If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t),)$ with Hamiltonian $H$ given by $H=\hbar\omega \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 2 \\ 0 & 2 & -1 \end{pmatrix}$ With $\psi_t(0)=(1,0,0)^T$ So I proceeded to find the stationary states of $H$ by finding it's eigenvectors and eigenvalues. $H$ has eigenvalues and eigenvectors: $3\hbar\omega,0,-3\hbar\omega$ $\psi_+=\frac{1}{3}(2,2,1)^T,\psi_0=\frac{1}{3}(2,-1,-2)^T,\psi_-=\frac{1}{3}(1,-2,2)^T$ Respectively.
Could anyone explain to me how to go from this to a general time dependent solution, and compute probabilities of location? I have only ever encountered $\Psi=\Psi(x,y,z,t)$ before, so I am extremely confused by this matrix format.
I would be extremely grateful for any help! Many thanks in advance