# 薛丁格繪景

${\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle }$

${\displaystyle U(t,t_{0})=e^{-iH(t-t_{0})/\hbar }}$

## 時間演化算符

### 定義

${\displaystyle |\psi (t)\rangle \ {\stackrel {def}{=}}\ U(t,\,t_{0})|\psi (t_{0})\rangle }$

${\displaystyle \langle \psi (t)|=\langle \psi (t_{0})|U^{\dagger }(t,\,t_{0})}$

### 性質

#### 幺正性

${\displaystyle \langle \psi (t)|\psi (t)\rangle =\langle \psi (t_{0})|\psi (t_{0})\rangle }$

${\displaystyle \langle \psi (t)|\psi (t)\rangle =\langle \psi (t_{0})|U^{\dagger }(t,\,t_{0})U(t,\,t_{0})|\psi (t_{0})\rangle }$

${\displaystyle U^{\dagger }(t,\,t_{0})U(t,\,t_{0})=I}$ ;

#### 單位性

${\displaystyle |\psi (t_{0})\rangle =U(t_{0},\,t_{0})|\psi (t_{0})\rangle }$

#### 閉包性

${\displaystyle U(t,\,t_{0})=U(t,\,t_{1})U(t_{1},\,t_{0})}$

${\displaystyle |\psi (t_{1})\rangle =U(t_{1},\,t_{0})|\psi (t_{0})\rangle }$
${\displaystyle |\psi (t)\rangle =U(t,\,t_{1})|\psi (t_{1})\rangle }$

${\displaystyle |\psi (t)\rangle =U(t,\,t_{1})U(t_{1},\,t_{0})|\psi (t_{0})\rangle }$

${\displaystyle |\psi (t)\rangle =U(t,\,t_{0})|\psi (t_{0})\rangle }$

${\displaystyle U(t,\,t_{0})=U(t,\,t_{1})U(t_{1},\,t_{0})}$

### 時間演化算符的微分方程式

${\displaystyle i\hbar {\partial \over \partial t}|\psi (t)\rangle =H|\psi (t)\rangle }$

${\displaystyle i\hbar {\partial \over \partial t}U(t)|\psi (0)\rangle =HU(t)|\psi (0)\rangle }$

${\displaystyle i\hbar {\partial \over \partial t}U(t)=HU(t)}$

${\displaystyle U(t)=e^{-iHt/\hbar }}$

${\displaystyle e^{-iHt/\hbar }=1-{\frac {iHt}{\hbar }}-{\frac {1}{2}}\left({\frac {Ht}{\hbar }}\right)^{2}+\cdots }$

${\displaystyle |\psi (t)\rangle =e^{-iHt/\hbar }|\psi (0)\rangle }$

${\displaystyle |\psi (t)\rangle =e^{-iEt/\hbar }|\psi (0)\rangle }$

${\displaystyle U(t)=\exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right)}$

${\displaystyle U(t)=T\exp \left({-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'}\right)}$

${\displaystyle U(t)=1+\sum _{n=1}^{\infty }\left({\frac {-i}{\hbar }}\right)^{n}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\dots \int _{0}^{t_{n-1}}dt_{n}H(t_{1})H(t_{2})\dots H(t_{n})}$

## 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 ${\displaystyle \rho _{\mathcal {I}}(t)=e^{iH_{0}t/\hbar }\rho _{S}(t)e^{-iH_{0}t/\hbar }}$ ${\displaystyle \rho _{\mathcal {S}}(t)=e^{-iHt/\hbar }\rho _{\mathcal {S}}(0)e^{iHt/\hbar }}$

## 參考文獻

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. ^ Robert D. Klauber. Student Friendly Quantum Field Theory: Basic Principles and Quantum Electrodynamics (PDF). Sandtrove Press. 2013 [2015-12-13]. ISBN 978-0-9845139-3-2. （原始内容存档 (PDF)于2015-12-22）.