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=Schrőder-Bernstein property=
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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.


The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. If ''X'' is similar to a part of ''Y'' and at the same time ''Y'' is similar to a part of ''X'' then ''X'' and ''Y'' are similar. In order to be specific one should decide
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).
* what kind of mathematical objects are ''X'' and ''Y'',
* what is meant by "a part",
* what is meant by "similar".
 
In the classical Cantor–Bernstein–Schroeder theorem
* ''X'' and ''Y'' are [[Set (mathematics)|sets]] (maybe infinite),
* "a part" is interpreted as a [[subset]],
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].
 
Not all statements of this form are true. For example, let
* ''X'' and ''Y'' are [[triangle]]s,
* "a part" means a triangle inside the given triangle,
* "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need no be similar.
 
==Notes==
{{reflist}}
 
==References==
 
{{Citation
| last = Srivastava
| first = S.M.
| title = A Course on Borel Sets
| year = 1998
| publisher = Springer
}}. See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
 
{{Citation
| last = Gowers
| first = W.T.
| year = 1996
| title = A solution to the Schroeder-Bernstein problem for Banach spaces
| journal = Bull. London Math. Soc.
| volume = 28
| pages = 297–304
| url = http://blms.oxfordjournals.org/content/28/3/297
}}.
 
{{Citation
| last = Casazza
| first = P.G.
| year = 1989
| title = The Schroeder-Bernstein property for Banach spaces
| journal = Contemp. Math.
| volume = 85
| pages = 61–78
| url = http://www.ams.org/mathscinet-getitem?mr=983381
}}.
 
==External links==
 
[http://sbseminar.wordpress.com/2007/10/30/theme-and-variations-schroeder-bernstein/ Theme and variations: Schroeder-Bernstein]
 
[http://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold When does Cantor Bernstein hold?]

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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