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I read before that one can't derive a more fundamental theory, and Quantum Mechanics is a more fundamental theory than classical physics.

I understand the equation

$$E=\frac{1}{2}mv^2+U$$

to be a classical equation as in Quantum Mechanics the concept of velocity doesn't make sense as the concept of velocity doesn't make sense in Quantum Mechanics. Yet I saw it get used in the derivation of the Schrodinger Equation

For Relativistic Quantum Mechanics I also saw the Dirac Equation Derived from the equation

$$E^2=\left(mc^2\right)^2+(pc)^2$$

which I understand that in Non Quantum Special Relativity the equation

$$E^2=\left(mc^2\right)^2+(pc)^2$$

comes from the laws of physics being the same in all inertial reference frames, and from the speed of light being the same for all inertial reference frames.

I understand inertial reference frames in terms of classical physics, in the sense that an object is in an inertial reference frame if it is moving at constant velocity, and the net force acting on it is $0$. This means that I understand the meaning of inertial reference frames in terms of classical physics, but I'm not sure if the concept of inertial reference frames would extend to Quantum Mechanics.

To me deriving the Schrodinger Equation from $$E=\frac{1}{2}mv^2+U$$ seems to go against the concept that I cannot derive an equation in a more fundamental theory from an equation in a less fundamental theory.

My question is why would we use classical equations for energy to derive equations in Quantum Mechanics, or if they aren't classical equations of energy then how do we derive them in Quantum Mechanics?

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    $\begingroup$ Partially related answer of mine: physics.stackexchange.com/q/555301 $\endgroup$
    – user87745
    Commented Nov 23, 2021 at 4:52
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    $\begingroup$ I don't think one is able to "rigorously" derive the time-dependent Schrodinger equation or the Dirac equation anymore than one can rigorously derive F=ma. Any "derivations" of it will be heuristic ones, and in a heuristic derivarion they're only trying to make the result plausible to you. To that end they'll use things you already know, like classical energy equations. $\endgroup$
    – flevinBombastus
    Commented Nov 23, 2021 at 4:57
  • $\begingroup$ Dvij D.C. is kind of saying this in his answer, but it seemed worth stating explicitly. $\endgroup$
    – flevinBombastus
    Commented Nov 23, 2021 at 4:58
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    $\begingroup$ The quantum phenomena could not be fitted with the classical equations. Because they were involved with the dimensions of the microcosm , for continuity it was anticipated that all the well validated classical physics equations should "emerge" from a successful quantum theory at the phase space of the variables of the classical observations. Making a wave differential equation out of the classical equations was an intuition that payed off. $\endgroup$
    – anna v
    Commented Nov 23, 2021 at 7:44

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