Observables correspond to hermitian operators from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
Observables in quantum mechanics are represented by physical quantities that are measurable, such as position, momentum, energy, and angular momentum. In the mathematical formalism of quantum mechanics, observables are associated with operators. These operators act on the wave function of a system and yield the values of the corresponding observable when applied to the wave function. Hermitian operators play a crucial role in quantum mechanics because they are associated with observables. The eigenvalues of a Hermitian operator correspond to the possible outcomes of measurements of the observable it represents. When a measurement is performed on a system in a particular state, the system will collapse into one of the eigenstates of the Hermitian operator, with the corresponding eigenvalue being the result of the measurement. The requirement for observables to correspond to Hermitian operators stems from the postulates of quantum mechanics. According to these postulates, the physical state of a system is described by a wave function, and observables are represented by operators that act on the wave function. The operators corresponding to observables must be Hermitian in order to ensure that the eigenvalues are real and that the probabilities of measurement outcomes are properly defined. Hermitian operators possess several important properties that make them suitable for representing observables in quantum mechanics. For instance, the eigenstates of a Hermitian operator form a complete orthonormal set, which means that any wave function can be expressed as a linear combination of the eigenstates. This property allows us to decompose the wave function into its constituent parts and determine the probabilities of different measurement outcomes. The correspondence between observables and Hermitian operators is a fundamental principle of quantum mechanics that underpins the probabilistic nature of quantum measurements. By associating observables with Hermitian operators, we can make predictions about the outcomes of measurements and calculate probabilities of different results. This correspondence ensures that the formalism of quantum mechanics is consistent with experimental observations and provides a solid foundation for understanding the behavior of quantum systems.
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