User:Boris Tsirelson/Sandbox1: Difference between revisions

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In [[Euclidean geometry]], a '''line''' (sometimes called, more explicitly, a '''straight line''') is an abstract concept that models the common notion of a curve that does not bend, has no thickness and extends infinitely in both directions.  
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


It is closely related to other basic concepts of geometry, especially, distance: it provides the shortest path between any two of its points. Moreover, in space it can also be described as the intersection of two planes.
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
 
Assuming an (intuitive or physical) idea of the geometry of a plane, "line" can be defined in terms of distances, orthogonality, coordinates etc. (as we shall do below).
 
In a more abstract approach (vector spaces) lines are defined as one-dimensional affine subspaces.
 
In an axiomatic approach, basic concepts of elementary geometry, such as "point" and "line", are undefined primitives.
 
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In Euclidean geometry, a line (sometimes called a straight line) is a straight curve having no thickness and extending infinitely in both directions. Line, together with point, is a basic concept of elementary geometry. It is closely related to other basic concepts, especially, distance: it provides the shortest path between any two of its points. "Line" can be defined in terms of distances, orthogonality, coordinates etc. In the axiomatic approach it is an undefined primitive. In a more abstract approach a line is defined as a one-dimensional affine subspace.

Latest revision as of 03:25, 22 November 2023


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The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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