91影视

Consider a time dependent potential, we can solve the Schrodinger equation for this potential in a two state system if we split the Hamiltonian into a time independent part $H^{掳}$and time dependent part $H^{\text{'}}$, that is:

$H=H^{掳}+H^1$

This is done in the section 9.1 to get the solution.

$$\begin{gathered} \mathrm{We}\mathrm{have},\mathrm{Zero}\mathrm{order}:c_{伪}^{\left(0\right)}\left(t\right)=伪,c_b^{\left(0\right)}\left(t\right)=b \\ \mathrm{The}\mathrm{coefficients}\mathrm{are}\mathrm{solutions}\mathrm{of}\mathrm{the}\mathrm{couples}\mathrm{ODE}鈥�\mathrm{s},\mathrm{that}\mathrm{is}: \\ \mathrm{The}\mathrm{equation}9.13is{\stackrel{.}{c}}_a=-\frac{i}{鈩�}{H}_{ab}^{\text{'}}{e}^{-i{蝇}_{0}t}{c}_b,鈥�鈥�鈥�{\stackrel{.}{c}}_b=-\frac{i}{鈩�}{H}_{ab}^{\text{'}}{e}^{-i{蝇}_{0}t}{c}_{伪} \\ \mathrm{First}\mathrm{order};\left\{\begin{array}{l}{\stackrel{.}{\mathrm{c}}}_{\mathrm{a}}=-\frac{\mathrm{i}}{鈩�}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}{\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}\mathrm{b}鈬�{\mathrm{c}}_{\mathrm{伪}}^{\left(1\right)}\left(\mathrm{t}\right)=\mathrm{伪}-\frac{\mathrm{ib}}{鈩�}{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\\ {\stackrel{.}{\mathrm{c}}}_{\mathrm{a}}=-\frac{\mathrm{i}}{鈩�}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}{\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}\mathrm{伪}鈬�{\mathrm{c}}_{\mathrm{b}}^{\left(1\right)}\left(\mathrm{t}\right)=\mathrm{b}-\frac{\mathrm{颈伪}}{鈩�}{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\end{array}\right. \\ \mathrm{Second}\mathrm{order}:{\stackrel{.}{\mathrm{c}}}_{\mathrm{a}}=-\frac{\mathrm{i}}{鈩�}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}{\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}\left[\mathrm{b}-\frac{\mathrm{颈伪}}{鈩�}{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\right] \\ {\mathrm{c}}_{\mathrm{伪}}^{\left(2\right)}\left(\mathrm{t}\right)=\mathrm{伪}-\frac{\mathrm{ib}}{鈩�}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}-\frac{\mathrm{伪}}{{鈩�}^2}{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\left[{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ba}}^{\text{'}}\left(\mathrm{t}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\right]{\mathrm{dt}}^{\text{'}} \\ \mathrm{To}\mathrm{get}{\mathrm{c}}_{\mathrm{b}}\mathrm{just}\mathrm{switch}\mathrm{a}鈫�\mathrm{b}(\mathrm{which}\mathrm{entails}\mathrm{also}\mathrm{changing}\mathrm{the}\mathrm{sign}\mathrm{of}{\mathrm{蝇}}_0) \\ {\mathrm{c}}_{\mathrm{b}}^{\left(2\right)}\left(\mathrm{t}\right)=\mathrm{b}-\frac{\mathrm{ia}}{鈩�}{\mathrm{H}}_{\mathrm{ba}}^{\text{'}}\left({\mathrm{t}}^{\text{'}}\right){\mathrm{e}}^{{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}-\frac{\mathrm{b}}{{鈩�}^2}{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ab}}^{\text{'}}\left({\mathrm{t}}^{\text{'}}\right){\mathrm{e}}^{{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\left[{鈭�}_0^{\mathrm{t}}{\mathrm{H}}_{\mathrm{ba}}^{\text{'}}\left({\mathrm{t}}^{\text{'}}\right){\mathrm{e}}^{-{\mathrm{颈蝇}}_0\mathrm{t}}{\mathrm{dt}}^{\text{'}}\right]{\mathrm{dt}}^{\text{'}} \end{gathered}$$