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Historically, where did the concept of operators in quantum mechanics come from?
How did people first understand that momentum operator should be of the form of $i \hbar \frac{{\rm d}}{{\rm d}x}$?
Also how did they find the expression for kinetic energy operator?
The idea of $p\to \frac{\partial }{\partial q}$ can be found in the canonical Hamilton-Jacobi theory, where $$ p=\frac{\partial S}{\partial q}\, . $$ This was (apparently) the inspiration for Schrodinger. In fact, on the first page of his original paper, one can find the equations \begin{align} H\left(q,\frac{\partial S}{\partial q}\right)&=E\, ,\\ S&=K\log\psi\, ,\\ H\left(q,\frac{K}{\psi}\frac{\partial \psi}{\partial q}\right)&= E \end{align}
It follows immediately that the kinetic energy is proportional to the second derivative of $\psi$ w/r to the position $q$.
Here's the translation of that first page:
This was a gradual development started by Heisenberg's insight. He invented (infinite) matrices (without any prior knowledge of matrix multiplication). This was followed by by Born, Jordan and Dirac. Dirac's book Principles of quantum mechanics (1930) explains in great detail where the operators come from. Two years later a rigorous mathematical theory of self-adjoint operators needed in quantum mechanics was developed by von Neumann.
Representation of momentum operator comes from the analogy with classical Hamiltonian mechanics, as explained in the answer by ZeroTheHero.
Comes from the general quantization rule: you take a classical expression and substitute the operators in it instead of coordinates and momenta.
EDIT. To address the question asked in a comment to the answer of ZeroTheHero: There is a book by B. L. van der Waerden, Sources in quantum mechanics, where many papers of the early period are translated into English, with comments. Unfortunately, Schroedinger's papers are not there. But the main papers by Heisenberg, Born, Jordan and Dirac of 1925-26 (and many earlier papers) are in the book.
Max Jammer in The conceptual development of Qm, 1967, (chap. 5.2) has a written a dozen page history about operators, an idea that was developped in the study of differential equations. He starts with a Leibniz paper from 1710 (note 87) Symbolismus memorabilis, mentions Lagrange and piles up scores of obscure names. In 1903 Ludwig Silberstein published Eine theorie der physikalischen operatoren in Otswalds Annalen that might have been known by later authors. As a careful historian Jammer refrains from pointing who has been the first, but it appears rather unambiguosly that he would accord priority to the tandem Born-Wiener. He writes that in work published 1925-6 "Born and Wiener, in generalizing matrix mechanics, introduced the operator calculus".
