# 相互作用繪景

（重定向自交互作用繪景

## 定義

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

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

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

### 態向量

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

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

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

### 算符

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

（請注意，${\displaystyle A_{\mathcal {S}}(t)\,\!}$ 通常不含時間，可以重寫為${\displaystyle A_{\mathcal {S}}\,\!}$ 。反例，對於時變外電場的狀況，哈密頓算符${\displaystyle H_{\mathcal {S}}(t)\,\!}$ 含時。）

#### 哈密頓算符

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

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

#### 密度矩陣

{\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}}\,\!}

## 時間演化方程式

### 量子態的時間演化

{\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}}}

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

### 算符的時間演化

{\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}}\,\!}

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

### 密度矩陣的時間演化

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

## 各種繪景比較摘要

 演化 海森堡繪景 交互作用繪景 薛丁格繪景 右矢 常定 ${\displaystyle |\psi (t)\rangle _{\mathcal {I}}=e^{iH_{0}t/\hbar }|\psi (t)\rangle _{\mathcal {S}}}$ ${\displaystyle |\psi (t)\rangle _{\mathcal {S}}=e^{-iHt/\hbar }|\psi (0)\rangle _{\mathcal {S}}}$ 可觀察量 ${\displaystyle A_{\mathcal {H}}(t)=e^{iHt/\hbar }A_{\mathcal {S}}e^{-iHt/\hbar }}$ ${\displaystyle A_{\mathcal {I}}(t)=e^{iH_{0}t/\hbar }A_{\mathcal {S}}e^{-iH_{0}t/\hbar }}$ 常定 密度算符 常定 ${\displaystyle \rho _{\mathcal {I}}(t)=e^{iH_{0}t/\hbar }\rho _{S}(t)e^{-iH_{0}/\hbar }}$ ${\displaystyle \rho _{\mathcal {S}}(t)=e^{-iHt/\hbar }\rho _{\mathcal {S}}(0)e^{iHt/\hbar }}$

## 註釋

1. ^ 在狄拉克繪景裏，${\displaystyle H_{0,\,{\mathcal {S}}}\,\!}$ 也可能含時。假設${\displaystyle H_{0,\,{\mathcal {S}}}\,\!}$ 含時並且對易，則時間演化算符${\displaystyle U(t)\,\!}$ 的公式不再是[1]:70-71
${\displaystyle U(t)=e^{\pm iH_{0,\,{\mathcal {S}}}\,t/\hbar }\,\!}$
而應改為
${\displaystyle U(t)=e^{-i/\hbar \int \limits _{0}^{t}H(t^{'})\,dt^{'}}\,\!}$

## 註釋

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.

## 參考文獻

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