# 位置空间与动量空间

## 经典力学中的位置空间与动量空间

### 拉格朗日力学

${\displaystyle {\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}={\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\,,\quad {\dot {q}}_{i}\equiv {\frac {dq_{i}}{dt}}\,.}$

${\displaystyle p_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}\,,}$

${\displaystyle {\dot {p}}_{i}={\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\,.}$

${\displaystyle d{\mathcal {L}}=\sum _{i=1}^{n}\left({\frac {\partial {\mathcal {L}}}{\partial q_{i}}}dq_{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}dt=\sum _{i=1}^{n}({\dot {p}}_{i}dq_{i}+p_{i}d{\dot {q}}_{i})+{\frac {\partial {\mathcal {L}}}{\partial t}}dt\,,}$

${\displaystyle {\dot {p}}_{i}dq_{i}=d(q_{i}{\dot {p}}_{i})-q_{i}d{\dot {p}}_{i}}$
${\displaystyle p_{i}d{\dot {q}}_{i}=d({\dot {q}}_{i}p_{i})-{\dot {q}}_{i}dp_{i}}$

${\displaystyle d\left[{\mathcal {L}}-\sum _{i=1}^{n}(q_{i}{\dot {p}}_{i}+{\dot {q}}_{i}p_{i})\right]=-\sum _{i=1}^{n}({\dot {q}}_{i}dp_{i}+q_{i}d{\dot {p}}_{i})+{\frac {\partial {\mathcal {L}}}{\partial t}}dt\,.}$

${\displaystyle d{\mathcal {L}}'=\sum _{i=1}^{n}\left({\frac {\partial {\mathcal {L}}'}{\partial p_{i}}}dp_{i}+{\frac {\partial L'}{\partial {\dot {p}}_{i}}}d{\dot {p}}_{i}\right)+{\frac {\partial {\mathcal {L}}'}{\partial t}}dt}$

${\displaystyle {\mathcal {L}}'={\mathcal {L}}-\sum _{i=1}^{n}(q_{i}{\dot {p}}_{i}+{\dot {q}}_{i}p_{i})\,,\quad -{\dot {q}}_{i}={\frac {\partial {\mathcal {L}}'}{\partial p_{i}}}\,,\quad -q_{i}={\frac {\partial {\mathcal {L}}'}{\partial {\dot {p}}_{i}}}\,.}$

${\displaystyle {\frac {d}{dt}}{\frac {\partial {\mathcal {L}}'}{\partial {\dot {p}}_{i}}}={\frac {\partial {\mathcal {L}}'}{\partial p_{i}}}\,.}$

### 哈密顿力学

${\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}\,,\quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}\,.}$

## 量子力学中的位置空间与动量空间

### 位置空间中的函数与算符

${\displaystyle \psi (\mathbf {r} )=\sum _{j}\phi _{j}\psi _{j}(\mathbf {r} )}$

${\displaystyle \psi (\mathbf {r} )=\int _{\mathbf {k} }\phi (\mathbf {k} )\psi _{\mathbf {k} }(\mathbf {r} ){\rm {d}}^{3}\mathbf {k} }$

${\displaystyle \mathbf {\hat {p}} =-i\hbar {\frac {\partial }{\partial \mathbf {r} }}}$

${\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{({\sqrt {2\pi }})^{3}}}e^{i\mathbf {k} \cdot \mathbf {r} }}$

${\displaystyle \psi (\mathbf {r} )={\frac {1}{({\sqrt {2\pi }})^{3}}}\int _{\mathbf {k} }\phi (\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }{\rm {d}}^{3}\mathbf {k} }$

### 动量空间中的函数与算符

${\displaystyle \phi (\mathbf {k} )=\sum _{j}\psi _{j}\phi _{j}(\mathbf {k} )}$

${\displaystyle \phi (\mathbf {k} )=\int _{\mathbf {r} }\psi (\mathbf {r} )\phi _{\mathbf {r} }(\mathbf {k} ){\rm {d}}^{3}\mathbf {r} }$

${\displaystyle \mathbf {\hat {r}} =i\hbar {\frac {\partial }{\partial \mathbf {p} }}=i{\frac {\partial }{\partial \mathbf {k} }}}$

${\displaystyle \phi _{\mathbf {r} }(\mathbf {k} )={\frac {1}{({\sqrt {2\pi }})^{3}}}e^{-i\mathbf {k} \cdot \mathbf {r} }}$

${\displaystyle \phi (\mathbf {k} )={\frac {1}{({\sqrt {2\pi }})^{3}}}\int _{\mathbf {r} }\psi (\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }{\rm {d}}^{3}\mathbf {r} }$

## 倒易空间与晶体

k与晶格动量有关时，k空间这一概念仍然相当重要，但与上文讨论的晶体之外的k空间有所不同。晶体的k空间中有一个无限点集——倒易点阵。其中所有点的k = 0。类似地，k空间中还存在一个体积有限的布里渊区，区域中所有的k值相等。

## 参考文献

1. ^ Eisberg, R.; Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles 2nd. John Wiley & Sons. 1985. ISBN 978-0-471-873730 （英语）.
2. ^ Hand, L. N.; Finch, J. D. Analytical Mechanics. Cambridge University Press. 1998: 190. ISBN 9780521575720 （英语）.
3. ^ Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. Quantum Mechanics (Schaum's Outline Series) 2nd. McGraw Hill. 2010. ISBN 978-0-071-623582 （英语）.
4. ^ Abers, E. Quantum Mechanics. Addison Wesley, Prentice Hall Inc. 2004. ISBN 978-0-131-461000 （英语）.
5. R. Penrose. The Road to Reality. Vintage books. 2007. ISBN 0-679-77631-1 （英语）.