角动量算符

角动量促使在旋转方面的运动得以数量化。在孤立系统里，如同能量和动量，角动量是守恒的。在量子力学里，角动量算符的概念是必要的，因为角动量的计算实现于描述量子系统的波函数，而不是经典地实现于一点或一刚体。在量子尺寸世界，分析的对象都是以波函数或量子幅来描述其概率性行为，而不是命定性(deterministic)行为。

在经典力学里，角动量 ${\displaystyle \mathbf {L} =(L_{x},\ L_{y},\ L_{z})\,\!}$  定义为位置 ${\displaystyle \mathbf {r} =(x,\ y,\ z)\,\!}$  与动量 ${\displaystyle \mathbf {p} =(p_{x},\ p_{y},\ p_{z})\,\!}$  的叉积：

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

在量子力学里，对应的角动量算符 ${\displaystyle {\hat {\mathbf {L} }}\,\!}$  定义为位置算符 ${\displaystyle {\hat {\mathbf {r} }}\,\!}$  与动量算符 ${\displaystyle {\hat {\mathbf {p} }}\,\!}$  的叉积：

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

由于动量算符的形式为

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

角动量算符的形式为

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

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

在量子力学里，每一个可观察量所对应的算符都是厄米算符。角动量是一个可观察量，所以，角动量算符应该也是厄米算符。让我们现在证明这一点，思考角动量算符的 x-分量 ${\displaystyle {\hat {L}}_{x}\,\!}$  ：

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

其伴随算符 ${\displaystyle L_{x}^{\dagger }\,\!}$  为

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

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

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

由于 ${\displaystyle {\hat {p}}_{z}\,\!}$  与 ${\displaystyle {\hat {y}}\,\!}$  之间、${\displaystyle {\hat {p}}_{y}\,\!}$  与 ${\displaystyle {\hat {z}}\,\!}$  之间分别相互对易，所以，

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

因此，${\displaystyle {\hat {L}}_{x}\,\!}$  是一个厄米算符。类似地，${\displaystyle {\hat {L}}_{y}\,\!}$  与 ${\displaystyle {\hat {L}}_{z}\,\!}$  都是厄米算符。总结，角动量算符是厄米算符。

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

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

其伴随算符 ${\displaystyle ({\hat {L}}^{2})^{\dagger }\,\!}$  为

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

由于 ${\displaystyle {\hat {L}}_{x}^{2}\,\!}$  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。

两个算符 ${\displaystyle {\hat {A}}\,\!}$  与 ${\displaystyle {\hat {B}}\,\!}$  的交换算符 ${\displaystyle [{\hat {A}},\ {\hat {B}}]\,\!}$  ，表示出它们之间的对易关系。

角动量算符与自己的对易关系

思考 ${\displaystyle {\hat {L}}_{x}\,\!}$  与 ${\displaystyle {\hat {L}}_{y}\,\!}$  的交换算符，

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

由于两者的对易关系不等于 0 ， ${\displaystyle L_{x}\,\!}$  与 ${\displaystyle L_{y}\,\!}$  彼此是不相容可观察量。${\displaystyle {\hat {L}}_{x}\,\!}$  与 ${\displaystyle {\hat {L}}_{y}\,\!}$  绝对不会有共同的基底量子态。一般而言，${\displaystyle {\hat {L}}_{x}\,\!}$  的本征态与 ${\displaystyle {\hat {L}}_{y}\,\!}$  的本征态不同。

给予一个量子系统，量子态为 ${\displaystyle |\psi \rangle \,\!}$  。对于可观察量算符 ${\displaystyle {\hat {L}}_{x}\,\!}$  ，所有本征值为 ${\displaystyle \ell _{xi}\,\!}$  的本征态 ${\displaystyle |f_{i}\rangle ,\quad i=1,\ 2,\ 3,\ \cdots \,\!}$  ，形成了一组基底量子态。量子态 ${\displaystyle |\psi \rangle \,\!}$  可以表达为这基底量子态的线性组合：${\displaystyle |\psi \rangle =\sum _{i}\ |f_{i}\rangle \langle f_{i}|\psi \rangle \,\!}$  。对于可观察量算符 ${\displaystyle {\hat {L}}_{y}\,\!}$  ，所有本征值为 ${\displaystyle \ell _{yi}\,\!}$  的本征态 ${\displaystyle |g_{i}\rangle ,\quad i=1,\ 2,\ 3,\ \cdots \,\!}$  ，形成了另外一组基底量子态。量子态 ${\displaystyle |\psi \rangle \,\!}$  可以表达为这基底量子态的线性组合：${\displaystyle |\psi \rangle =\sum _{i}\ |g_{i}\rangle \langle g_{i}|\psi \rangle \,\!}$  。

根据哥本哈根诠释，量子测量可以用量子态坍缩机制来诠释。假若，我们测量可观察量 ${\displaystyle L_{x}\,\!}$  ，得到的测量值为其本征值 ${\displaystyle \ell _{xi}\,\!}$  ，则量子态概率地坍缩为本征态 ${\displaystyle |f_{i}\rangle \,\!}$  。假若，我们立刻再测量可观察量 ${\displaystyle L_{x}\,\!}$  ，得到的答案必定是 ${\displaystyle \ell _{xi}\,\!}$  ，量子态仍旧处于 ${\displaystyle |f_{i}\rangle \,\!}$  。可是，假若，我们改为测量可观察量 ${\displaystyle L_{y}\,\!}$  ，则量子态不会停留于本征态 ${\displaystyle |f_{i}\rangle \,\!}$  ，而会坍缩为 ${\displaystyle {\hat {L}}_{y}\,\!}$  的本征态。假若，得到的测量值为其本征值 ${\displaystyle \ell _{yj}\,\!}$  ，则量子态概率地坍缩为本征态 ${\displaystyle |g_{j}\rangle \,\!}$  。

根据不确定性原理，

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

${\displaystyle L_{x}\,\!}$  的不确定性与 ${\displaystyle L_{y}\,\!}$  的不确定性的乘积 ${\displaystyle \Delta L_{x}\ \Delta L_{y}\,\!}$  ，必定大于或等于 ${\displaystyle {\frac {\hbar |\langle L_{z}\rangle |}{2}}\,\!}$  。

${\displaystyle L_{x}\,\!}$  与 ${\displaystyle L_{z}\,\!}$  之间，${\displaystyle L_{y}\,\!}$  与 ${\displaystyle L_{z}\,\!}$  之间，也有类似的特性。

角动量平方算符与角动量算符之间的对易关系

思考 ${\displaystyle {\hat {L}}^{2}\,\!}$  与 ${\displaystyle {\hat {L}}_{z}\,\!}$  的交换算符，

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

${\displaystyle {\hat {L}}^{2}\,\!}$  与 ${\displaystyle {\hat {L}}_{z}\,\!}$  是对易的，${\displaystyle L^{2}\,\!}$  与 ${\displaystyle L_{z}\,\!}$  彼此是相容可观察量，两个算符有共同的本征态。根据不确定性原理，我们可以同时地测量到 ${\displaystyle L^{2}\,\!}$  与 ${\displaystyle L_{z}\,\!}$  的本征值。

类似地，

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

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

${\displaystyle {\hat {L}}^{2}\,\!}$  与 ${\displaystyle {\hat {L}}_{x}\,\!}$  之间、${\displaystyle {\hat {L}}^{2}\,\!}$  与 ${\displaystyle {\hat {L}}_{y}\,\!}$  之间，都分别拥有类似的物理特性。

哈密顿算符与角动量算符之间的对易关系

思考哈密顿算符 ${\displaystyle {\hat {H}}\,\!}$  与 ${\displaystyle {\hat {L}}_{z}\,\!}$  的交换算符，

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

${\displaystyle {\hat {H}}\,\!}$  与 ${\displaystyle {\hat {L}}_{z}\,\!}$  是对易的，${\displaystyle H\,\!}$  与 ${\displaystyle L_{z}\,\!}$  彼此是相容可观察量，两个算符拥有共同的本征态。根据不确定性原理，我们可以同时地测量到 ${\displaystyle H\,\!}$  与 ${\displaystyle L_{z}\,\!}$  的同样的本征值。

类似地，

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

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

${\displaystyle {\hat {H}}\,\!}$  与 ${\displaystyle {\hat {L}}_{x}\,\!}$  之间，${\displaystyle {\hat {H}}\,\!}$  与 ${\displaystyle {\hat {L}}_{y}\,\!}$  之间，都分别拥有类似的物理特性。

在经典力学里的对易关系

在经典力学里，角动量算符也遵守类似的对易关系：

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

其中，${\displaystyle \{\ ,\ \}\,\!}$  是泊松括号，${\displaystyle \epsilon _{ijk}\,\!}$  是列维-奇维塔符号，${\displaystyle i\,\!}$  、${\displaystyle j\,\!}$  、${\displaystyle k\,\!}$  ，代表直角坐标 ${\displaystyle (x,\ y,\ z)\,\!}$  。

采用球坐标。展开角动量算符的方程：

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

其中，${\displaystyle \mathbf {e} _{r}\,\!}$  、${\displaystyle \mathbf {e} _{\theta }\,\!}$  、${\displaystyle \mathbf {e} _{\phi }\,\!}$  ，分别为径向单位向量、天顶角单位向量、与方位角单位向量。

转换回直角坐标，

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

其中，${\displaystyle \mathbf {e} _{x}\,\!}$  、${\displaystyle \mathbf {e} _{y}\,\!}$  、${\displaystyle \mathbf {e} _{z}\,\!}$  ，分别为 x-单位向量、y-单位向量、与 z-单位向量。

所以，${\displaystyle {\hat {L}}_{x}\,\!}$  、${\displaystyle {\hat {L}}_{y}\,\!}$  、${\displaystyle {\hat {L}}_{z}\,\!}$  分别是

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

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

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

角动量平方算符是

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

其中，

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

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

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

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

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

经过一番繁杂的运算，终于得到想要的方程^[2]:169

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

满足算符 ${\displaystyle {\hat {L}}^{2}\,\!}$  的本征函数是球谐函数 ${\displaystyle Y_{\ell m}\,\!}$  ：

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

其中，本征值 ${\displaystyle \ell \,\!}$  是正整数。

球谐函数也是满足算符 ${\displaystyle {\hat {L}}_{z}\,\!}$  微分方程的本征函数：

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

其中，本征值 ${\displaystyle m\,\!}$  是整数，${\displaystyle -\ell \leq m\leq 0\,\!}$  。

因为这两个算符的正则对易关系是 0 ，它们可以有共同的本征函数。

球谐函数 ${\displaystyle Y_{\ell m}\,\!}$  表达为

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

其中，${\displaystyle i\,\!}$  是虚数单位，${\displaystyle P_{\ell m}(\cos {\theta })\,\!}$  是伴随勒让德多项式，用方程定义为

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

而 ${\displaystyle P_{\ell }(x)\,\!}$  是 ${\displaystyle \ell }$  阶勒让德多项式，可用罗德里格公式表示为：

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

球谐函数满足正交归一性：

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

这样，角动量算符的本征函数，形成一组单范正交基。任意波函数 ${\displaystyle \psi (\theta ,\,\phi )\,\!}$  都可以表达为这单范正交基的线性组合：

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

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

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1. ^ Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, ISBN 0-13-111892-7 

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