# 自旋-軌道作用

## 電子的自旋-軌道作用

### 磁場

$\mathbf{B} = - \,\frac{\mathbf{v} \times \mathbf{E}}{c^2}\,\!$(1)

$\mathbf{E}=\frac{Ze}{4\pi\epsilon_0 r^2}\hat{\mathbf{r}} =\frac{Ze}{4\pi\epsilon_0 r^3}\mathbf{r} \,\!$

$\mathbf{p}=m\mathbf{v}\,\!$

$\mathbf{B} = \frac{Ze}{4\pi\epsilon_0 m c^2 r^3}\,\mathbf{r}\times\mathbf{p}= \frac{Ze}{4\pi\epsilon_0 m c^2 r^3}\,\mathbf{L}\,\!$(2)

$\mathbf{B}\,\!$ 是一個正值因子乘以 $\mathbf{L}\,\!$ ，也就是說，磁場與電子的軌道角動量平行。

### 磁矩

$\boldsymbol{\mu} = \gamma\,\mathbf{S}\,\!$

$\boldsymbol{\mu} = - \frac{e}{m}\mathbf{S}\,\!$(3)

### 哈密頓量微擾項目

$H'= - \boldsymbol{\mu}\cdot\mathbf{B}\,\!$

$H'=\frac{Ze^2}{4\pi\epsilon_0 m^2 c^2}\ \frac{\mathbf{L}\cdot\mathbf{S}}{r^3}\,\!$

$H'=\frac{Ze^2}{8\pi\epsilon_0 m^2 c^2}\ \frac{\mathbf{L}\cdot\mathbf{S}}{r^3}\,\!$

### 能級位移

$\mathbf{J}=\mathbf{L}+\mathbf{S}\,\!$

$\mathbf J^2=\mathbf L^2+\mathbf S^2+2\mathbf{L}\cdot \mathbf{S}\,\!$

$\mathbf{L}\cdot\mathbf{S}= {1\over 2}(\mathbf{J}^2 - \mathbf{L}^2 - \mathbf{S}^2)\,\!$

\begin{align} \langle n,j,l,s\,|\,\mathbf{L}\cdot\mathbf{S} \,|\,n,j,l,s\rangle & ={1\over 2}(\langle\mathbf{J}^2\rangle - \langle\mathbf{L}^2\rangle - \langle\mathbf{S}^2\rangle) \\ & ={\hbar^2\over 2}[j(j+1) - l(l+1) - s(s+1)] \\ & ={\hbar^2\over 2}[j(j+1) - l(l+1) - 3/4] \\ \end{align}\,\!

$\langle n,j,l,s\,|\, r^{ - 3}\, |\,n,j,l,s\rangle=\frac{2Z^3}{a_0^3 n^3 l(l+1)(2l+1)}\,\!$

$E_n^{(1)}=\frac{Z^4 e^2 \hbar^2}{8\pi\epsilon_0 m^2 c^2 a_0^3}\ \frac{[j(j+1) - l(l+1) - 3/4]}{n^3\, l(l+1)(2l+1)}\,\!$

$E_n^{(1)}=\frac{(E_n^{(0)})^2}{mc^2}\ \frac{2n[j(j+1) - l(l+1) - 3/4]}{l(l+1)(2l+1)}\,\!$

$Y_0^0=\frac{1}{\sqrt{4\pi}}\,\!$

## 參考文獻

1. ^ French, A. P. Special Relativity (The M.I.T Introductory Physics Series). W. W. Norton & Company, Inc. 1968: pp. 237–250. ISBN 0748764224.
2. ^ Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 266–276. ISBN 0-13-111892-7.
• E. U. Condon and G. H. Shortley. The Theory of Atomic Spectra. Cambridge University Press. 1935. ISBN 0-521-09209-4.