# 電子磁偶極矩

## 單一電子磁矩

${\displaystyle {\boldsymbol {\mu }}={\frac {-e}{2m_{e}}}\,\mathbf {L} .}$

me代表的是電子不變質量，請注意角動量L在此可以是自旋角動量，軌道角動量，或是總角動量。經典自旋磁矩的結果受比例因子的影響，因此，經典的結果需要乘上一個無量綱量g因子進行校正。

${\displaystyle {\boldsymbol {\mu }}=g{\frac {-e}{2m_{e}}}\mathbf {L} .}$

${\displaystyle {\boldsymbol {\mu }}=-g\mu _{B}{\frac {\mathbf {L} }{\hbar }}.}$

### 旋轉磁偶極矩

${\displaystyle {\boldsymbol {\mu }}_{S}=-g_{S}\mu _{B}{\frac {\mathbf {S} }{\hbar }}.}$

${\displaystyle \mu _{S}\approx 2{\frac {e}{2m_{e}}}{\frac {\hbar }{2}}=\mu _{B}.}$

${\displaystyle ({\boldsymbol {\mu }}_{S})_{z}=-g_{S}\mu _{B}m_{S}}$

2.00231930419922 ± (1.5 × 10−12).[3]

-928.476377 × 10−26 ± 0.000023 × 10−26 J·T−1.[4]

### g值的經典理論

${\displaystyle \rho _{e}(r)=eN_{e}e^{-r^{2}/r_{e}^{2}}}$

${\displaystyle \rho _{m}(r)=m_{e}N_{m}e^{-r^{2}/r_{m}^{2}}}$

${\displaystyle g=\left({\frac {r_{e}}{r_{m}}}\right)^{8}}$.

${\displaystyle \left({\frac {r_{e}}{r_{m}}}\right)\approx 1.09051}$

### 軌道磁偶極矩

${\displaystyle {\boldsymbol {\mu }}_{L}=-g_{L}\mu _{B}{\frac {\mathbf {L} }{\hbar }}.}$

### 總磁偶極矩

${\displaystyle {\boldsymbol {\mu }}_{J}=g_{J}\mu _{B}{\frac {\mathbf {J} }{\hbar }}.}$

g值 gJ是著名的朗德g因子，和gLgS 有關的相關內容請看朗德g因子

## 實例: 氫原子

${\displaystyle \mu _{L}=-g_{L}{\frac {\mu _{B}}{\hbar }}\langle \Psi _{n,\ell ,m}|L|\Psi _{n,\ell ,m}\rangle =-\mu _{B}{\sqrt {\ell (\ell +1)}}.}$

${\displaystyle (\mathbf {\mu _{L}} )_{z}=-\mu _{B}m_{\ell }.\,}$

## 包利和狄拉克理論的電子自旋

${\displaystyle H={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\right]^{2}+e\phi .}$

${\displaystyle H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}+e\phi -{\frac {e\hbar }{2mc}}{\boldsymbol {\sigma }}\cdot \mathbf {B} .}$

${\displaystyle \left[-i\gamma ^{\mu }\left(\partial _{\mu }+ieA_{\mu }\right)+m\right]\psi =0\,}$

${\displaystyle {\begin{pmatrix}(mc^{2}-E+e\phi )&c\sigma \cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\\-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)&\left(mc^{2}+E-e\phi \right)\end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}.}$

${\displaystyle (E-e\phi )\psi _{+}-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\psi _{-}=mc^{2}\psi _{+}}$
${\displaystyle -(E-e\phi )\psi _{-}+c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\psi _{+}=mc^{2}\psi _{-}}$

${\displaystyle E-e\phi \approx mc^{2}}$
${\displaystyle p\approx mv}$

${\displaystyle \psi _{-}\approx {\frac {1}{2mc}}{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\psi _{+}}$

${\displaystyle (E-mc^{2})\psi _{+}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\right]^{2}\psi _{+}+e\phi \psi _{+}}$

## 註釋

1. ^ B Odom et al. 2006 Phys. Rev. Lett. 97 030801.
2. ^ A. Mahajan and A. Rangwala. Electricity and Magnetism, p. 419 (1989). Via Google Books.
3. ^ http://physics.nist.gov/cgi-bin/cuu/Value?eqae%7Csearch_for=electron+magnetic+moment
4. ^ http://physics.nist.gov/cgi-bin/cuu/Value?muem%7Csearch_for=magnetic+moment+electron
5. ^ Source: Journal of Mathematical Physics, 52, 082303 (2011) (存檔副本. [2012-04-26]. （原始內容存檔於2012-07-18）. or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )
6. ^ Polykarp Kusch, H. M. Foley
7. ^ intrinsic moment of the electron