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Lekeitio is also the birthplace of Resurrección María de Azkue, one of the most important Basque scholars of the 19th century.

Lekeitio:2882710

Lekeitio is also the birthplace of Resurrección María de Azkue, one of the most important Basque scholars of the 19th century.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

The festivities are in honor of the patron of the town, San Antolin, and are celebrated from 1 to 8 September. One of the most popular parts of the festival is the goose pulling event.

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

The festivities are in honor of the patron of the town, San Antolin, and are celebrated from 1 to 8 September. One of the most popular parts of the festival is the goose pulling event.

Lekeitio is also the birthplace of Resurrección María de Azkue, one of the most important Basque scholars of the 19th century.

The festivities are in honor of the patron of the town, San Antolin, and are celebrated from 1 to 8 September. One of the most popular parts of the festival is the goose pulling event.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

The festivities are in honor of the patron of the town, San Antolin, and are celebrated from 1 to 8 September. One of the most popular parts of the festival is the goose pulling event.

The celebration of San Pedro takes place from 29 June, saints day, to 1 July. It begins with a mass in honor of the saint and a procession with his image. The mass is celebrated in the church of Santa Maria, and from there the procession starts to walk the streets of the town.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

The celebration of San Pedro takes place from 29 June, saints day, to 1 July. It begins with a mass in honor of the saint and a procession with his image. The mass is celebrated in the church of Santa Maria, and from there the procession starts to walk the streets of the town.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

The celebration of San Pedro takes place from 29 June, saints day, to 1 July. It begins with a mass in honor of the saint and a procession with his image. The mass is celebrated in the church of Santa Maria, and from there the procession starts to walk the streets of the town.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

A 15 m-long panel of etchings was discovered in the Armintxe Cave in 2016. Two of the etchings were of lions - the first seen in Basque Country. The art dates from 12,000 to 14,500 years ago.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

The celebration of San Pedro takes place from 29 June, saints day, to 1 July. It begins with a mass in honor of the saint and a procession with his image. The mass is celebrated in the church of Santa Maria, and from there the procession starts to walk the streets of the town.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

This day has been documented since the 5th century and it is said that its origins are older. This act has been celebrated since 1877.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

A 15 m-long panel of etchings was discovered in the Armintxe Cave in 2016. Two of the etchings were of lions - the first seen in Basque Country. The art dates from 12,000 to 14,500 years ago.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

The festivities are in honor of the patron of the town, San Antolin, and are celebrated from 1 to 8 September. One of the most popular parts of the festival is the goose pulling event.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

Lekeitio:2882710

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

This is held on 5 September, the aim being to hold on for as long as possible to a goose that is hanging from a rope that crosses the harbor from one dock to the other. The rope has one side fixed and on the other side there is a group of men pulling the rope to raise and lower it. In the middle of the rope there is a goose drenched in oil (in previous times the bird was alive). There are a lot of boats that take part and all of them have to go, in the order assigned through a random lottery in the morning, to the place the goose is and one participant from each boat has to grab the goose by the neck as strongly as possible. Once the boat has advanced to the front, the men at the end of the rope start pulling it, lifting the goose with the member holding it. Once the participants reach the top, the men let him drop from there, before rapidly pulling him up again. They continue like that until the participant lets the goose go or until the neck of the goose breaks. The one who makes most elevations wins the competition.

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

The most important monument is the church of "Santa María", a gothic basilica from the 15th century.

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

The goose pulling event was also played on dry land, (and today is still held in Markina-Xemein). Several changes have taken place in this festival because in the past all the boats were sailed by 13 sailors. Only sailors were allowed to participate and there were strict regulations governing the speed and size of the boat, which had to be manned by 12 oarsmen and the captain. If there was any doubt about the winner they arranged a race that went to the island near Lekeitio.

The festival is controversial, especially among animal rights activists, who have called for it to be stopped. During the festival people wear denim work clothes, often combined with a white shirt.

The celebration of San Pedro takes place from 29 June, saints day, to 1 July. It begins with a mass in honor of the saint and a procession with his image. The mass is celebrated in the church of Santa Maria, and from there the procession starts to walk the streets of the town.

Uncertainty principle:31883

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.

Uncertainty principle:31883

but it was not always obvious what formula_69 precisely meant. The problem is that the time at which the particle has a given state is not an operator belonging to the particle, it is a parameter describing the evolution of the system. As Lev Landau once joked "To violate the time–energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!"

Uncertainty principle:31883

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

Uncertainty principle:31883

Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit" ("indeterminacy"),

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.

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

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.

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 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.

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

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.

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

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

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

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

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

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

Uncertainty principle:31883

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

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

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

Uncertainty principle:31883

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

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.

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

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.

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

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

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

Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

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

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

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

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

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

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.

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

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.

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

Bohr was present when Einstein proposed the thought experiment which has become known as Einstein's box. Einstein argued that "Heisenberg's uncertainty equation implied that the uncertainty in time was related to the uncertainty in energy, the product of the two being related to Planck's constant." Consider, he said, an ideal box, lined with mirrors so that it can contain light indefinitely. The box could be weighed before a clockwork mechanism opened an ideal shutter at a chosen instant to allow one single photon to escape. "We now know, explained Einstein, precisely the time at which the photon left the box." "Now, weigh the box again. The change of mass tells the energy of the emitted light. In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."

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

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

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

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

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

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.

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

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

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. 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 , where 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 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.

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. 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 , where is the wavenumber.

Such variable pairs are known as complementary variables or canonically conjugate variables; and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

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.

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

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.

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.

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

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

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

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

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

but it was not always obvious what formula_69 precisely meant. The problem is that the time at which the particle has a given state is not an operator belonging to the particle, it is a parameter describing the evolution of the system. As Lev Landau once joked "To violate the time–energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!"

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. 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

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

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.

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

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. 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

formula_47 or formula_48. However, when formula_41 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

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

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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 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.

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

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

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

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

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.

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

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 coherent state is a right eigenstate of the annihilation operator,

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.

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

In the case of the single-moded plane wave, 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.

Another common misconception is that the energy–time uncertainty principle says that the conservation of energy can be temporarily violated—energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time. Although this agrees with the "spirit" of relativistic quantum mechanics, it is based on the false axiom that the energy of the universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is "violated" when quantum field theory uses temporary electron–positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of "all histories" must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.

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.

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

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

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

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

Another common misconception is that the energy–time uncertainty principle says that the conservation of energy can be temporarily violated—energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time. Although this agrees with the "spirit" of relativistic quantum mechanics, it is based on the false axiom that the energy of the universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is "violated" when quantum field theory uses temporary electron–positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of "all histories" must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.

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

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.

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

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

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

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

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

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.

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 formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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

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

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

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

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 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.

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

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

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

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

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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 formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

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.)

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

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.

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

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".

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 and that "x" and "p" are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.

One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_4 is a probability density function for position, we calculate its standard deviation.

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

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

with formula_7 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_7 is the "Fourier transform" of formula_9 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.

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

For an arbitrary Hermitian operator formula_22 we can associate a standard deviation

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

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

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

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

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

According to the de Broglie hypothesis, every object in the universe is a wave, i.e., a situation which gives rise to this phenomenon. The position of the particle is described by a wave function formula_1. The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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. Click the "show" button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

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