Calculus 2 Equations: Expert Solutions and More

We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space ${\displaystyle {\mathbb{C}}^2}$. We denote by |0⟩ and |1⟩ the standard basis elements ${\displaystyle {(1,0)}^T}$ and ${\displaystyle {(0,1)}^T}$. Let the Hamiltonian of the system in this basis be given by
$H=\omega \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)$
and assume that for ${\displaystyle t=0}$ the state of the system is just given by ${\displaystyle \psi (t=0)=\mid 0⟩}$. In the following, we also assume natural units in which ${\displaystyle h\overline{=}1}$.
We expand the state at time t in the basis |0⟩, |1⟩ so:
${\displaystyle \mid Psi\left(t\right)⟩={\alpha }_0\left(t\right)\mid 0⟩+{\alpha }_1\left(t\right)\mid 1⟩}$
Problems: Use Schrödinger's equation in order to derive a differential equations for ${\displaystyle {\alpha }_0\text{and}{\alpha }_1}$:
(i) Find a solution given the initial conditions.
(ii) What is the probability that the system can be measured in |1⟩ at some time t?

How do you call this system?
Is there a specific name for a dynamical system that depends on the relative indexation ${\displaystyle i\pm k}$ for some k? For example, consider the following dynamical system defined on a ring of cells by
$\begin{array}{rl}{\dot{u}}_i& =u_i+2(v_{i-1}+v_{i+1})\\ {\dot{v}}_i& =u_i-v_i\end{array}$
for each cell i, where the derivative is with respect to time t.
The main reason I ask this is because I won't to compare this kind of systems with systems involving spatial coordinates, u(x,t), as in reaction-diffusion equations.

find a point on the plane x+y+z=1 that is closest to the point (2,0,-3)