I like this explanation, which puts the concept of operator at center and has some nice computational relations, and is related to the physical world by the not

I like this explanation, which puts the concept of operator at center and has some nice computational relations, and is related to the physical world by the notion of an observable, with an eigenfunction being a special kind of operator-function relationship. At the end, there's a nice non-QM application (Fourier transforms). source: PW Atkins & RS Friedman, "Molecular Quantum Mechanics 3rd Ed" > An observable is any dynamical variable that can be measured... in classical mechanics... observables are represented by functions* (such as position as a function of time), in quantum mechanics they are represented by mathematical operators... We shall not in general distinguish between the observable and the operator that represents that observable (such as the position of a particle along the x-axis)" > "An operator is a symbol for an instruction to carry out some action, an opeeation, on a function...in certain cases, the outcome of an operation is the same function, multiplied by a constant" [that constant being the eigenvalue] Form: (operator) (eigenfunction) = (eigenvalue) (eigenfunction) > "An important point is that a general function can be expanded in terms of all the eigenfunctions of an operator, a so-called complete set of functions... then a general function can be expressed as the linear combination [a sum over a complete set of functions, each function having its own coefficient]" Finally, we get to a practical real-world example (non-QM): > "...for instance, the straight line g = ax can be recreated over a certain range by superimposing an infinite number of sine functions, each of which is an eigenfunction of the [differentiation] operator, d2/dx2. Alternatively, the same function may be constructed from an infinite number of exponential functions, which are eigenfunctions of d/dx." Extending this general concept, we can go into the famous Fourier Transform, used widely for all kinds of classical problems in converting waveforms into frequency peaks, such as in musical analysis, electrical engineering, etc. The wiki page on Fourier transforms has a very brief mention of this view, i.e. The Fourier transform decomposes a function into eigenfunctions for the group of translations. Here's what looks like a deep dive into this approach to the Fourier Transform (2008): http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/PPVIETEeigenFT.pdf http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...