# 相互作用繪景

## 定義

$U(t)=e^{ - i/\hbar \int\limits _0^t H(t^{'})\, dt^{'}}\,\!$

### 態向量

$| \psi_{I}(t) \rang = e^{i H_{0,\,S}\,t / \hbar} | \psi_{S}(t) \rang\,\!$(1)

$| \psi_{S}(t) \rang = e^{ - iH_S\,t / \hbar} | \psi_{S}(0) \rangle\,\!$

$| \psi_{I}(t) \rang = e^{ - i H_{1,\,S}\,t / \hbar} | \psi_{S}(0) \rang\,\!$

### 算符

$A_{I}(t) = e^{i H_{0,\,S}\,t / \hbar} A_{S}(t)\, e^{ - i H_{0,\,S}\,t / \hbar}\,\!$

（請注意，$A_S(t)\,\!$通常不含時間，可以重寫為$A_S\,\!$。反例，當算符代表隨時間變化的外電場的時候，$A_S(t)\,\!$會含時間。）

#### 哈密頓算符

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

$U(t) =e^{ - i/\hbar \int\limits _0^t H(t^{'})\, dt^{'}}\,\!$

$H_{1,\,I}(t) = e^{i H_{0,\,S}\,t / \hbar} H_{1,\,S}\, e^{ - i H_{0,\,S}\,t / \hbar}\,\!$(2)

#### 密度矩陣

\begin{align}\rho_I(t) & = \sum_n p_n|\psi_{n,\,I}(t)\rang \lang \psi_{n,\,I}(t)| \\ & = \sum_n p_n\, e^{i H_{0,\,S}\,t / \hbar}|\psi_{n,\,S}(t)\rang \lang \psi_{n,\,S}(t)|e^{ - i H_{0,\,S}\,t / \hbar} \\ & = e^{i H_{0,\,S}\,t / \hbar} \rho_S(t)\,e^{ - i H_{0,\,S}\,t / \hbar} \\ \end{align}\,\!

## 時間演化方程式

### 量子態的時間演化

\begin{align} i \hbar \frac{d}{dt} | \psi_{I} (t) \rang & = e^{i H_{0,\,S}\,t / \hbar}\left[ - H_{0,\,S}| \psi_{s}(t) \rang +i \hbar \frac{d}{dt} | \psi_{S} (t) \rang\right]\\ & =e^{i H_{0,\,S}\,t / \hbar}\left[ - H_{0,\,S}| \psi_{S}(t) \rang +H_S | \psi_{s} (t) \rang\right] \\ & =e^{i H_{0,\,S}\,t / \hbar} H_{1,\,S}| \psi_{S}(t) \rang \\ \end{align}

$i \hbar \frac{d}{dt} | \psi_{I} (t) \rang= e^{i H_{0,\,S}\,t / \hbar}H_{1,\,S}\,e^{ - i H_{0,\,S}\,t / \hbar}| \psi_{I}(t) \rang \,\!$

$i \hbar \frac{d}{dt} | \psi_{I}(t)\rang =H_{1,\,I}| \psi_{I}(t)\rang \,\!$

### 算符的時間演化

\begin{align} i\hbar\frac{d}{dt}A_I(t) & =i\hbar\frac{d}{dt}( e^{i H_{0,\,S}\,t / \hbar} A_{S}\,e^{ - i H_{0,\,S}\,t / \hbar}) \\ & = - H_{0,\,S}\,e^{i H_{0,\,S}\,t / \hbar} A_{S}\,e^{ - i H_{0,\,S}\,t / \hbar} + e^{i H_{0,\,S}\,t / \hbar} A_{S}\,e^{ - i H_{0,\,S}\,t / \hbar} H_{0,\,S} \\ & =A_I(t)H_{0,\,S} - H_{0,\,S}A_I(t) \\ & =\left[A_I(t),\,H_0\right] \\ \end{align}\,\!

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

### 密度矩陣的時間演化

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

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