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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 $\times$, $+$!

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Weyl exponential form of the Canonical Commutation Relations

Here $\hat{q},\hat{p}$ are operators of position and momentum with canonical commutation relation $[\hat{q},\hat{p}]=i\hbar$. …
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