Consider the Hamiltonian \[ H= \alpha H_0+\beta(t) H_1\] where \begin{equation} H_0= \begin{pmatrix}1&0&0\\0&2&0 \\0&0&3\end{pmatrix}\qquad \text{and} \qquad H_1 = \begin{pmatrix}0&0&1\\0&0&0\\1&0&-2 \end{pmatrix}\end{equation}
and where the time dependent function \(\beta(t)\) is given by \(\beta(t)=\alpha\) for \(t\le 0\) and zero for \(t>0\). Find \(\big|\langle\Psi(t>0)\vert\Psi(t<0)\rangle\big|^2\) where \(\vert\Psi(t<0)\rangle\) is thenormalized ground state wave function and \(\vert\Psi(t>0)\rangle\) is the ground state of the system at \(t>0\).
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4727:Diamond Point
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