How did quantum mechanics operators come into being?

Question

Now I am starting to learn Quantum Mechanics. In the class I am taught about operators, postulates and all other basic stuff.

I understand operators to be +, -, /, etc; but quantum mechanical operators are entirely different; to understand them, I think, I need to know the historical development of the physics operators. So, I want to know how these operators were discovered/invented; some of the historical figures on this subject would also help along with some first text books from the original authors (modern texts are also okay).

Answer 1

It is a good question, how such a weird theory could be invented:-)

In its present form the theory was developed by Dirac (physicist) and von Neumann (mathematician). von Neumann essentially developed the mathematical operator theory which is needed. They both wrote books on quantum mechanics which explain the motivation for operators. (You may prefer one book or another depending on your background: whether you are a physicist or a mathematician.) But they gave the theory its final form, and the history is really complicated. To understand some history, I recommend van der Waerden, Sources in the history of quantum mechanics, which is a collection of original articles, translated into English, with his commentaries. To go deeper in the history, you have to learn German, first of all.

(Some people who knew von Neumann say that he was an alien, extraterrestrial: a human mind could not invent such things.)

EDIT. As I said, discovery of quantum mechanics is a very long story, and this is a result of collective effort. It starts at least from the discovery by Balmer of the empirical formula for the spectral lines of hydrogen, which triggered the whole process. (But really the story can be traced back to Newton.) Then Rydberg and Ritz should be mentioned, and Planck and Einstein. A milestone was Bohr's theory which is sometimes called "old quantum mechanics". It describes many phenomena correctly, but there are no operators yet. Then Heisenberg, Schrödinger, Born, Jordan and Dirac made a "quantum leap", and gave it the more or less modern form. Operators were introduced by Dirac but without a rigorous mathematical justification. Then von Neumann developed the rigorous mathematical theory that was required. On the early stage (which can be called pre-history, that is before Bohr), there is an excellent exposition in the book by Shlomo Sternberg, Group theory and physics, Appendix F “A history of 19th century spectroscopy”. It covers the development from Newton to Bohr. I am not familiar with any exposition of similar quality and clarity for the period between Bohr and Heisenberg. Sommerfeld's book Atombau und Spektrallinien is close to it, but he was more interested in explaining the current state of the theory, rather than history.

Answer 2

The turn to operators occurred in Heisenberg's 1925 paper, and it was a compromise between positions taken by Bohr and Born. Bohr wanted minimal modifications to classical mechanics, like his quantization rules for the hydrogen atom, while Born wanted a new discrete mechanics governed by difference equations. Heisenberg's idea was to keep equations of motion more or less classical, but to reinterpret symbols in them. Those symbols represent observables (position, momentum, energy, etc.), and in Hamiltonian dynamics they are functions of positions $x$ and momenta $p$, which form the phase space. Applying this idea Heisenberg discovered that his symbols would not commute, for example $xp-px=i\hbar$ with the Planck constant $\hbar$, so they could not be numbers. A mathematician friend of his pointed out that matrices do not commute, and could satisfy such relations, and this gave Heisenberg's proposal its name, matrix mechanics. Except that since electrons in an atom have infinitely many energy levels these "matrices" had to be of infinite size.

At about the same time Schrodinger was working from a different perspective, he wanted to reduce quantum effects to wave dynamics. So he represented states of quantum systems by wave functions $\psi$ on the phase space and looked for equations of motion. His idea was that quantum particles are wave packets, and he originally thought of $|\psi|^2$ as charge density, only later Born identified it with probability density. After matrix mechanics came out Schrodinger realized that he needed to introduce observables into his picture, but they could not be mere symbols. Since states are wave functions on the phase space rather than its points, observables can not be functions of them, they have to act on wave functions like matrices act on vectors. After some experimenting in 1926 he came up with representing position by a multiplication operator $\psi\mapsto x\psi$, and momentum by differentiation $\psi\mapsto-i\hbar\frac{\partial \psi}{\partial x}$. The big clue was that if we think of $x,p$ as these operators then $(xp-px)\psi=i\hbar\,\psi$. Then we can form other observables from these ones. Classical kinetic energy is $p^2/2m$, replacing $p$ by its operator we get $-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$ from the Schrodinger equation. The total energy of a classical oscillator is $p^2/2m+kx^2/2$, replacing gives the Hamiltonian of the quantum oscillator $-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{kx^2}2$. And so on.

In 1930 Dirac conceptualized this picture in his textbook Principles of Quantum Mechanics (1930), where he introduced the bra-ket notation among other things. Then von Neumann gave an axiomatic formulation based on the notion of abstract Hilbert space that he introduced, and established that matrix and wave mechanics were talking about the same thing in two different ways. In Schrodinger's picture the states became elements of a Hilbert space, namely the space $L^2$ of square integrable (wave) functions on the position space. The observables became self-adjoint operators on it. Upon choice of an orthonormal basis in this space however the states become (infinite) vectors, and the operators become (infinite) matrices. This is the Heisenberg's picture.

There is a relevant thread on Physics SE How did the operators come about? Landsman gives a nice short survey of historical developments in Between Classical and Quantum. More comprehensive references are Kragh's Quantum Generations, and Jammer's classic Conceptual Development of Quantum Mechanics.