# 角動量算符

## 數學定義

${\displaystyle \mathbf {L} \ {\stackrel {def}{=}}\ \mathbf {r} \times \mathbf {p} \,\!}$

${\displaystyle {\hat {\mathbf {L} }}\ {\stackrel {def}{=}}\ {\hat {\mathbf {r} }}\times {\hat {\mathbf {p} }}\,\!}$

${\displaystyle {\hat {\mathbf {p} }}=-i\hbar \nabla \,\!}$

${\displaystyle {\hat {\mathbf {L} }}=-i\hbar ({\hat {\mathbf {r} }}\times \nabla )\,\!}$

## 角動量是厄米算符

${\displaystyle {\hat {L}}_{x}={\hat {y}}{\hat {p}}_{z}-{\hat {z}}{\hat {p}}_{y}\,\!}$

${\displaystyle {\hat {L}}_{x}^{\dagger }=({\hat {y}}{\hat {p}}_{z}-{\hat {z}}{\hat {p}}_{y})^{\dagger }={\hat {p}}_{z}^{\dagger }{\hat {y}}^{\dagger }-{\hat {p}}_{y}^{+}{\hat {z}}^{\dagger }\,\!}$

${\displaystyle {\hat {L}}_{x}^{\dagger }={\hat {p}}_{z}{\hat {y}}-{\hat {p}}_{y}{\hat {z}}\,\!}$

${\displaystyle {\hat {L}}_{x}^{\dagger }={\hat {y}}{\hat {p}}_{z}-{\hat {z}}{\hat {p}}_{y}={\hat {L}}_{x}\,\!}$

${\displaystyle {\hat {L}}^{2}={\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2}\,\!}$

${\displaystyle ({\hat {L}}^{2})^{\dagger }=({\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2})^{\dagger }=({\hat {L}}_{x}^{2})^{\dagger }+({\hat {L}}_{y}^{2})^{\dagger }+({\hat {L}}_{z}^{2})^{\dagger }\,\!}$

${\displaystyle ({\hat {L}}^{2})^{\dagger }={\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2}={\hat {L}}^{2}\,\!}$

## 對易關係

### 角動量算符與自己的對易關係

{\displaystyle {\begin{aligned}\left.\right.[{\hat {L}}_{x},\ {\hat {L}}_{y}]&=[{\hat {y}}{\hat {p}}_{z}-{\hat {z}}{\hat {p}}_{y},\ {\hat {z}}{\hat {p}}_{x}-{\hat {x}}{\hat {p}}_{z}]\\&=[{\hat {y}}{\hat {p}}_{z},\ {\hat {z}}{\hat {p}}_{x}]-[{\hat {z}}{\hat {p}}_{y},\ {\hat {z}}{\hat {p}}_{x}]-[{\hat {y}}{\hat {p}}_{z},\ {\hat {x}}{\hat {p}}_{z}]+[{\hat {z}}{\hat {p}}_{y},\ {\hat {x}}{\hat {p}}_{z}]\\&=i\hbar ({\hat {x}}{\hat {p}}_{y}-{\hat {y}}{\hat {p}}_{x})\\&=i\hbar {\hat {L}}_{z}\\\end{aligned}}\,\!}

${\displaystyle \Delta L_{x}\ \Delta L_{y}\geq \left|{\frac {\langle [{\hat {L}}_{x},\ {\hat {L}}_{y}]\rangle }{2i}}\right|={\frac {\hbar |\langle {\hat {L}}_{z}\rangle |}{2}}\,\!}$

${\displaystyle L_{x}\,\!}$ 的不確定性與 ${\displaystyle L_{y}\,\!}$ 的不確定性的乘積 ${\displaystyle \Delta L_{x}\ \Delta L_{y}\,\!}$ ，必定大於或等於 ${\displaystyle {\frac {\hbar |\langle L_{z}\rangle |}{2}}\,\!}$

${\displaystyle L_{x}\,\!}$${\displaystyle L_{z}\,\!}$ 之間，${\displaystyle L_{y}\,\!}$${\displaystyle L_{z}\,\!}$ 之間，也有類似的特性。

### 角動量平方算符與角動量算符之間的對易關係

{\displaystyle {\begin{aligned}\left.\right.[{\hat {L}}^{2},\ {\hat {L}}_{z}]&=[{\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2},\ {\hat {L}}_{z}]\\&={\hat {L}}_{x}{\hat {L}}_{x}{\hat {L}}_{z}-{\hat {L}}_{z}{\hat {L}}_{x}{\hat {L}}_{x}+{\hat {L}}_{y}{\hat {L}}_{y}{\hat {L}}_{z}-{\hat {L}}_{z}{\hat {L}}_{y}{\hat {L}}_{y}\\&={\hat {L}}_{x}({\hat {L}}_{z}{\hat {L}}_{x}-i\hbar {\hat {L}}_{y})-({\hat {L}}_{x}{\hat {L}}_{z}+i\hbar {\hat {L}}_{y}){\hat {L}}_{x}+{\hat {L}}_{y}({\hat {L}}_{z}{\hat {L}}_{y}+i\hbar {\hat {L}}_{x})-({\hat {L}}_{y}{\hat {L}}_{z}-i\hbar {\hat {L}}_{x}){\hat {L}}_{y}\\&=0\\\end{aligned}}\,\!}

${\displaystyle {\hat {L}}^{2}\,\!}$${\displaystyle {\hat {L}}_{z}\,\!}$對易的${\displaystyle L^{2}\,\!}$${\displaystyle L_{z}\,\!}$ 彼此是相容可觀察量，兩個算符有共同的本徵態。根據不確定性原理，我們可以同時地測量到 ${\displaystyle L^{2}\,\!}$${\displaystyle L_{z}\,\!}$ 的本徵值。

${\displaystyle [{\hat {L}}^{2},\ {\hat {L}}_{x}]=0\,\!}$
${\displaystyle [{\hat {L}}^{2},\ {\hat {L}}_{y}]=0\,\!}$

${\displaystyle {\hat {L}}^{2}\,\!}$${\displaystyle {\hat {L}}_{x}\,\!}$ 之間、${\displaystyle {\hat {L}}^{2}\,\!}$${\displaystyle {\hat {L}}_{y}\,\!}$ 之間，都分別擁有類似的物理特性。

### 哈密頓算符與角動量算符之間的對易關係

${\displaystyle [{\hat {H}},\ {\hat {L}}_{z}]=\left[i\hbar {\frac {\partial }{\partial t}},\ {\hat {x}}{\hat {p}}_{y}-{\hat {y}}{\hat {p}}_{x}\right]=0\,\!}$

${\displaystyle {\hat {H}}\,\!}$${\displaystyle {\hat {L}}_{z}\,\!}$對易的${\displaystyle H\,\!}$${\displaystyle L_{z}\,\!}$ 彼此是相容可觀察量，兩個算符擁有共同的本徵態。根據不確定性原理，我們可以同時地測量到 ${\displaystyle H\,\!}$${\displaystyle L_{z}\,\!}$ 的同樣的本徵值。

${\displaystyle [{\hat {H}},\ {\hat {L}}_{x}]=0\,\!}$
${\displaystyle [{\hat {H}},\ {\hat {L}}_{y}]=0\,\!}$

${\displaystyle {\hat {H}}\,\!}$${\displaystyle {\hat {L}}_{x}\,\!}$ 之間，${\displaystyle {\hat {H}}\,\!}$${\displaystyle {\hat {L}}_{y}\,\!}$ 之間，都分別擁有類似的物理特性。

### 在經典力學裏的對易關係

${\displaystyle \{L_{i},\ L_{j}\}=\epsilon _{ijk}L_{k}\,\!}$

## 本徵值與本徵函數

{\displaystyle {\begin{aligned}{\hat {\mathbf {L} }}&={\frac {\hbar }{i}}{\hat {\mathbf {r} }}\times \nabla \\&={\frac {\hbar }{i}}r\mathbf {e} _{r}\times \left(\mathbf {e} _{r}{\frac {\partial }{\partial r}}+\mathbf {e} _{\theta }{\frac {1}{r}}{\frac {\partial }{\partial \theta }}+\mathbf {e} _{\phi }{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \phi }}\right)\\&={\frac {\hbar }{i}}\left(-\mathbf {e} _{\theta }{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}+\mathbf {e} _{\phi }{\frac {\partial }{\partial \theta }}\right)\\\end{aligned}}\,\!}

${\displaystyle {\hat {\mathbf {L} }}={\frac {\hbar }{i}}\left[\mathbf {e} _{x}\left(-\sin \phi {\frac {\partial }{\partial \theta }}-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\right)+\mathbf {e} _{y}\left(\cos \phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi {\frac {\partial }{\partial \phi }}\right)+\mathbf {e} _{z}{\frac {\partial }{\partial \phi }}\right]\,\!}$

${\displaystyle {\hat {L}}_{x}={\frac {\hbar }{i}}\left(-\sin \phi {\frac {\partial }{\partial \theta }}-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\right)\,\!}$
${\displaystyle {\hat {L}}_{y}={\frac {\hbar }{i}}\left(\cos \phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi {\frac {\partial }{\partial \phi }}\right)\,\!}$
${\displaystyle {\hat {L}}_{z}={\frac {\hbar }{i}}{\frac {\partial }{\partial \phi }}\,\!}$

${\displaystyle {\hat {L}}^{2}={\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2}\,\!}$

{\displaystyle {\begin{aligned}{\hat {L}}_{x}^{2}&=-\hbar ^{2}\left(-\sin \phi {\frac {\partial }{\partial \theta }}-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\right)\left(-\sin \phi {\frac {\partial }{\partial \theta }}-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\right)\\&=-\hbar ^{2}\left(\sin ^{2}\phi {\frac {\partial ^{2}}{\partial \theta ^{2}}}+\cot \theta \cos ^{2}\phi {\frac {\partial }{\partial \theta }}+\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}-\csc ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}\right.\\\end{aligned}}\,\!}
${\displaystyle \left.+\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}-\cot ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}+\cot ^{2}\theta \cos ^{2}\phi {\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)\,\!}$
{\displaystyle {\begin{aligned}{\hat {L}}_{y}^{2}&=-\hbar ^{2}\left(\cos \phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi {\frac {\partial }{\partial \phi }}\right)\left(\cos \phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi {\frac {\partial }{\partial \phi }}\right)\\&=-\hbar ^{2}\left(\cos ^{2}\phi {\frac {\partial ^{2}}{\partial \theta ^{2}}}+\cot \theta \sin ^{2}\phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}+\csc ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}\right.\\\end{aligned}}\,\!}
${\displaystyle \left.-\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}+\cot ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}+\cot ^{2}\theta \sin ^{2}\phi {\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)\,\!}$
${\displaystyle {\hat {L}}_{z}^{2}=-\hbar ^{2}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\,\!}$

${\displaystyle {\hat {L}}^{2}=-\hbar ^{2}\left({\frac {\partial ^{2}}{\partial \theta ^{2}}}+\cot \theta {\frac {\partial }{\partial \theta }}+(1+\cot ^{2}\theta ){\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)=-\hbar ^{2}\left({\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)\,\!}$

${\displaystyle {\hat {L}}^{2}Y_{\ell m}=-\ell (\ell +1)\hbar ^{2}Y_{\ell m}\,\!}$

${\displaystyle {\hat {L}}_{z}Y_{\ell m}=m\hbar Y_{\ell m}\,\!}$

${\displaystyle Y_{\ell m}(\theta ,\ \phi )=(i)^{m+|m|}{\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell m}(\cos {\theta })\,e^{im\phi }\,\!}$

${\displaystyle P_{\ell m}(x)=(1-x^{2})^{|m|/2}\ {\frac {d^{|m|}}{dx^{|m|}}}P_{\ell }(x)\,}$

${\displaystyle P_{\ell }(x)\,\!}$${\displaystyle \ell }$勒讓德多項式，可用羅德里格公式表示為：

${\displaystyle P_{\ell }(x)={1 \over 2^{\ell }\ell !}{d^{\ell } \over dx^{\ell }}(x^{2}-1)^{\ell }}$

${\displaystyle \int _{0}^{2\pi }\int _{0}^{\pi }\ Y_{\ell _{1}m_{1}}Y_{\ell _{2}m_{2}}\sin(\theta )d\theta d\phi =\delta _{\ell _{1}\ell _{2}}\delta _{m_{1}m_{2}}\,\!}$

${\displaystyle \psi (\theta ,\,\phi )=\sum _{\ell ,m}\ A_{\ell m}Y_{\ell m}(\theta ,\,\phi )\,\!}$

## 參考文獻

1. ^ Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, ISBN 0201547155
2. ^ Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7

## 外部連結

• 圣地牙哥加州大学物理系量子力学視聽教學：角動量加法