Origin of operators in quantum mechanics

Historically, where did the concept of operators in quantum mechanics come from?

1. How did people first understand that momentum operator should be of the form of $$i \hbar \frac{{\rm d}}{{\rm d}x}$$?

2. 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: • Actually, even before Schrodinger, Heisenberg found that many properties of "quantum" systems can be described by taking the $x$ and $p$ to be infinite-dimenisonal matrices with which $[x,p]=xp-px=\hbar$.
– Prahar
Nov 14 '17 at 21:51
• Is there an English translation of Schrodinger's paper you referenced? I'm a bit confused that there are seemingly two different variables $S$ taken to be the same, one being the action and the other being entropy, but that could very well be a lack of knowledge on my side. Nov 15 '17 at 4:35
• @danielunderwood easy: just OCR that image and feed it to translate.google.com :-) Nov 15 '17 at 12:20
• @danielunderwood actually there is a collection of English translations of early QM papers by Schrodinger and others. I have the book somewhere but saw it pop up when searching for a pdf of the original.
– user6552
Nov 15 '17 at 12:38
• I did find this site that seems to have it under Quantization as an Eigenvalue Problem I. They are pretty low quality scans, but they're there and in English. Nov 15 '17 at 20:38
1. From where did the concept of operator in quantum mechanics came, historically?

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.

1. Momentum operator.

Representation of momentum operator comes from the analogy with classical Hamiltonian mechanics, as explained in the answer by ZeroTheHero.

1. Kinetic energy

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".