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In 1932 Dirac offered a precise definition and derivation of the time–energy uncertainty relation in a relativistic quantum theory of "events".

One "false" formulation of the energy–time uncertainty principle says that measuring the energy of a quantum system to an accuracy formula_70 requires a time interval formula_71. 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. The time formula_69 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.

In 1932 Dirac offered a precise definition and derivation of the time–energy uncertainty relation in a relativistic quantum theory of "events".

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

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

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 is performed, then the system is in a particular eigenstate of that observable. However, the particular eigenstate of the observable need not be an eigenstate of another observable : If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

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:

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

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.

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

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. 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.

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, "x", and momentum, "p", can be predicted from initial conditions.

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

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 non-trivial bounds on the sum of the variances for two incompatible observables. (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref. due to Huang.) For two non-commuting observables formula_31 and formula_32 the first stronger uncertainty relation is given by

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

Suppose we consider a quantum particle on a ring, where the wave function depends on an angular variable formula_73, which we may take to lie in the interval formula_74. Define "position" and "momentum" operators formula_25 and formula_26 by

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

where we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

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

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

There is reason to believe that violating the uncertainty principle also strongly implies the violation of the second law of thermodynamics. See Gibbs paradox.

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,

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:

The many-worlds 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 many-worlds interpretation follows from each observer within any universe having no knowledge of what goes on in the other universes.

Using the standard rules for creation and annihilation operators on the energy eigenstates,

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,

For the usual position and momentum operators formula_99 and formula_100 on the real line, no such counterexamples can occur. As long as formula_101 and formula_102 are defined in the state formula_84, the Heisenberg uncertainty principle holds, even if formula_84 fails to be in the domain of formula_105 or of formula_106.

In particular, the above Kennard bound is saturated for the ground state , for which the probability density is just the normal distribution.

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:

In particular, the above Kennard bound is saturated for the ground state , for which the probability density is just the normal distribution.

we can conclude the following: (the right most equality holds only when Ω = "ω") .

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

we can conclude the following: (the right most equality holds only when Ω = "ω") .

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 we impose periodic boundary conditions on formula_26. The definition of formula_25 depends on our choice to have formula_73 range from 0 to formula_82. These operators satisfy the usual commutation relations for position and momentum operators, formula_83.

where Ω describes the width of the initial state but need not be the same as ω. Through integration over the , we can solve for the -dependent solution. After many cancelations, the probability densities reduce to

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 , we can solve for the -dependent solution. After many cancelations, the probability densities reduce to

Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

where we have used the notation formula_117 to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

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 we have used the notation formula_117 to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

On the other hand, consider a wave function that is a sum of many waves, which we may write this as

we can conclude the following: (the right most equality holds only when Ω = "ω") .

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

we can conclude the following: (the right most equality holds only when Ω = "ω") .

Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that formula_84 is not in the domain of the operator formula_94, since multiplication by formula_73 disrupts the periodic boundary conditions imposed on formula_26. Thus, the derivation of the Robertson relation, which requires formula_97 and formula_98 to be defined, does not apply. (These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations.)

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

for some convenient polynomial and real positive definite matrix of type .

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

The non-negative eigenvalues then imply a corresponding non-negativity condition on the determinant,

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

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.

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

This was later improved as follows: if formula_128 is such that

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

There is reason to believe that violating the uncertainty principle also strongly implies the violation of the second law of thermodynamics. See Gibbs paradox.

then, if , while if , then there is a polynomial of degree such that

where is a polynomial of degree and is a real positive definite matrix.

then, if , while if , then there is a polynomial of degree such that

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

then, if , while if , then there is a polynomial of degree such that

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

then, if , while if , then there is a polynomial of degree such that

Since this positivity condition is true for "all" "a", "b", and "c", it follows that all the eigenvalues of the matrix are non-negative.

then, if , while if , then there is a polynomial of degree such that

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

then, if , while if , then there is a polynomial of degree such that

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

This was later improved as follows: if formula_128 is such that

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

This was later improved as follows: if formula_128 is such that

As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle.

This was later improved as follows: if formula_128 is such that

then, if , while if , then there is a polynomial of degree such that

This was later improved as follows: if formula_128 is such that

where the brackets formula_24 indicate an expectation value. For a pair of operators formula_25 and formula_26, we may define their "commutator" as

This was later improved as follows: if formula_128 is such that

for some convenient polynomial and real positive definite matrix of type .

This was later improved as follows: if formula_128 is such that

The principle is quite counter-intuitive, 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.

where is a polynomial of degree and is a real positive definite matrix.

This was later improved as follows: if formula_128 is such that

where is a polynomial of degree and is a real positive definite matrix.

Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

where is a polynomial of degree and is a real positive definite matrix.

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

where is a polynomial of degree and is a real positive definite matrix.

where we have used the notation formula_117 to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

where is a polynomial of degree and is a real positive definite matrix.

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

where is a polynomial of degree and is a real positive definite matrix.

One "false" formulation of the energy–time uncertainty principle says that measuring the energy of a quantum system to an accuracy formula_70 requires a time interval formula_71. 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. The time formula_69 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.

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

for some convenient polynomial and real positive definite matrix of type .

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

Using the standard rules for creation and annihilation operators on the energy eigenstates,

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit" ("uncertainty"). When the English-language version of Heisenberg's textbook, "The Physical Principles of the Quantum Theory", was published in 1930, however, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter.

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

This was later improved as follows: if formula_128 is such that

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

In the case of the single-moded plane wave, formula_4 is a uniform distribution. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet.

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

This result was stated in Beurling's complete works without proof and proved in Hörmander (the case formula_131) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case 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

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

then, if , while if , then there is a polynomial of degree such that

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

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

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

for some convenient polynomial and real positive definite matrix of type .

then, if , while if , then there is a polynomial of degree such that

for some convenient polynomial and real positive definite matrix of type .

Wolfgang Pauli called Einstein's fundamental objection to the uncertainty principle "the ideal of the detached observer" (phrase translated from the German):

for some convenient polynomial and real positive definite matrix of type .

A full description of the case as well as the following extension to Schwartz class distributions appears in ref.

for some convenient polynomial and real positive definite matrix of type .

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. 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.

for some convenient polynomial and real positive definite matrix of type .

The mathematician G. H. Hardy formulated the following uncertainty principle: it is not possible for and to both be "very rapidly decreasing". Specifically, if in formula_123 is such that

for some convenient polynomial and real positive definite matrix of type .

Now let formula_84 be any of the eigenstates of formula_26, which are given by formula_86. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator formula_25 is bounded, since formula_73 ranges over a bounded interval. Thus, in the state formula_84, the uncertainty of formula_32 is zero and the uncertainty of formula_31 is finite, so that

Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.

"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."

Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.

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

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit" ("uncertainty"). When the English-language version of Heisenberg's textbook, "The Physical Principles of the Quantum Theory", was published in 1930, however, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

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

Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

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

Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

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 formula_51 with star product ★ and a function "f", the following is generally true:

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

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

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states.

In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. 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".

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

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states.

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the "Schrödinger uncertainty relation",

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad hoc old quantum theory with modern quantum mechanics. The central premise was that the classical concept of motion does not fit at the quantum level, as electrons in an atom do not travel on sharply defined orbits. Rather, their motion is smeared out in a strange way: the Fourier transform of its time dependence only involves those frequencies that could be observed in the quantum jumps of their radiation.

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, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle:

The form of formula_42 implies that the right-hand side of the new uncertainty relation

where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states.

Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.

where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states.

On the other hand, consider a wave function that is a sum of many waves, which we may write this as