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In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

Technically a synonym for function, "operator" in physics almost always means either

  1. A linear mapping between two function spaces (defined on, say, $\mathbb R^n$), or

  2. A square matrix (representing a linear mapping from a vector space to another according to a certain basis).

Some examples:

  1. The so-called operator, $D \colon C^1\to C^0$ such that $f(x)\mapsto -i\hbar \nabla_xf(x)$ defines an operator $D$ as typically thought of in physics, and

  2. The Fourier transform $\mathcal{F}\colon{L}^1(\mathbb R^n)\to{L}^\infty(\mathbb R^n)$.

In , observables are Hermitian operators acting on the of quantum states, together with a "recipe" for physically interpreting them, namely: straight after an observation, the quantum state $\psi$ is projected onto a random eigenfunction $\psi_j$ of the operator $F$ with probability (or probability density) $|\langle\psi|\psi_j\rangle|^2$, and the $M^\text{th}$ moment of this probability distribution given quantum state $\psi$ is $|\langle\psi|F^M\psi\rangle|^2$.

Another common physics concept of an operator, more in line with its mathematical meaning, is a state transition or operator in quantum mechanics $U(t) = \exp(i\hat{H}t)$ that maps a quantum state $\psi$ into its value $U(t)\psi$ a time $t$ later in the Schrödinger picture, where $\hat{H}$ is the (energy) observable.