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# 相互作用绘景

## 定义

${\displaystyle H_{\mathcal {S))=H_{0,\,{\mathcal {S))}+H_{1,\,{\mathcal {S))}\,\!}$

${\displaystyle U(t)=e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }\,\!}$

${\displaystyle U(t)=e^{-{\frac {i}{\hbar ))\int \limits _{0}^{t}H_{0,\,{\mathcal {S))}(t^{'})\,dt^{'))\,\!}$

### 态向量

${\displaystyle |\psi (t)\rangle _{\mathcal {I)){\stackrel {def}{=))e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }|\psi (t)\rangle _{\mathcal {S))\,\!}$

${\displaystyle |\psi (t)\rangle _{\mathcal {S))=e^{-iH_{\mathcal {S))\,t/\hbar }|\psi (0)\rangle _{\mathcal {S))\,\!}$

${\displaystyle |\psi (t)\rangle _{\mathcal {I))=e^{-iH_{1,\,{\mathcal {S))}\,t/\hbar }|\psi (0)\rangle _{\mathcal {S))\,\!}$

### 算符

${\displaystyle A_{\mathcal {I))(t)=e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }A_{\mathcal {S))(t)\,e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }\,\!}$

（请注意，${\displaystyle A_{\mathcal {S))(t)\,\!}$通常不含时间，可以重写为${\displaystyle A_{\mathcal {S))\,\!}$。反例，对于时变外电场的状况，哈密顿算符${\displaystyle H_{\mathcal {S))(t)\,\!}$含时。）

#### 哈密顿算符

${\displaystyle H_{0,\,{\mathcal {I))}(t)=e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }H_{0,\,{\mathcal {S))}\,e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }=H_{0,\,{\mathcal {S))}\,\!}$

${\displaystyle H_{1,\,{\mathcal {I))}(t)=e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }H_{1,\,{\mathcal {S))}\,e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }\,\!}$

#### 密度矩阵

{\displaystyle {\begin{aligned}\rho _{\mathcal {I))(t)&=\sum _{n}p_{n}|\psi _{n}(t)\rangle _{\mathcal {I))\,{}_{\mathcal {I))\langle \psi _{n}(t)|\\&=\sum _{n}p_{n}\,e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }|\psi _{n}(t)\rangle _{\mathcal {S))\,{}_{\mathcal {S))\langle \psi _{n}(t)|e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }\\&=e^{iH_{0,\,{\mathcal {S))}\,t/\hbar }\rho _{\mathcal {S))(t)\,e^{-iH_{0,\,{\mathcal {S))}\,t/\hbar }\\\end{aligned))\,\!}

## 时间演化方程式

### 量子态的时间演化

{\displaystyle {\begin{aligned}i\hbar {\frac {d}{dt))|\psi (t)\rangle _{\mathcal {I))&=e^{iH_{0}\,t/\hbar }\left[-H_{0}|\psi (t)\rangle _{\mathcal {S))+i\hbar {\frac {d}{dt))|\psi (t)\rangle _{\mathcal {S))\right]\\&=e^{iH_{0}\,t/\hbar }\left[-H_{0}|\psi (t)\rangle _{\mathcal {S))+H_{\mathcal {S))|\psi (t)\rangle _{\mathcal {S))\right]\\&=e^{iH_{0}\,t/\hbar }H_{1,\,{\mathcal {S))}|\psi (t)\rangle _{\mathcal {S))\\&=e^{iH_{0}\,t/\hbar }H_{1,\,{\mathcal {S))}\,e^{-iH_{0}\,t/\hbar }|\psi (t)\rangle _{\mathcal {I))\\\end{aligned))}

${\displaystyle i\hbar {\frac {d}{dt))|\psi (t)\rangle _{\mathcal {I))=H_{1,\,{\mathcal {I))}|\psi (t)\rangle _{\mathcal {I))\,\!}$

### 算符的时间演化

{\displaystyle {\begin{aligned}i\hbar {\frac {d}{dt))A_{\mathcal {I))(t)&=i\hbar {\frac {d}{dt))(e^{iH_{0}\,t/\hbar }A_{\mathcal {S))\,e^{-iH_{0}\,t/\hbar })\\&=-H_{0}\,e^{iH_{0}\,t/\hbar }A_{\mathcal {S))\,e^{-iH_{0}\,t/\hbar }+e^{iH_{0}\,t/\hbar }A_{\mathcal {S))\,e^{-iH_{0}\,t/\hbar }H_{0}\\&=A_{\mathcal {I))(t)H_{0}-H_{0}A_{\mathcal {I))(t)\\&=\left[A_{\mathcal {I))(t),\,H_{0}\right]\\\end{aligned))\,\!}

${\displaystyle i\hbar {\frac {d}{dt))A_{\mathcal {H))(t)=\left[A_{\mathcal {H))(t),\,H\right]\,\!}$

### 密度矩阵的时间演化

${\displaystyle i\hbar {\frac {d}{dt))\rho _{\mathcal {I))(t)=\left[H_{1,\,{\mathcal {I))}(t),\rho _{\mathcal {I))(t)\right]\,\!}$

## 各种绘景比较摘要

 演化 海森堡绘景 交互作用绘景 薛丁格绘景 右矢 常定 ${\displaystyle |\psi (t)\rangle _{\mathcal {I))=e^{iH_{0}t/\hbar }|\psi (t)\rangle _{\mathcal {S))}$ ${\displaystyle |\psi (t)\rangle _{\mathcal {S))=e^{-iHt/\hbar }|\psi (0)\rangle _{\mathcal {S))}$ 可观察量 ${\displaystyle A_{\mathcal {H))(t)=e^{iHt/\hbar }A_{\mathcal {S))e^{-iHt/\hbar ))$ ${\displaystyle A_{\mathcal {I))(t)=e^{iH_{0}t/\hbar }A_{\mathcal {S))e^{-iH_{0}t/\hbar ))$ 常定 密度算符 常定 ${\displaystyle \rho _{\mathcal {I))(t)=e^{iH_{0}t/\hbar }\rho _{S}(t)e^{-iH_{0}/\hbar ))$ ${\displaystyle \rho _{\mathcal {S))(t)=e^{-iHt/\hbar }\rho _{\mathcal {S))(0)e^{iHt/\hbar ))$

## 注释

1. ^ 在狄拉克绘景里，${\displaystyle H_{0,\,{\mathcal {S))}\,\!}$也可能含时。假设${\displaystyle H_{0,\,{\mathcal {S))}\,\!}$含时并且对易，则时间演化算符${\displaystyle U(t)\,\!}$的公式不再是[1]:70-71
${\displaystyle U(t)=e^{\pm iH_{0,\,{\mathcal {S))}\,t/\hbar }\,\!}$
而应改为
${\displaystyle U(t)=e^{-i/\hbar \int \limits _{0}^{t}H(t^{'})\,dt^{'))\,\!}$

## 注释

1. Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
2. ^ Parker, C.B. McGraw Hill Encyclopaedia of Physics 2nd. Mc Graw Hill. 1994: 786, 1261. ISBN 0-07-051400-3.
3. ^ Y. Peleg, R. Pnini, E. Zaarur, E. Hecht. Quantum mechanics. Schuam's outline series 2nd. McGraw Hill. 2010: 70. ISBN 9-780071-623582.
4. ^ Ian J R Aitchison; Anthony J.G. Hey. Gauge Theories in Particle Physics: A Practical Introduction, Volume 1: From Relativistic Quantum Mechanics to QED, Fourth Edition. CRC Press. 17 December 2012. ISBN 978-1-4665-1302-0.

## 参考文献

• Townsend, John S. A Modern Approach to Quantum Mechanics, 2nd ed.. Sausalito, CA: University Science Books. 2000. ISBN 1-891389-13-0.