What is the difference between a functional and an operator? When we define an operator in physics, e.g. the momentum operator as $\hat{p} = i \frac{d}{dx}$, it is said this operator acts on the wave functions. But isn't something that takes a function as an argument also called a functional? Why do we call $\hat{p}$ momentum operator and not momentum functional?
Answer
Loosely, an operator (acting on a function space) takes functions to functions (e.g., $f(x)$ to $-i f'(x)$). On the other hand, a functional takes functions to numbers (think about a certain integral, or the derivative evaluated at a certain point).
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