User:Boris Tsirelson/Sandbox1: Difference between revisions

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[http://en.wikipedia.org/wiki/Hilbert%27s_axioms Hilberts axioms]
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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.


[http://en.wikipedia.org/wiki/Axiomatization axiomatization]
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).
 
[http://www.dpmms.cam.ac.uk/~wtg10/definition.html definition]
 
[[chess]]
 
[[Euclidean space]]
 
[[Euclidean plane]]
 
[[Circle (mathematics)]]
 
=Plane=
 
==Non-axiomatic approach==
 
===Definitions===
 
====A remark====
 
To define a plane is more complicated than it may seem.
 
It is tempting to define a plane as a surface with zero curvature (or something like that). However, this is not a good idea, since the notions of surface and curvature are much more complicated than the notion of plane. In fact, several different notions of surface are introduced by topology and differential geometry, and several different notions of curvature are introduced by differential geometry; these are far beyond elementary mathematics. Fortunately, it is possible to define a plane via more elementary notions, and this way is preferred in mathematics. Still, some problems remain, see "axiomatic approach" below.
 
Several equivalent definitions of plane given below may be compared with the definition of [[Circle (mathematics)|circle]] as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center). A circle is a set of points chosen according to their relation to some given parameters (center and radius). Similarly, a plane is a set of points chosen according to their relation to some given objects (points, lines etc). However, a circle determines its center and radius uniquely; for a plane, the situation is different.
 
====Definition via distances====
 
Let two different points ''A'' and ''B'' be given (in the three-dimensional space). The set of all points ''C'' that are equally far from ''A'' and ''B'' (that is, <math>|AC|=|BC|</math>) is a plane.
 
It is the plane orthogonal to the line ''AB'' through the middle point of the line segment ''AB''.
 
====Definition via [[Right angle (geometry)|right angles]] (orthogonality)====
 
Let two different points ''A'' and ''B'' be given (in the three-dimensional space). The set of all points ''C'' such that the lines ''AB'' and ''AC'' are orthogonal (that is, the angle ''BAC'' is right) is a plane.
 
It is the plane orthogonal to the line ''AB'' through the point ''A''.
 
====Definition via [[Line (geometry)|straight lines]]====
 
Plane via [[cartesian coordinates]]:
 
==Axiomatic approach==
 
==Modern approach==

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


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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