Do operators always give a number after operating?

Question

I am having some doubts regarding operators. In QM, when operators work on a wave function, will it always give a number times the wave function? Suppose I applied it on any normal function of x. Will it always give me a number times the function as a result or it may give another function of x in the place of a number? And how to differentiate between the results?

Answer 1

An operator $A: V \to W$ is a linear map between two vector spaces $V$ and $W$, e.g. two function spaces. In quantum mechanics, $V$ and $W$ are typically the Hilbert space $L^2(\mathbb{R}^n)$ of square-integrable functions (or a dense subspace of $L^2(\mathbb{R}^n)$). Thus, an operator takes an element of its domain (e.g. a function) and maps it to an element of its range (e.g. another function).

If an element $f\in V$ satisfies $Af = \lambda f$ for some $\lambda \in \mathbb{C}$, then $f$ is called an eigenvector of $f$ with eigenvalue $\lambda$. Eigenvectors are very special elements of $V$. It may happen that an operator has no eigenvectors at all.

Answer 2

No. In general, an operator acting on a function or a vector gives another, different function or vector.

Only for the special case of eigenfunctions or eigenvectors does the action of an operator on an object give a scalar multiple of same object.