If we devise a physical system and perform an observation of some physical quantity, how can we infer that this quantity is related to the eigenvalues of the momentum operator -ih d/dx ? Another way to look at it. Suppose you were handed the theory of quantum mechanics and that you already had an understanding of how to measure the fundamental quantities mass, length and time. How would you design an apparatus to measure particle momentum, one that you were confident was associated with the eigenvalues of -ih d/dx ? Furthermore, what were the early quantum theorists talking about when they said "particle's momentum?" When de Broglie proposed that the momentum of a particle is proportional to its wavelength, he wasn't defining momentum as p=hk. He had some prior idea of a (measurable) momentum in mind that he was relating to wavelength. What was it? Previously posed this question back 2009 here https://www.physicsforums.com/threads/what-is-the-definition-of-a-momentum-measurement.328164/ but never got an answer.