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
  

 

 

 

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
    

 

 

Dirac wave equation[[search?q=dirac&parent id=6452627|]]

• Dirac equation
  

 

 

history

• 1924 Pauli
• 1925 Uhlenbeck and Goudsmit
• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

books

• MICHELA MASSIMI  Pauli's Exclusion Principle: The Origin and Validation of a Scientific Principle 10.1093/bjps/axn056 [1]http://bjps.oxfordjournals.org/content/60/1/235.extract
• The evolution of Pauli’s exclusion principle
  Gordon N. Fleming http://www.personal.psu.edu/gnf1/blogs/flemin/Pauli%20Exclusion%20Prin.2007a.pdf
• 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=