# 朗道量子化

（重定向自朗道能级

## 推导

${\displaystyle {\hat {H}}={\frac {1}{2m}}({\hat {\mathbf {p} }}-q{\hat {\mathbf {A} }}/c)^{2}.}$

${\displaystyle \mathbf {B} =\mathbf {\nabla } \times {\hat {\mathbf {A} }}.\,}$

${\displaystyle {\hat {\mathbf {A} }}={\begin{pmatrix}0\\Bx\\0\end{pmatrix}}.}$

${\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2m}}\left({\hat {p}}_{y}-{\frac {qB{\hat {x}}}{c}}\right)^{2}.}$

${\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2}}m\omega _{c}^{2}\left({\hat {x}}-{\frac {\hbar k_{y}}{m\omega _{c}}}\right)^{2}.}$

${\displaystyle E_{n}=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right),\quad n\geq 0~.}$

${\displaystyle \Psi (x,y)=e^{ik_{y}y}\phi _{n}(x-x_{0})~.}$

## 朗道能级

${\displaystyle k_{y}={\frac {2\pi N}{L_{y}}}}$ ,

${\displaystyle 0\leq N<{\frac {m\omega _{c}L_{x}L_{y}}{2\pi \hbar }}.}$

${\displaystyle {\frac {ZBL_{x}L_{y}}{(hc/e)}}=Z{\frac {\Phi }{\Phi _{0}}},}$

${\displaystyle D=Z(2S+1){\frac {\Phi }{\Phi _{0}}}~.}$

## 对称规范中的朗道能级

${\displaystyle {\hat {\mathbf {A} }}={\frac {1}{2}}{\begin{pmatrix}-By\\Bx\\0\end{pmatrix}}}$

${\displaystyle {\hat {H}}={\frac {1}{2}}\left[\left(-i{\frac {\partial }{\partial x}}-{\frac {y}{2}}\right)^{2}+\left(-i{\frac {\partial }{\partial y}}+{\frac {x}{2}}\right)^{2}\right]}$

${\displaystyle {\hat {a}}={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]}$
${\displaystyle {\hat {a}}^{\dagger }={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]}$
${\displaystyle {\hat {b}}={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]}$
${\displaystyle {\hat {b}}^{\dagger }={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]}$

${\displaystyle [{\hat {a}},{\hat {a}}^{\dagger }]=[{\hat {b}},{\hat {b}}^{\dagger }]=1}$ .

${\displaystyle {\hat {H}}={\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}}$

${\displaystyle {\hat {L}}_{z}=-i\hbar {\frac {\partial }{\partial \theta }}=-\hbar ({\hat {b}}^{\dagger }{\hat {b}}-{\hat {a}}^{\dagger }{\hat {a}})}$

${\displaystyle {\hat {H}}|n,m\rangle =E_{n}|n,m\rangle }$
${\displaystyle E_{n}=\left(n+{\frac {1}{2}}\right)}$
${\displaystyle |n,m\rangle ={\frac {({\hat {b}}^{\dagger })^{m+n}}{\sqrt {(m+n)!}}}{\frac {({\hat {a}}^{\dagger })^{n}}{\sqrt {n!}}}|0,0\rangle }$

${\displaystyle \psi _{n,m}(x,y)=\left({\frac {\partial }{\partial w}}-{\frac {\bar {w}}{4}}\right)^{n}w^{n+m}e^{-|w|^{2}/4}}$

## 规范变换的影响

${\displaystyle {\vec {A}}\to {\vec {A}}'={\vec {A}}+{\vec {\nabla }}\lambda ({\vec {x}})}$

${\displaystyle {\hat {\pi }}={\hat {\mathbf {p} }}-q{\hat {\mathbf {A} }}/c}$

${\displaystyle \langle \alpha |{\hat {x}}|\alpha \rangle =\langle \alpha '|{\hat {x}}|\alpha '\rangle }$
${\displaystyle \langle \alpha |{\hat {\pi }}|\alpha \rangle =\langle \alpha '|{\hat {\pi '}}|\alpha '\rangle }$
${\displaystyle \langle \alpha |\alpha \rangle =\langle \alpha '|\alpha '\rangle }$

${\displaystyle {\mathcal {G}}^{\dagger }{\hat {x}}{\mathcal {G}}={\hat {x}}}$
${\displaystyle {\mathcal {G}}^{\dagger }\left({\hat {p}}-{\frac {e{\hat {A}}}{c}}-{\frac {e{\vec {\nabla }}\lambda (x)}{c}}\right){\mathcal {G}}={\hat {p}}-{\frac {e{\hat {A}}}{c}}}$
${\displaystyle {\mathcal {G}}^{\dagger }{\mathcal {G}}=1}$

${\displaystyle {\mathcal {G}}=\exp \left({\frac {ie\lambda ({\vec {x}})}{\hbar c}}\right)}$

## 参考文献

1. 黄昆; 韩汝琦. 《固体物理学》. 北京: 高等教育出版社. : 255–274. ISBN 978-7-04-001025-1 （中文（中国大陆））.
2. ^ Landau, L. D. Diamagnetismus der metalle. Zeitschrift für Physik. 1930, 64 (9-10): 629–637. doi:10.1007/BF01397213 （德语）.
3. ^ Л·Д·朗道; Е·М·栗弗席兹; 严肃（译）; 喀兴林（校）. 《理论物理学教程第三卷·量子力学（非相对论理论）》. 北京: 高等教育出版社. : 416–420. ISBN 978-7-04-024306-2 （中文（中国大陆））.
4. ^ Mikhailov, S. A. A new approach to the ground state of quantum Hall systems. Basic principles. Physica B: Condensed Matter. 2001, 299: 6. doi:10.1016/S0921-4526(00)00769-9 （英语）.