formulasearchengine - User contributions [en]

Glucuronate reductase

2014-06-25T18:41:28Z

150.244.204.147:

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Panel data

2013-12-13T18:38:20Z

150.244.131.81: /* Data sets which have a panel design */

{{Unreferenced stub|auto=yes|date=December 2009}}
'''Vector decomposition''' is the decomposition of a [[Vector (geometric)|vector]] of '''R'''''n'' into several vectors, all [[linearly independent]] (in mutually distinct directions in the ''n''-dimensional space).

==Vector decomposition in two dimensions==
In two dimensions, a vector can be decomposed in many ways. In the [[Cartesian coordinate system]], the vector is decomposed into a portion along the <math>\hat{x}</math> or <math>\hat{i}</math> and the <math>\hat{y}</math> or <math>\hat{j}</math> directions.

One of the most common situations is when given a vector with magnitude and direction (or given in [[polar coordinate system|polar]] form), it can be converted into the [[Vector (geometric)#Addition and subtraction|sum]] of two perpendicular vectors (or converted to a Cartesian coordinate). In order to do this it makes use of trigonometry, such as sine and cosine.

==Application in physics==
Vector decomposition is used in physics to help adding vectors and hence solve many mechanical problems involving [[force]], [[work (physics)|work]], [[momentum]], etc.

==See also==
* [[Coordinate system]]
* [[Helmholtz decomposition]] (decomposition of a [[vector field]])

[[Category:Vectors]]

{{linear-algebra-stub}}

Finite potential well

2013-10-21T16:44:05Z

150.244.37.45: /* Finding wavefunctions for the bound state */ minor edit fixed typo

In [[theoretical physics|theoretical]] [[condensed matter physics]], '''Bosonization''' is a mathematical procedure by which a system of interacting [[fermions]] in [[dimension|(1+1) dimensions]] can be transformed to a system of massless, non-interacting [[bosons]].<ref name=gogolin>{{cite book|last=Gogolin|first=Alexander O.|title=Bosonization and Strongly Correlated Systems|year=2004|publisher=Cambridge University Press|isbn=0-521-61719-7|url=http://books.google.com/books?id=BZDfFIpCoaAC&dq=Bosonization+and+Strongly+Correlated+Systems&lr=&source=gbs_navlinks_s}}</ref> The method of bosonization was conceived independently by particle physicists [[Sidney Coleman]] and [[Stanley Mandelstam]]; and condensed matter physicists [[Daniel Mattis]] and [[Alan Luther]] in 1975.<ref name=gogolin />

The basic physical idea behind bosonization is that [[Dirac sea|particle-hole excitations]] are bosonic in character. However, it was shown by [[Sin-Itiro Tomonaga|Tomonaga]] in 1950 that this principle is only valid in one-dimensional systems.<ref name=senechal>{{cite journal|last=Sénéchal|first=David|title=An Introduction to Bosonization|journal=Theoretical Methods for Strongly Correlated Electrons|year=1999|series=CRM Series in Mathematical Physics|doi=10.1007/0-387-21717-7_4|url=http://www.springerlink.com/content/q8p7094320733111/}}</ref> Bosonization is an [[effective field theory]] that focuses on low-energy excitations.<ref name=glazman>{{cite book|last=Sohn|first=Lydia (ed.)|title=Mesoscopic electron transport|year=1997|publisher=Springer|isbn=0-7923-4737-4|url=http://arxiv.org/pdf/cond-mat/9610037v1.pdf}}</ref>

Two complex fermions <math>\psi,\bar\psi</math> are written as functions of a boson <math>\phi</math>
:<math>\bar\psi_-\psi_+ = :\exp(i\phi):,\qquad \bar\psi_-\psi_+ = :\exp(-i\phi):</math><ref>
In actuality, there is a [[cocycle]] prefactor to give correct (anti-)commutation relations with other fields under consideration.</ref>
while the inverse map is given by
:<math>\partial\phi=:\bar\psi\psi:</math>
All equations are [[normal-order]]ed. The changed statistics arises from [[anomalous dimension]]s of the fields.

==References==
<references />

[[Category:Quantum field theory]]
[[Category:Condensed matter physics]]

{{quantum-stub}}

Critical point (thermodynamics)

2012-07-24T10:39:32Z

150.244.210.161: /* See also */ added supercritical fluid extraction

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