About

Technically a synonym for function, "operator" in physics almost always means either (i) a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions or (ii) a square matrix (standing for a linear mapping from $\mathbb{C}^N\to\mathbb{C}^N$). For example, $D:C^1\to C^1:\;D f(x) = -i\,\hbar\,d_x\,f(x)$ defines an operator $D$ as typically thought of in physics. Another example is the Fourier transform $\mathfrak{F}:\mathbf{L}^2\to\mathbf{L}^2$ In quantum mechanics, observables are (i) Hermitian operators operating on the Hilbert space of quantum states, together with *(ii)* 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) $|\left<\psi|\psi_j\right>|^2$, and the $M^{th}$ moment of this probability distribution given quantum state $\psi$ is $|\left<\psi|F^M\,\psi\right>|^2$. Another common physics concept of an operator, more in line with its mathematical meaning, is a state transition or time evolution 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 Hamiltonian (energy) observable.

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