含时微扰和跃迁

含时微扰

在相互作用表象下，薛定谔方程写为：

\(i \hbar \dot U_I = V_I U_I，U_I(t = 0) = I\)

于是将\(U_I\)积出来，得到：

\[\begin{aligned} U_I &= I + \frac{1}{i \hbar}\int_{0}^{t} V_IU_Idt \\\\&= I + \frac{1}{i \hbar}\int_{0}^{t} V_Idt \\\\&+ (\frac{1}{i \hbar})^2\int_{0}^{t}\int_{0}^{t} V_I(t)V_I(t')U_Idtdt' \\\\\approx&= I + \frac{1}{i \hbar}\int_{0}^{t} V_Idt \\\\&+ (\frac{1}{i \hbar})^2\int_{0}^{t}\int_{0}^{t} V_I(t)V_I(t')dtdt' \end{aligned}\]

令\(\left|\psi \right> = \sum_n C_n(t) e^{iE_n t/\hbar} \left|n \right>\)，则\(\left|\psi \right>_I = \sum_n C_n(t) \left|n \right>\),

\({\left|\psi \right>}_I (t) = U_I \sum_n C_n(t) \left| n \right>\)

\(=\sum_n C_n \left| n \right>+\sum_m \sum_n C_n(0) \frac{1}{i\hbar} \left| m \right>\int^t_0\left< m|V_I |n\right>\)

当\(C_n(0) = \delta_{nk}\)时，原式化简为：

\(=\left| k \right> + \sum_m \frac{1}{i\hbar} \int^t_0 \left< m|V_I |k\right> \left| m \right>\)

\(=\left| k \right> + \sum_m \frac{1}{i\hbar} \int^t_0 e^{i(E_m-E_k)t/\hbar}\left< m|V_S |k\right> \left| m \right>\)

为了便于计算，转换到了薛定谔表象下，定义\(t\)时刻的从k态跃迁到m态的几率为：

\[ P_{k\rightarrow m} = \left| C_m(t)\right| = \left|\frac{1}{i\hbar} \int^t_0 e^{i(E_m-E_k)t/\hbar}\left< m|V_S |k\right> \right|^2 \]

常微扰和简谐微扰

简谐微扰\(V = W(x) e^{iwt} +W^* (x)e^{-iwt}\)，计算跃迁频率表达式：

\[\begin{aligned} &\frac{1}{\hbar^2} \left| \int_0^{t} \left<f|V(x) |i\right>e^{i(E_m-E_k)t/\hbar}dt \right|^2 \\\\=& \frac{1}{\hbar^2} | t \left<f|W(x) |i\right> e^{i(w+w_{fi}t)/2}\frac{\sin[(w+w_{fi})t/2]}{(w+w_{fi})t/2} \\\\=& t \left<f|W^*(x) |i\right> e^{i(w-w_{fi}t)/2}\frac{\sin[(w-w_{fi})t/2]}{(w-w_{fi})t/2} |^2 \end{aligned}\]

当\(w+w_{fi} \approx 0\)时，第一项起主要影响，再由delta函数的弱极限定义，得到

\[ P \approx \frac{2\pi t}{\hbar}|W_fi|^2 \delta(E_f-E_i-\hbar w) \]

同理\(w- w_{fi} \approx 0\)，\(P \approx \frac{2\pi t}{\hbar}|W^*_fi|^2 \delta(E_f-E_i+\hbar w)\)

所以共振跃迁的条件为 \(\hbar w = E_f -E_i\)