[UPSC CSE 2026] Foundations of Quantum Mechanics (Q3, 10M)

A particle is in the normalized state $\psi$ which is a superposition of the energy eigenstates $\psi_1$ and $\psi_2$ with energies $10\text{ eV}$ and $30\text{ eV}$ , respectively. The average value of the energy of the particle in the state $\psi$ is $24\text{ eV}$ . Find the state $\psi$ in terms of $\psi_1$ and $\psi_2$ .