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\begin{document}

Tutorial for solving the time-dependent Schr\"{o}dinger equation with
Scientific Workplace.%
\[
i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}|\psi\rangle
\]

Diagonalizing the Hamiltonian
\begin{align*}
\hat{H} &  =\left(
\begin{array}
[c]{cc}%
0 & \frac{1}{2}\Omega\\
\frac{1}{2}\Omega & \Delta
\end{array}
\right)  \\
\hat{E} &  =\left(
\begin{array}
[c]{cc}%
\frac{1}{2}\Delta+\frac{1}{2}\sqrt{\Delta^{2}+\Omega^{2}} & 0\\
0 & \frac{1}{2}\Delta-\frac{1}{2}\sqrt{\Delta^{2}+\Omega^{2}}%
\end{array}
\right)  \\
\hat{U} &  =\left(
\begin{array}
[c]{cc}%
-\frac{\Delta-\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega} & -\frac{\Delta
+\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega}\\
1 & 1
\end{array}
\right)
\end{align*}%
\[
\hat{U}^{-1}\hat{H}\hat{U}=\left(
\begin{array}
[c]{cc}%
-\frac{\Delta-\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega} & -\frac{\Delta
+\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega}\\
1 & 1
\end{array}
\right)  ^{-1}\left(
\begin{array}
[c]{cc}%
0 & \frac{1}{2}\Omega\\
\frac{1}{2}\Omega & \Delta
\end{array}
\right)  \left(
\begin{array}
[c]{cc}%
-\frac{\Delta-\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega} & -\frac{\Delta
+\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega}\\
1 & 1
\end{array}
\right)  =\hat{E}
\]
Solving the time-dependent Schr\"{o}dinger equation
\begin{align*}
|\psi(t)\rangle & =e^{-i\hat{H}t/\hbar}|\psi(0)\rangle\\
& =e^{-i\hat{U}\hat{E}t\hat{U}^{-1}/\hbar}|\psi(0)\rangle\\
& =\hat{U}e^{-i\hat{E}t/\hbar}\hat{U}^{-1}|\psi(0)\rangle\\
\left(
\begin{array}
[c]{c}%
a_{1}(t)\\
a_{2}(t)
\end{array}
\right)    & =\left(
\begin{array}
[c]{cc}%
-\frac{\Delta-\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega} & -\frac{\Delta
+\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega}\\
1 & 1
\end{array}
\right)  \left(
\begin{array}
[c]{cc}%
e^{-it\left(  \frac{1}{2}\Delta+\frac{1}{2}\sqrt{\Delta^{2}+\Omega^{2}%
}\right)  } & 0\\
0 & e^{-it\left(  \frac{1}{2}\Delta-\frac{1}{2}\sqrt{\Delta^{2}+\Omega^{2}%
}\right)  }%
\end{array}
\right)  \left(
\begin{array}
[c]{cc}%
-\frac{\Delta-\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega} & -\frac{\Delta
+\sqrt{\Delta^{2}+\Omega^{2}}}{\Omega}\\
1 & 1
\end{array}
\right)  ^{-1}\left(
\begin{array}
[c]{c}%
1\\
0
\end{array}
\right)
\end{align*}

etc..

\end{document}