"Spin system and Pauli exclusion principle"의 두 판 사이의 차이

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<h5>introduction</h5>
  
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*  the simplest example of quantum mechanical system<br>
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*  quantization of the angular momentum<br>
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*  measures as being some multiple of Planck's constant divided by 2pi<br>
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<h5 style="margin: 0px; line-height: 2em;">representation theory</h5>
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*  concept from the representation of  <math>SU(2)</math><br>
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*  half of highest weight is called the spin of the module<br>
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**  Casimir operator can also detect this number.<br>
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*  spin <math>1/2</math> is the most important case since they are the matter particles<br>
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*  this is why we have half-integral spin although those representation are integral highest weight representations.<br>
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<h5>operator formulation</h5>
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*  파울리 행렬 ([http://pythagoras0.springnote.com/pages/3063330 해밀턴의 사원수] 참조)<br><math>\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} </math><br><math>\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}  </math><br><math>\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}</math><br>
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*  raising and lowering 연산자<br><math>\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})</math><br><math>\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}</math><br><math>\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}</math><br><math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math><br>
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<h5 style="line-height: 2em; margin: 0px;">sl(2)</h5>
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*  3차원 리대수<br><math>E=</math><br><math>F</math><br><math>H=\begin{pmatrix} 1&0\ 0&-1 \end{pmatrix}</math><br>
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*  commutator<br><math>[E,F]=H</math><br><math>[H,E]=2E</math><br><math>[H,F]=-2F</math><br>
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<h5>spin particle statstics</h5>
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*  Bosons<br>
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* photon
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*  vector boson<br>
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*  Gluon<br>
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*  follows Bose-Einstein statistics<br>
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*  force-transmitting particles<br>
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*  
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*  Fermions = spin- <math>1/2</math> particles<br>
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**  quarks and leptons<br>
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**  follows Fermi-Dirac statistics<br>
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**  matter particles<br>
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*  
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<h5>history</h5>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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<h5>related items</h5>
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
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* http://en.wikipedia.org/wiki/
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* http://www.scholarpedia.org/
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* http://www.proofwiki.org/wiki/
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
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<h5>books</h5>
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* [[2010년 books and articles]]<br>
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* http://gigapedia.info/1/
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* http://gigapedia.info/1/
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
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<h5>expositions</h5>
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
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* http://www.ams.org/mathscinet
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* http://www.zentralblatt-math.org/zmath/en/
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* http://arxiv.org/
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* http://www.pdf-search.org/
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* http://pythagoras0.springnote.com/
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
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* http://dx.doi.org/
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<h5>question and answers(Math Overflow)</h5>
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* http://mathoverflow.net/search?q=
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* http://mathoverflow.net/search?q=
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<h5>blogs</h5>
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*  구글 블로그 검색<br>
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**  http://blogsearch.google.com/blogsearch?q=<br>
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** http://blogsearch.google.com/blogsearch?q=
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* http://ncatlab.org/nlab/show/HomePage
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<h5>experts on the field</h5>
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* http://arxiv.org/
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<h5>links</h5>
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* http://functions.wolfram.com/

2010년 9월 22일 (수) 18:40 판

introduction
  • the simplest example of quantum mechanical system
  • quantization of the angular momentum
  • measures as being some multiple of Planck's constant divided by 2pi

 

 

 

representation theory
  • concept from the representation of  \(SU(2)\)
  • half of highest weight is called the spin of the module
    • Casimir operator can also detect this number.
  • spin \(1/2\) is the most important case since they are the matter particles
  • this is why we have half-integral spin although those representation are integral highest weight representations.

 

 

operator formulation
  • 파울리 행렬 (해밀턴의 사원수 참조)
    \(\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \)
    \(\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \)
    \(\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\)
  • raising and lowering 연산자
    \(\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})\)
    \(\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\)
    \(\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}\)
    \([\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}\)

 

 

sl(2)
  • 3차원 리대수
    \(E=\)
    \(F\)
    \(H=\begin{pmatrix} 1&0\ 0&-1 \end{pmatrix}\)
  • commutator
    \([E,F]=H\)
    \([H,E]=2E\)
    \([H,F]=-2F\)

 

 

spin particle statstics
  • Bosons
  • photon
  • vector boson
  • Gluon
  • follows Bose-Einstein statistics
  • force-transmitting particles
  •  

 

  • Fermions = spin- \(1/2\) particles
    • quarks and leptons
    • follows Fermi-Dirac statistics
    • matter particles
  •  

 

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions

 

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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