Show that if the energy levels are discrete, the general solution of the Schrödinger equation

$$i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \nabla^{2} \psi+V(\mathbf{x}) \psi$$

is a linear superposition of stationary states

$$\psi(\mathbf{x}, t)=\sum_{n=1}^{\infty} a_{n} \chi_{n}(\mathbf{x}) \exp \left(-i E_{n} t / \hbar\right)$$

where $\chi_{n}(\mathbf{x})$ is a solution of the time-independent Schrödinger equation and $a_{n}$ are complex coefficients. Can this general solution be considered to be a stationary state? Justify your answer.

A linear operator $\hat{O}$ acts on the orthonormal energy eigenfunctions $\chi_{n}$ as follows:

$$\begin{aligned} &\hat{O} \chi_{1}=\chi_{1}+\chi_{2} \\ &\hat{O} \chi_{2}=\chi_{1}+\chi_{2} \\ &\hat{O} \chi_{n}=0, \quad n \geqslant 3 \end{aligned}$$

Obtain the eigenvalues of $\hat{O}$. Hence, find the normalised eigenfunctions of $\hat{O}$. In an experiment a measurement is made of $\hat{O}$ at $t=0$ yielding an eigenvalue of 2 . What is the probability that a measurement at some later time $t$ will yield an eigenvalue of 2 ?