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The '''Heisenberg Uncertainty Principle''' states that it is impossible to measure [[precision|precisely]] at the same time both the position and momentum of a particle. The more precisely we wish to measure one observable, the less precisely we can measure the other at that time.  This is fundamental to [[quantum mechanics]].

The '''Heisenberg Uncertainty Principle''' is a consequence of [[quantum mechanics]], perhaps the consequence best known by lay people. It states that it is impossible to measure [[precision|precisely]] at the same time both the position and momentum of a particle. The more precisely we wish to measure one observable, the less precisely we can measure the other at that time.

In mathematical terms, the principle is stated as follows:  for any two [[quantum]] observables which do not commute, the product of the [[standard deviation]]s of the measurements of those observables must be greater than or equal to a positive constant, related to [[Planck's constant]] (<math>\hbar</math>).  

In mathematical terms, the principle is stated as follows:  for any two [[quantum]] observables which do not commute, the product of the [[standard deviation]]s of the measurements of those observables must be greater than or equal to a positive constant, related to [[Planck's constant]] (<math>\hbar</math>).  

Contrary to popular belief, this is ''not'' merely a measurement issue. While it is often stated that it is not possible to know the precise position and momentum of a particle at the same time, this is misleading; this statement implies that the particle has precisely defined position and momentum, but that information is unavailable to us. In fact, the Uncertainty Principle tells us that a particle cannot have precisely defined position or momentum.

Contrary to popular belief, this is ''not'' merely a measurement issue. While it is often stated that it is not possible to know the precise position and momentum of a particle at the same time, this is misleading; this statement implies that the particle has precisely defined position and momentum, but that information is unavailable to us. In fact, the Uncertainty Principle tells us that a particle cannot have precisely defined position or momentum.

A similar example is [[time]] and [[energy]], the product of which also has a lower limit. Quantum fluctuations are a result of this, where for short time, there is enough energy in "empty space" to create a pair of particle and antiparticle, such as [[electron]] and [[positron]]. Although this appears to be a strange or extreme process, it is the only one to explain some properties of black holes. Laser physics with ultrashort [[laser]] pulses is another field, where the limit in the product of time and energy plays an important role.

 

A similar example is [[time]] and [[energy]] (also a conjugate pair), the product of which also has a lower limit. Quantum fluctuations are a result of this, where for short time, there is enough energy in "empty space" to create a pair of particle and antiparticle, such as [[electron]] and [[positron]]. Although this appears to be a strange or extreme process, it is the only one to explain some properties of black holes. Laser physics with ultrashort [[laser]] pulses is another field, where the limit in the product of time and energy plays an important role.

==Derivation==

==Derivation==