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Linear Operators: Motivation in Quantum Mechanics
This lecture introduces the motivation for studying linear operators, focusing on their essential role in Quantum Mechanics (QM). It covers the properties of linear operators in inner product spaces, the representation of measurable quantities by self-adjoint linear operators in QM, and the estimation of probabilities for measuring eigenvalues. Examples of observables in QM, such as position, momentum, and energy operators, are discussed. Additionally, the importance of linear operators in solving linear ODEs and PDEs, identifying spectra and eigenvectors, and computing matrix exponentials is highlighted.