User:Boris Tsirelson/Sandbox1: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Boris Tsirelson
No edit summary
 
(685 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.
 
====Definition via distances====
 
Plane via [[Right angle (geometry)|right angles]] (orthogonality):
 
Plane 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).

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