particles. The mass-energy and lifetime of these particles are related by the energy–time uncertainty relation. The energy of a 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.

The energy–time uncertainty principle does not temporarily violate conservation of energy; it does not imply that energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time.^{[25]}^{: 145 } The energy of the universe is not an exactly known parameter at all times.^{[1]} When events transpire at very short time intervals, there is uncertainty in the energy of these events.

Historically, the uncertainty principle has been confused^{[26]}^{[27]} with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system,^{[28]}^{[29]} that is, without changing something in a system. Heisenberg used such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.^{[30]} It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems,^{[31]} and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects.^{[32]} Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.^{[33]}

Starting with Kennard's derivation of position-momentum uncertainty, Howard Percy Robertson developed^{[34]}^{[1]} a formulation for arbitrary Hermitian operator operators
expressed in terms of their standard deviation
where the brackets indicate an expectation value. For a pair of operators and , define their *commutator* as

and the Robertson uncertainty relation is given by

Erwin Schrödinger^{[35]} showed how to allow for correlation between the operators, giving a stronger inequality, known as the *Robertson-Schrödinger uncertainty relation*,^{[36]}^{[1]}

where the *anticommutator*, is used.

The derivation shown here incorporates and builds off of those shown in Robertson,^{[34]} Schrödinger^{[36]} and standard textbooks such as Griffiths.^{[25]}^{: 138 } For any Hermitian operator , based upon the definition of variance, we have
we let and thus

Similarly, for any other Hermitian operator in the same state for

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^{[37]} which is defined as
and thus Equation (**1**) can be written 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 and using the fact that and are Hermitian operators, we find

Similarly it can be shown that

Thus, we have and

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^{[38]} 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.^{[39]} (See the counterexample section below.) This issue can be overcome by using a variational method for the proof,^{[40]}^{[41]} or by working with an exponentiated version of the canonical commutation relations.^{[39]}

Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators and are self-adjoint 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^{[42]} for a detailed discussion of this important but technical distinction.)

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

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 Lorenzo Maccone and Arun K. Pati give non-trivial bounds on the sum of the variances for two incompatible observables.^{[43]} (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.^{[44]} due to Yichen Huang.) For two non-commuting observables and the first stronger uncertainty relation is given by
where , , is a normalized vector that is orthogonal to the state of the system and one should choose the sign of to make this real quantity a positive number.

The second stronger uncertainty relation is given by where is a state orthogonal to . The form of implies that the right-hand side of the new uncertainty relation is nonzero unless is an eigenstate of . One may note that can be an eigenstate of without being an eigenstate of either or . However, when is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero unless is an eigenstate of both.

The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components in any decomposition of the density matrix given as
Here, for the probabilities and hold. Then, using the relation
for ,
it follows that^{[45]}
where the function in the bound is defined
The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation
where on the right-hand side there is a concave roof over the decompositions of the density matrix.
The improved relation above is saturated by all single-qubit quantum states.^{[45]}

With similar arguments, one can derive a relation with a convex roof on the right-hand side^{[45]}
where denotes the quantum Fisher information and the density matrix is decomposed to pure states as
The derivation takes advantage of the fact that the quantum Fisher information is the convex roof of the variance times four.^{[46]}^{[47]}

A simpler inequality follows without a convex roof^{[48]}
which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have
while for pure states the equality holds.

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:^{[49]}

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 non-negative.

The non-negative eigenvalues then imply a corresponding non-negativity condition on the determinant, or, explicitly, after algebraic manipulation,

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.

- For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter
^{[50]}^{[51]}

The derivation of the Robertson inequality for operators and requires and to be defined. There are quantum systems where these conditions are not valid.^{[52]}
One example is a quantum particle on a ring, where the wave function depends on an angular variable in the interval . Define "position" and "momentum" operators and by
and
with periodic boundary conditions on . The definition of depends the range from 0 to . These operators satisfy the usual commutation relations for position and momentum operators, . More precisely, whenever both and are defined, and the space of such is a dense subspace of the quantum Hilbert space.^{[53]}

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
The Robertson uncertainty principle does not apply in this case: is not in the domain of the operator , since multiplication by disrupts the periodic boundary conditions imposed on .^{[39]}

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 .^{[54]}

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 beam-splitter) 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.^{[55]} 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.^{[56]}

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−encompassing both systematic and statistical errors—holds:^{[27]}

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.^{[57]}^{[58]}
Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors and . There is increasing experimental evidence^{[31]}^{[59]}^{[60]}^{[61]} 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^{[62]} *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:
Defining:
as the *inaccuracy* in the measured values of the variable *A* and
as the *resulting fluctuation* in the conjugate variable *B*, Kazuo Fujikawa^{[63]} established an uncertainty relation similar to the Heisenberg original one, but valid both for *systematic and statistical errors*:

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.^{[41]}^{[64]}^{[65]}^{[66]} 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.^{[67]} This conjecture, also studied by I. I. Hirschman^{[68]} and proven in 1975 by W. Beckner^{[69]} and by Iwo Bialynicki-Birula and Jerzy Mycielski^{[70]} is that, for two normalized, dimensionless Fourier transform pairs *f*(*a*) and *g*(*b*) where

- and

the Shannon information entropies and 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 *x*_{0} and *p*_{0} 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 wavefunction *φ*(*p*), the above constraint can be written for the corresponding entropies as

where h is the Planck constant.

Depending on one's choice of the *x _{0} p_{0}* product, the expression may be written in many ways. If

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
where h is interpreted as a dimensionless number equal to the value of the Planck constant in the chosen system of units. Note that these inequalities can be extended to multimode quantum states, or wavefunctions in more than one spatial dimension.^{[71]}

The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities^{[72]}
(equivalently, from the fact that normal distributions maximize the entropy of all such with a given variance), it readily follows that this entropic uncertainty principle is *stronger than the one based on standard deviations*, because

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 two-bin measurement, Partitioning the uniform spatial distribution into two equal bins is straightforward. We set the offset 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. |

For a particle of spin- the following uncertainty relation holds
where are angular momentum components. The relation can be derived from
and
The relation can be strengthened as^{[45]}^{[73]}
where is the quantum Fisher information.

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:^{[74]}^{[75]}^{[76]}

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 *time-bandwidth* or *duration-bandwidth* product satisfies
where and are the standard deviations of the time and frequency energy or power (i.e. squared) representations respectively.^{[77]} The minimum is attained for a Gaussian-shaped pulse (Gabor wavelet) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitude ; squaring reduces each by a factor .] Another common measure is the product of the time and frequency full width at half maximum (of the power/energy), which for the Gaussian equals (see bandwidth-limited pulse).

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 short-time 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 trade-off.

Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) 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.

Let be a sequence of *N* complex numbers and be its discrete Fourier transform.

Denote by the number of non-zero elements in the time sequence and by the number of non-zero 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 time-limiting operator and band-limiting operators, respectively. Then
where the norm is the operator norm of operators on the Hilbert space of functions on the integers modulo *N*. This inequality has implications for signal reconstruction.^{[78]}

When *N* is a prime number, a stronger inequality holds:
Discovered by Terence Tao, this inequality is also sharp.^{[79]}

Amrein–Berthier^{[80]} and Benedicks's theorem^{[81]} intuitively says that the set of points where f is non-zero and the set of points where ƒ̂ is non-zero 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^{[82]}^{[83]}

One expects that the factor *Ce*^{C|S||Σ|} may be replaced by *Ce*^{C(|S||Σ|)1/d}, which is only known if either S or Σ is convex.

The mathematician G. H. Hardy formulated the following uncertainty principle:^{[84]} it is not possible for f and ƒ̂ to both be "very rapidly decreasing". Specifically, if f in is such that
and
( an integer),
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
then
where P is a polynomial of degree (*N* − *d*)/2 and A is a real *d* × *d* positive definite matrix.

This result was stated in Beurling's complete works without proof and proved in Hörmander^{[85]} (the case ) and Bonami, Demange, and Jaming^{[86]} 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.^{[87]}

A full description of the case *ab* < 1 as well as the following extension to Schwartz class distributions appears in ref.^{[88]}

**Theorem** — If a tempered distribution is such that
and
then
for some convenient polynomial P and real positive definite matrix A of type *d* × *d*.

In 1925 Heisenberg published the *Umdeutung* (reinterpretation) paper where he showed that central aspect of quantum theory was the non-commutativity: the theory implied that the relative order of position and momentum measurement was significant. Working with Max Born and Pascual Jordan, he continued to develop matrix mechanics, that would become the first modern quantum mechanics formulation.^{[89]}

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. Writing to Wolfgang Pauli in February 1927, he worked out the basic concepts.^{[90]}

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 Δ*p*. 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^{[91]} 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 fo