# 自旋

## 自旋量子數

### 基本粒子的自旋

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

## 自旋的方向

### 自旋投影量子數與自旋多重態

$\hbar s_z, \qquad s_z=-s, -s+1 \cdots, s-1, s$

## 自旋與磁矩

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

## 量子力學中關於自旋的數學表示

### 自旋算符

$[S_i, S_j ] = i \hbar \epsilon_{ijk} S_k$

$S^2 |s,m\rangle = \hbar^2 s(s + 1)|s,m\rangle S_z |s,m\rangle = \hbar m |s,m\rangle.$

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

### 自旋與包立不相容原理

$\psi(\,...\, ;\,\mathbf r_i,\sigma_i\,;\, ...\,;\mathbf r_j,\sigma_j\,;\,...)\stackrel{!}{=}(-1)^{2S}\cdot \psi ( \,...\, ;\,\mathbf r_j,\sigma_j\,;\, ...\,;\mathbf r_i,\sigma_i\,;\,...)\,.$

### 自旋與旋轉

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

$\sum_{m=-j}^{j} a_m^* b_m = \sum_{m=-j}^{j} (\sum_{n=-j}^j U_{nm} a_n)^* (\sum_{k=-j}^j U_{km} b_k)$
$\sum_{n=-j}^j \sum_{k=-j}^j U_{np}^* U_{kq} = \delta_{pq}.$

$\begin{pmatrix} a_{1/2}' \\ a_{-1/2}' \end{pmatrix} = \exp{(i \sigma_z \gamma / 2)} \exp{(i \sigma_y \beta / 2)} \exp{(i \sigma_x \alpha / 2)} \begin{pmatrix} a_{1/2} \\ a_{-1/2} \end{pmatrix}$

### 自旋與勞侖茲變換

$\psi' = \exp{\left(\frac{1}{8} \omega_{\mu\nu} [\gamma_{\mu}, \gamma_{\nu}]\right)} \psi$

$\langle\psi|\phi\rangle = \bar{\psi}\phi = \psi^{\dagger}\gamma_0\phi$

### 包立矩陣和自旋算符

$S_x = {\hbar \over 2} \sigma_x$
$S_y = {\hbar \over 2} \sigma_y$
$S_z = {\hbar \over 2} \sigma_z$

$\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$
$\sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$
$\sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

### 沿x, y和z軸的自旋測量

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

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

### 沿任意方向的自旋測量

$\frac{1}{\sqrt{2+2u_z}}\begin{bmatrix} 1+u_z \\ u_x+iu_y \end{bmatrix}.$

### 自旋測量的相容性

$\mid \langle \psi_{x+/-} \mid \psi_{y+/-} \rangle \mid ^ 2 = \mid \langle \psi_{x+/-} \mid \psi_{z+/-} \rangle \mid ^ 2 = \mid \langle \psi_{y+/-} \mid \psi_{z+/-} \rangle \mid ^ 2 = \frac{1}{2}$

## 參考資料

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