角動量算符

角動量促使在旋轉方面的運動得以數量化。在孤立系統裏，如同能量和動量，角動量是守恆的。在量子力學裏，角動量算符的概念是必要的，因為角動量的計算實現於描述量子系統的波函數，而不是經典地實現於一點或一剛體。在量子尺寸世界，分析的對象都是以波函數或量子幅來描述其機率性行為，而不是命定性()行為。

在經典力學裏，角動量 ${\displaystyle \mathbf {L} =(L_{x},\ L_{y},\ L_{z})\,\!}$ 定義為位置 ${\displaystyle \mathbf {r} =(x,\ y,\ z)\,\!}$ 與動量 ${\displaystyle \mathbf {p} =(p_{x},\ p_{y},\ p_{z})\,\!}$ 的叉積：

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

在量子力學裏，對應的角動量算符 ${\displaystyle {\hat {\mathbf {L} }}\,\!}$ 定義為位置算符 ${\displaystyle {\hat {\mathbf {r} }}\,\!}$ 與動量算符 ${\displaystyle {\hat {\mathbf {p} }}\,\!}$ 的叉積：

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

由於動量算符的形式為

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

角動量算符的形式為

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

其中，${\displaystyle \nabla \,\!}$ 是梯度算符。

在量子力學裏，每一個可觀察量所對應的算符都是厄米算符。角動量是一個可觀察量，所以，角動量算符應該也是厄米算符。讓我們現在證明這一點，思考角動量算符的 x-分量 ${\displaystyle {\hat {L}}_{x}\,\!}$ ：

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

其伴隨算符 ${\displaystyle L_{x}^{\dagger }\,\!}$ 為

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

由於 ${\displaystyle {\hat {y}}\,\!}$ 、${\displaystyle {\hat {z}}\,\!}$ 、${\displaystyle {\hat {p}}_{y}\,\!}$ 、${\displaystyle {\hat {p}}_{z}\,\!}$ ，都是厄米算符，

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

由於 ${\displaystyle {\hat {p}}_{z}\,\!}$ 與 ${\displaystyle {\hat {y}}\,\!}$ 之間、${\displaystyle {\hat {p}}_{y}\,\!}$ 與 ${\displaystyle {\hat {z}}\,\!}$ 之間分別相互對易，所以，

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

因此，${\displaystyle {\hat {L}}_{x}\,\!}$ 是一個厄米算符。類似地，${\displaystyle {\hat {L}}_{y}\,\!}$ 與 ${\displaystyle {\hat {L}}_{z}\,\!}$ 都是厄米算符。總結，角動量算符是厄米算符。

再思考 ${\displaystyle {\hat {L}}^{2}\,\!}$ 算符，

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

其伴隨算符 ${\displaystyle ({\hat {L}}^{2})^{\dagger }\,\!}$ 為

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

由於 ${\displaystyle {\hat {L}}_{x}^{2}\,\!}$ 算符、${\displaystyle {\hat {L}}_{y}^{2}\,\!}$ 算符、${\displaystyle {\hat {L}}_{z}^{2}\,\!}$ 算符，都是厄米算符，

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

所以，${\displaystyle {\hat {L}}^{2}\,\!}$ 算符是厄米算符。

兩個算符 ${\displaystyle {\hat {A}}\,\!}$ 與 ${\displaystyle {\hat {B}}\,\!}$ 的交換算符 ${\displaystyle [{\hat {A}},\ {\hat {B}}]\,\!}$ ，表示出它們之間的對易關係。

採用球坐標。展開角動量算符的方程式：

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

其中，${\displaystyle \mathbf {e} _{r}\,\!}$ 、${\displaystyle \mathbf {e} _{\theta }\,\!}$ 、${\displaystyle \mathbf {e} _{\phi }\,\!}$ ，分別為徑向單位向量、天頂角單位向量、與方位角單位向量。

轉換回直角坐標，

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

其中，${\displaystyle \mathbf {e} _{x}\,\!}$ 、${\displaystyle \mathbf {e} _{y}\,\!}$ 、${\displaystyle \mathbf {e} _{z}\,\!}$ ，分別為 x-單位向量、y-單位向量、與 z-單位向量。

所以，${\displaystyle {\hat {L}}_{x}\,\!}$ 、${\displaystyle {\hat {L}}_{y}\,\!}$ 、${\displaystyle {\hat {L}}_{z}\,\!}$ 分別是

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

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

${\displaystyle {\hat {L}}_{z}={\frac {\hbar }{i}}{\frac {\partial }{\partial \phi }}\,\!}$ 。

角動量平方算符是

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

其中，

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

${\displaystyle \left.+\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}-\cot ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}+\cot ^{2}\theta \cos ^{2}\phi {\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)\,\!}$、

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

${\displaystyle \left.-\cot \theta \sin \phi \cos \phi {\frac {\partial ^{2}}{\partial \theta \partial \phi }}+\cot ^{2}\theta \sin \phi \cos \phi {\frac {\partial }{\partial \phi }}+\cot ^{2}\theta \sin ^{2}\phi {\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)\,\!}$ 、

${\displaystyle {\hat {L}}_{z}^{2}=-\hbar ^{2}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\,\!}$ 。

經過一番繁雜的運算，終於得到想要的方程式[2]^:169

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

滿足算符 ${\displaystyle {\hat {L}}^{2}\,\!}$ 的本徵函數是球諧函數 ${\displaystyle Y_{\ell m}\,\!}$ ：

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

其中，本徵值 ${\displaystyle \ell \,\!}$ 是正整數。

球諧函數也是滿足算符 ${\displaystyle {\hat {L}}_{z}\,\!}$ 微分方程式的本徵函數：

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

其中，本徵值 ${\displaystyle m\,\!}$ 是整數，${\displaystyle -\ell \leq m\leq 0\,\!}$ 。

因為這兩個算符的正則對易關係是 0 ，它們可以有共同的本徵函數。

球諧函數 ${\displaystyle Y_{\ell m}\,\!}$ 表達為

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

其中，${\displaystyle i\,\!}$ 是虛數單位，${\displaystyle P_{\ell m}(\cos {\theta })\,\!}$ 是伴隨勒讓德多項式，用方程式定義為

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

而 ${\displaystyle P_{\ell }(x)\,\!}$ 是 ${\displaystyle \ell }$ 階勒讓德多項式，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 }} 。

球諧函數滿足正交歸一性：

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

這樣，角動量算符的本徵函數，形成一組單範正交基。任意波函數 ${\displaystyle \psi (\theta ,\,\phi )\,\!}$ 都可以表達為這單範正交基的線性組合：

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

其中，${\displaystyle A_{\ell m}=\int _{0}^{2\pi }\int _{0}^{\pi }\ Y_{\ell m}^{*}(\theta ,\,\phi )\psi (\theta ,\,\phi )\sin(\theta )d\theta d\phi \,\!}$ 。

• 氫原子
• 球對稱位勢
• 拉普拉斯-龍格-冷次向量

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

• 圣地牙哥加州大学物理系量子力学視聽教學：角動量加法