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It dawned on me while revising the basics of quantum mechanics (the Schroedinger equation, de Broglie's matter waves, the quantum postulates etc. - essentially the basic first year undergrad schedule).

Since Sch.Eq was borne out of use of Hamilton's equations, isn't it a chicken-egg situation that QM then makes implications about the macro-world and finally state that the Expectation value (or average) of an operator is actually the observable (Ehrenfest thm.), out of which the operator itself is a derivation of the observable borne out of 'classical' physics formulation? Could there be a logical fallacy?

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  • $\begingroup$ What do you mean with implications about the macro world? $\endgroup$
    – Karim Chahine
    Commented Feb 8, 2021 at 16:04
  • $\begingroup$ I meant that QM then tries to give a foundation (or further insight/explanation) as from where classical physics is based upon. In a way, when we make gross observations, QM offers a fundamental reason about these observations, such as why microscopic effects are not detected; and should not be detected. $\endgroup$
    – plusunim
    Commented Feb 8, 2021 at 21:46

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While it is true that the Schrodinger Equation was motivated by classical mechanics, it was not derived from classical mechanics. Quantum mechanics is a 'new' theory and was not derived, but rather proposed.

The basic idea is that the 'new' physics being proposed had better reproduce classical physics in the 'classical regime'. This is called the correspondence principle. For example, one can check that the motion of the center of a wave packet for a large mass particle will follow the predictions of classical mechanics.

For more on how Schrodinger came up with his equation, you might find his own paper on the subject quite fascinating: (See Schrodinger's quite readable 1926 paper "An Undulatory Theory of the Mechanics of Atoms and Molecules https://journals.aps.org/pr/abstract/10.1103/PhysRev.28.1049)

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  • $\begingroup$ Thank for the link. I am still not entirely convinced yet. But I'll get into it. My main point is that classical physics' influences, thoughts and formulations ended up formualting (being the background of) QM, which in turn started offering explanations about the observable world - lo and behold a correspondence was affirmed. I do hope I am clear now. $\endgroup$
    – plusunim
    Commented Feb 8, 2021 at 21:51
  • $\begingroup$ But you cannot derive QM from classical physics, you need a new language and new interpretations. It's not like trying to prove something assuming the hypothesis. $\endgroup$
    – Karim Chahine
    Commented Feb 8, 2021 at 22:24

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