In this lecture, we tackle a classic problem in Quantum Mechanics focusing on the statistical interpretation of the state vector. Given a quantum system prepared in a linear superposition of orthonormal energy eigenstates ($|\psi\rangle = \sum c_n |\phi_n\rangle$), we rigorously calculate the specific probabilities of obtaining discrete energy values upon measurement ($P_n = |c_n|^2$). The procedure emphasizes the use of Dirac notation and inner products to find probability amplitudes, culminating in the formal calculation of the expectation value for the Hamiltonian operator ($\langle \hat{H} \rangle$).

In this lecture, we explore the mathematical foundations of quantum observables by rigorously proving the essential properties of Hermitian operators. Using Dirac bra-ket notation, we demonstrate that the eigenvalues of a Hermitian operator are strictly real, a mathematical necessity ensuring that physical observables yield real measurement values. Subsequently, we prove that eigenvectors corresponding to distinct eigenvalues are mutually orthogonal, forming the basis for the state space. This video serves as an excellent pedagogical bridge for students familiarizing themselves with abstract operator algebra and its direct physical implications in Quantum Mechanics.