91影视

$$\begin{gathered} \left.c_a=-\frac{i}{\sout{h}}\left[c_{a}H{\text{'}}_{aa}+c_{a}H{\text{'}}_{ab}{e}^{i{蝇}_{0}t}\right] \\ 鈬�c_a=-\frac{i}{\sout{h}}\left[c_{b}H{\text{'}}_{bb}+c_{a}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}\right]\right\} \end{gathered}$$

role="math" localid="1658555553364" $c_a=-\frac{i}{\sout{h}}\left[c_{a}H{\text{'}}_{aa}+c_{b}H{\text{'}}_{ab}{e}^{-i\left(E_b-E_a\right)t/\sout{h}}\right]$

$c_a=-\frac{i}{\sout{h}}H{\text{'}}_{ab}{e}^{i{蝇}_{0}t}{c}_b,c_b=-\frac{i}{\sout{h}}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}{c}_a$

role="math" localid="1658555600978" $c_b=-\frac{i}{\sout{h}}\left[c_{b}H{\text{'}}_{bb}+c_{a}H{\text{'}}_{ba}{e}^{-i\left(E_b-E_a\right)t/\sout{h}}\right]$

Initial conditions: role="math" localid="1658555900086" $c_a\left(0\right)=1,c_b\left(0\right)=0,$

Zero order: $c_a\left(0\right)=1,c_b\left(0\right)=0,$

First order:$\left\{\begin{array}{l}{c}_a=-\frac{i}{\sout{h}}H{\text{'}}_{aa}鈬�c_a\left(t\right)=1-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{aa}\left(t\text{'}\right)dt\text{'}\\ c_b=-\frac{i}{\sout{h}}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}鈬�c_b\left(t\right)=1-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{ba}\left(t\text{'}\right)e^{i{蝇}_{0}t}dt\text{'}\end{array}\right.$

$$\begin{gathered} {\left|c_a\right|}^2=\left[1-\frac{i}{\sout{\sout{h}}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\right]\left[1-\frac{i}{\sout{\sout{h}}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\right]=1+{\left[\frac{i}{\sout{\sout{h}}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\right]}^2=1\left(tofirstorderinH\text{'}\right) \\ {\left|c_b\right|}^2=\left[-\frac{i}{\sout{\sout{h}}}{鈭�}_0^t{H}_{ba}\left(t\text{'}\right)e^{i{蝇}_{0}t}dt\text{'}\right]\left[\frac{i}{\sout{\sout{h}}}{鈭�}_0^t{H}_{ab}\left(t\text{'}\right)e^{i{蝇}_{0}t}dt\text{'}\right]=0\left(tofirstorderinH\text{'}\right) \\ So{\left|c_a\right|}^2+{\left|c_b\right|}^2=1\mathbf{(}\mathbf{to}\mathbf{}\mathbf{first}\mathbf{}\mathbf{order}\mathbf{)} \end{gathered}$$

$d_b=-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\text{'}}\left(\frac{i}{\sout{h}}H{\text{'}}_{bb}\right)c_b+e^{\frac{i}{\sout{h}}{鈭�}_0^t{H}_{bb}\left(t\text{'}\right)dt\text{'}}{c}_{b}But{c}_a=-\frac{i}{\sout{h}}\left[c_{a}H{\text{'}}_{aa}+c_{b}H{\text{'}}_{ab}{e}^{-i{蝇}_{0}t}\right]$

Two terms cancel, leaving

$$\begin{gathered} d_b=-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\text{'}}{c}_b{H}_{ab}{e}^{-i{蝇}_{0}t}.But{c}_b=e^{-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{bb}\left(t\text{'}\right)dt\text{'}}{d}_b \\ =-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t\left[H{\text{'}}_{aa}\left(t\text{'}\right)-H{\text{'}}_{bb}\left(t\right)\right]dt}{H}_{ab}{e}^{i{蝇}_{0}t}{d}_b,or{d}_a=-\frac{i}{\sout{h}}{e}^{i蠒}H{\text{'}}_{ab}{e}^{i{蝇}_{0}t}{d}_b \end{gathered}$$

Similarly,$$\begin{gathered} d_b=-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\text{'}}\left(\frac{i}{\sout{h}}H{\text{'}}_{bb}\right)c_b=e^{-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{bb}\left(t\text{'}\right)dt\text{'}}{d}_b.But{c}_b=-\frac{i}{\sout{h}}\left[c_b{H}_{bb}+c_{a}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}\right] \\ =-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\text{'}}{c}_{a}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}.But{c}_a=e^{-\frac{i}{\sout{h}}{鈭�}_0^t{H}_{aa}\left(t\text{'}\right)dt\text{'}}{d}_a \\ =-\frac{i}{\sout{h}}{e}^{\frac{i}{\sout{h}}{鈭�}_0^t\left[H{\text{'}}_{bb}\left(t\text{'}\right)-H{\text{'}}_{aa}(t\text{'})\right]dt\text{'}}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}{d}_a=-\frac{i}{\sout{h}}{e}^{i蠒}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}{d}_a \end{gathered}$$

Initial conditions:$c_a\left(0\right)=1鈬�d_a\left(0\right)=1;c_b\left(0\right)=0鈬�d_b\left(0\right)=0$.

Zero order:$d_a\left(t\right)=1,d_b\left(t\right)=0$.

First order:${\stackrel{藱}{d}}_a=0鈬�d_a\left(t\right)=1鈬�c_a\left(t\right)=e^{-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{aa}(t\text{'})dt\text{'}}$

${\stackrel{藱}{d}}_b=-\frac{i}{\sout{h}}{e}^{-i蠒}H{\text{'}}_{ba}{e}^{i{蝇}_{0}t}鈬�d_b=-\frac{i}{\sout{h}}{鈭�}_0^t{e}^{-i蠒(t\text{'})}H{\text{'}}_{ba}(t\text{'})e^{i{蝇}_{0}t\text{'}}dt\text{'}$

These don鈥檛 look much like the results in (a), but remember, we鈥檙e only working to first order in $H\text{'}Hso{c}_a\left(t\right)鈮�1-\frac{i}{\sout{h}}{鈭�}_0^{t}H{\text{'}}_{aa}(t\text{'})dt\text{'}(\mathrm{to}\mathrm{this}\mathrm{order}),\mathrm{while}\mathrm{for}{\mathrm{c}}_{\mathrm{b}},\mathrm{the}\mathrm{factor}\mathrm{H}{\text{'}}_{\mathrm{ba}}$in the integral means it is already first order and hence both the exponential factor in front and $e-i蠒$should be replaced by 1. Then $c_b\left(t\right)鈮�1-{鈭�}_0^t\frac{i}{h}H{\text{'}}_{aa}\left(t\text{'}\right)e^{i{蝇}_{0}t\text{'}}dt\text{'}$, and we recover the results in (a).