A quantum system has a complete set of orthonormal eigenstates, $\psi_{n}(x)$, with nondegenerate energy eigenvalues, $E_{n}$, where $n=1,2,3 \ldots$ Write down the wave-function, $\Psi(x, t), t \geqslant 0$ in terms of the eigenstates.

A linear operator acts on the system such that

$$\begin{aligned} &A \psi_{1}=2 \psi_{1}-\psi_{2} \\ &A \psi_{2}=2 \psi_{2}-\psi_{1} \\ &A \psi_{n}=0, n \geqslant 3 \end{aligned}$$

Find the eigenvalues of $A$ and obtain a complete set of normalised eigenfunctions, $\phi_{n}$, of $A$ in terms of the $\psi_{n}$.

At time $t=0$ a measurement is made and it is found that the observable corresponding to $A$ has value 3. After time $t, A$ is measured again. What is the probability that the value is found to be 1 ?