Spin system and Pauli exclusion principle

수학노트
http://bomber0.myid.net/ (토론)님의 2011년 11월 8일 (화) 13:59 판
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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
  • \(\mathbf{L}=\mathbf{r}\times \mathbf{p}\)
  • 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

 

 

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

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

 

expositions

 

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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