[UPSC CSE 2023] Schrodinger’s Wave equation and applications (Q125, 20M)

Consider the potential V(x) = \begin{cases} 0, & 0 < x < a \\ \infty, & \text{elsewhere} \end{cases} (a) Estimate the energies of the ground state as well as those of the first and the second excited states for (i) an electron enclosed in a box of size a = 10^{-10}\text{ m}. (ii) a 1\text{ g} metallic sphere which is moving in a box of size a = 10\text{ cm}. (b) Discuss the importance of the Quantum effects for both of these systems. (c) Estimate the velocities of the electron and the metallic sphere using uncertainty principle.