# 自旋

## 自旋量子数

### 基本粒子的自旋

${\displaystyle S=\hbar \,{\sqrt {s(s+1))),}$

## 自旋的方向

### 自旋投影量子数与自旋多重态

${\displaystyle \hbar s_{z},\qquad s_{z}=-s,-s+1\cdots ,s-1,s}$

## 自旋与磁矩

${\displaystyle \mu =g\,{\frac {q}{2m))\,S}$

## 量子力学中关于自旋的数学表示

### 自旋算符

${\displaystyle [S_{i},S_{j}]=i\hbar \epsilon _{ijk}S_{k))$

${\displaystyle S^{2}|s,m\rangle =\hbar ^{2}s(s+1)|s,m\rangle }$
${\displaystyle S_{z}|s,m\rangle =\hbar m|s,m\rangle .}$

${\displaystyle S_{\pm }|s,m\rangle =\hbar {\sqrt {s(s+1)-m(m\pm 1)))|s,m\pm 1\rangle ,}$

### 自旋与泡利不相容原理

${\displaystyle \psi (\,...\,;\,\mathbf {r} _{i},\sigma _{i}\,;\,...\,;\mathbf {r} _{j},\sigma _{j}\,;\,...){\stackrel {!}{=))(-1)^{2S}\cdot \psi (\,...\,;\,\mathbf {r} _{j},\sigma _{j}\,;\,...\,;\mathbf {r} _{i},\sigma _{i}\,;\,...)\,.}$

### 自旋与旋转

${\displaystyle |a_{1/2}|^{2}+|a_{-1/2}|^{2}\,=1.}$

${\displaystyle \sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}(\sum _{n=-j}^{j}U_{nm}a_{n})^{*}(\sum _{k=-j}^{j}U_{km}b_{k})}$
${\displaystyle \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}$

${\displaystyle {\begin{pmatrix}a_{1/2}'\\a_{-1/2}'\end{pmatrix))=\exp {(i\sigma _{z}\gamma /2)}\exp {(i\sigma _{y}\beta /2)}\exp {(i\sigma _{x}\alpha /2)}{\begin{pmatrix}a_{1/2}\\a_{-1/2}\end{pmatrix))}$

### 自旋与洛伦兹变换

${\displaystyle \psi '=\exp {\left({\frac {1}{8))\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi }$

${\displaystyle \langle \psi |\phi \rangle ={\bar {\psi ))\phi =\psi ^{\dagger }\gamma _{0}\phi }$

### 泡利矩阵和自旋算符

${\displaystyle S_{x}={\hbar \over 2}\sigma _{x))$
${\displaystyle S_{y}={\hbar \over 2}\sigma _{y))$
${\displaystyle S_{z}={\hbar \over 2}\sigma _{z))$

${\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix))}$
${\displaystyle \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix))}$
${\displaystyle \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix))}$

### 沿x, y和z轴的自旋测量

${\displaystyle \psi _{x+}={\frac {1}{\sqrt {2))}{\begin{pmatrix}{1}\\{1}\end{pmatrix)),\psi _{x-}={\frac {1}{\sqrt {2))}{\begin{pmatrix}{1}\\{-1}\end{pmatrix))}$,
${\displaystyle \psi _{y+}={\frac {1}{\sqrt {2))}{\begin{pmatrix}{1}\\{i}\end{pmatrix)),\psi _{y-}={\frac {1}{\sqrt {2))}{\begin{pmatrix}{1}\\{-i}\end{pmatrix))}$,
${\displaystyle \psi _{z+}={\begin{pmatrix}1\\0\end{pmatrix)),\psi _{z-}={\begin{pmatrix}0\\1\end{pmatrix))}$.

${\displaystyle \psi ={\begin{pmatrix}{a+bi}\\{c+di}\end{pmatrix))}$

### 沿任意方向的自旋测量

${\displaystyle {\frac {1}{\sqrt {2+2u_{z)))){\begin{bmatrix}1+u_{z}\\u_{x}+iu_{y}\end{bmatrix)).}$

### 自旋测量的相容性

${\displaystyle \mid \langle \psi _{x+/-}\mid \psi _{y+/-}\rangle \mid ^{2}=\mid \langle \psi _{x+/-}\mid \psi _{z+/-}\rangle \mid ^{2}=\mid \langle \psi _{y+/-}\mid \psi _{z+/-}\rangle \mid ^{2}={\frac {1}{2))}$

## 参考资料

1. ^ electron g factor. The NIST Reference on Constants, Units, and Uncertainty. National Institute of Standards and Technology. 2006 [2008-10-18].