User:Boris Tsirelson/Sandbox1: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Boris Tsirelson
No edit summary
 
(675 intermediate revisions by one other user not shown)
Line 1: Line 1:
[http://en.wikipedia.org/wiki/Hilbert%27s_axioms Hilberts axioms]
{{AccountNotLive}}
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.
 
''Below, all points, lines and planes are situated in a three-dimensional Euclidean space, and by lines we mean [[Line (geometry)|straight lines]].''
 
====Definition via distances====
 
Let two different points ''A'' and ''B'' be given. The set of all points ''C'' that are equally far from ''A'' and ''B'' (that is, <math>|AC|=|BC|</math>) is a plane.
 
This is the plane orthogonal to the line ''AB'' through the middle point of the line segment ''AB''.
 
====Definition via right angles (orthogonality)====
 
Let two different points ''A'' and ''B'' be given. The set of all points ''C'' such that the lines ''AB'' and ''AC'' are orthogonal (that is, the angle ''BAC'' is [[Right angle (geometry)|right]]) is a plane.
 
This is the plane orthogonal to the line ''AB'' through the point ''A''.
 
====Definition via lines====
 
Let three points ''A'', ''B'' and ''C'' be given, not lying on a line. Consider the lines ''DE'' for all points ''D'' on the line ''AB'' different from ''B'' and all points ''E'' on the line ''BC'' different from ''B''. The [[union]] of all these lines, together with the point ''B'', is a plane.
 
This is the plane through ''A'', ''B'' and ''C''.
 
In other words, this plane is the set of all points ''F'' such that either ''F'' coincides with ''B'' or there exists a line through ''F'' that intersects both the line ''AB'' and the line ''BC'', and not at their intersection point ''B''.
 
====Definition 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).

Retrieved from "https://citizendium.org/wiki/index.php?title=User:Boris_Tsirelson/Sandbox1&oldid=904361"