Spin system and Pauli exclusion principle

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
  

 

 

classical angular momentum

• L=r\times p

• A classical electron moving around a nucleus in a circular orbit
  
  • orbital angular momentum, L=m_evr
  • magnetic dipole moment, \mu= -evr/2
  • where e, m_e, v, and r are the electron´s charge, mass, velocity, and radius, respectively.
• A classical electron of homogeneous mass and charge density rotating about a symmetry axis
  
  • angular momentum, L=(3/5)m_eR^2\Omega
  • magnetic dipole moment, \mu= -(3/10)eR^2\Omega, where R and \Omega are the electron´s classical radius and rotating frequency
• gyromagnetic ratio \(\gamma = \mu/L=-e/2m_e\)
  [/pages/7141159/attachments/4562863 I15-62-g20.jpg]
  
• pictures from Gyromagnetic Ratio and Anomalous Magnetic Moment

 

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)

• 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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

 

 

 

articles

 

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/