Can anyone help me solving the differential equation: $$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=H|\Psi(t)\rangle$$ $$i\hbar\frac{\partial}{\partial t}\begin{pmatrix}\alpha_0(t)\\\alpha_1(t)\end{pmatrix}=\begin{pmatrix}0&-i\\-i&0\end{pmatrix}\begin{pmatrix}\alpha_0(t)\\\alpha_1(t)\end{pmatrix}$$ Where we have that $$|\Psi(t)\rangle=\begin{pmatrix}\alpha_0(t)\\\alpha_1(t)\end{pmatrix}$$ and the condition, that expand the state at time t in the basis $|0\rangle, |1\rangle$, such that $$|\Psi(t)\rangle=\alpha_0(t)|0\rangle+\alpha_1(t)|1\rangle$$ the Hamiltonian (H) of the system in this basis be given by $$ H=\omega\begin{pmatrix} 0 &1 \\ 1 &0 \end{pmatrix}=\begin{pmatrix} 0 &-i \\ -i &0 \end{pmatrix} $$ and assume that for $t=0$ the state of the system is just given by $\psi(t=0)=\left . \left | 0 \right \rangle\right .$
Can anyone help me solving the equation given by the conditions? I'm not sure how to deal with matrix and complex number solving a differential equation? Hope anyone can help me?