An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory. They play a central role in describing observables (measurable quantities like energy, momentum, etc.).
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Quantum operator in the context of Observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.
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