In quantum mechanics, observables are represented by mathematical operators rather than simple variables as in classical mechanics. This is due to the fundamental principles and mathematical structure of quantum mechanics.
Quantum mechanics describes the behavior of physical systems at the microscopic level, where particles and waves exhibit both particle-like and wave-like properties. The state of a quantum system is represented by a mathematical object called a wave function, which contains all the information about the system's properties.
Observables in quantum mechanics correspond to physical quantities that can be measured, such as position, momentum, energy, and spin. Each observable is associated with a mathematical operator that acts on the wave function to extract the relevant information.
Operators in quantum mechanics are used to calculate the expected values of observables in a given quantum state. When an operator acts on a wave function, it yields a new wave function or a set of possible outcomes and their probabilities. The expectation value of an observable is obtained by calculating the average of these outcomes weighted by their respective probabilities.
The choice of operators for observables is not arbitrary; it is based on the mathematical framework of quantum mechanics, specifically the principles of linear algebra and the postulates of quantum mechanics. Operators in quantum mechanics have unique properties, such as linearity, Hermiticity, and eigenvalue equations, which allow them to represent physical observables accurately and consistently.
The use of operators to represent observables in quantum mechanics provides a powerful and elegant formalism for describing and predicting the behavior of quantum systems. It allows us to calculate probabilities, study the time evolution of states, and understand the relationship between different observables through commutation and uncertainty relations.