almost finsihed deriving analytical expression for third order coupling elements (not really elegant tbh...)

This commit is contained in:
Antoine Marie 2022-11-23 17:01:03 +01:00
parent 76e18eec87
commit abc1ae8e07

View File

@ -705,6 +705,31 @@ Therefore the differential equations can be simplified to
\end{align} \end{align}
and can be solved by integration of the right side using the previous analytic formula for Hamiltonian elements. and can be solved by integration of the right side using the previous analytic formula for Hamiltonian elements.
\begin{align}
\text{Part B}: &(\dv{\bW^{(3)}}{s})_{pR} = - \left[\left( \bF{}{(0)}\bF{}{(2)} + \bF{}{(2)}\bF{}{(0)} \right)\bW^{(1)}\right]_{pR} \notag \\
&= - \sum_q\left( \bF{}{(0)}\bF{}{(2)} + \bF{}{(2)}\bF{}{(0)} \right)_{pq}W^{(1)}_{qR} \notag \\
&= - \sum_q(\epsilon_p+\epsilon_q)F_{pq}^{(2)}W^{(1)}_{qR} \notag \\
&= - \sum_{qS} (\epsilon_p+\epsilon_q) \frac{\Delta_{pS}+\Delta_{qS}}{\Delta_{pS}^2+\Delta_{qS}^2}W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) W^{(1)}_{qR}(0) \notag \\
&\times (1 - e^{-(\Delta_{pS}^2+\Delta_{qS}^2)s})e^{-\Delta_{qR}^2s} \notag \\
\text{Part B}: & (\bW^{(3)}(s))_{pR} = - \sum_{qS} (\epsilon_p+\epsilon_q) \frac{\Delta_{pS}+\Delta_{qS}}{\Delta_{pS}^2+\Delta_{qS}^2}W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) W^{(1)}_{qR}(0) \notag \\
&\times \left(-\frac{e^{-\Delta_{qR}^2s} }{\Delta_{qR}^2} + \frac{e^{-(\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2)s}}{\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2}\right)
\end{align}
\begin{align}
\text{Part C}: &(\dv{\bW^{(3)}}{s})_{pR} = - \left[\bW^{(1)}\left( \bC{}{(0)}\bC{}{(2)} + \bC{}{(2)}\bC{}{(0)} \right)\right]_{pR} \notag \\
&= - \sum_S W^{(1)}_{pS} \left( \bD^{(0)}\bD^{(2)} + \bD^{(2)}\bD^{(0)} \right)_{SR}\notag \\
&= - \sum_SW^{(1)}_{pS}(D_R+D_S)D_{SR}^{(2)}\notag \\
&= - \sum_{Sq}W^{(1)}_{pS}(0)e^{-\Delta_{pS}^2s} (D_R+D_S)\frac{-\Delta_{qS}-\Delta_{qR}}{\Delta_{qS}^2+\Delta_{qR}^2} \notag \\
&\times W^{(1)}_{qS}(0)W^{(1)}_{qR}(0)(1-e^{-(\Delta_{qS}^2+\Delta_{qR}^2)s})) \notag \\
\text{Part C}: & (\bW^{(3)}(s))_{pR} = + \sum_{Sq} (D_R+D_S) \frac{\Delta_{qS}+\Delta_{qR}}{\Delta_{qS}^2+\Delta_{qR}^2} W^{(1)}_{pS}(0)W^{(1)}_{qS}(0) W^{(1)}_{qR}(0) \notag \\
&\times \left(-\frac{e^{-\Delta_{pS}^2s} }{\Delta_{pS}^2} + \frac{e^{-(\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2)s}}{\Delta_{pS}^2+\Delta_{qS}^2+\Delta_{qR}^2}\right) \notag
\end{align}
\begin{align}
\text{Part A}: &(\dv{\bW^{(3)}}{s})_{pR} = (2 \bF{}{(0)}\bW^{(1)}\bD^{(2)} + 2 \bF{}{(2)}\bW^{(1)}\bD^{(0)})_{pR} \notag \\
&= 2\sum_{qS} \epsilon_p\delta_{pq} W_{qS}^{(1)}D_{SR}^{(2)} + F_{pq}^{(2)}W_{qS}^{(1)}D_S\delta_{SR}
\end{align}
%///////////////////////////% %///////////////////////////%
\subsubsection{Forth order Hamiltonian elements} \subsubsection{Forth order Hamiltonian elements}
%///////////////////////////% %///////////////////////////%