Operator (physics)

In physics, an operator is a function over a space of physical states to another space of physical 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 a very useful tool in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian ${\displaystyle L(q,{\dot {q}},t)}$ or equivalently the Hamiltonian ${\displaystyle H(q,p,t)}$, a function of the generalized coordinates q, generalized velocities ${\displaystyle {\dot {q}}=\mathrm {d} q/\mathrm {d} t}$ and its conjugate momenta:

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