When I see physics formulas everything is perfect continuous everywhere, like momentum conserving Navier-Stokes equation, even like Schrödinger equation. But in atom scale everything is not continuous, even energy is not. Is that just because the discontinuity is too small to be observed?
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1$\begingroup$ "But in atom scale everything is not continuous", why exactly do you say this? $\endgroup$– jacob1729Jul 26 at 14:11
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1$\begingroup$ The question of how well continuous descriptions approximate a discrete system, or vice versa, is a research-level question in many fields! $\endgroup$– StenJul 26 at 14:36
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$\begingroup$ Discreteness and continuity are features of mathematical models: they don't exist in reality. Indeed, reality demands counterintuitive hybrids of discreteness and continuity in quantum theory. $\endgroup$– John DotyJul 26 at 15:56
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Originally, yes, the discrete nature of matter was much to small to be observed.
Right now, things like the Navier-Stokes equation are still useful, because trying to do discrete math with the number of particles that would be useful for fluids requires immense computational resources. It's much easier to do approximations to an unsolvable equation than to simulate moles of molecules in all their detail.
Also, even right now there would be a decent argument as to whether or not everything is actually discrete. Things like the electric field, which have a value at every point in space - are there minimum amounts of electric field? Is there a minimum amount of velocity? There probably is for angular velocity (it's $\hbar/2$). Nothing in our universe has less spin than that. We can generate measurements like Planck Distance from that, but it's also arguable that Planck Distance is just "the shortest distance over which it's useful to measure things" rather than a discrete block of distance from which we build the rest of space.
Fortunately, both continuous and discrete models can be very effective. Whether you like Schrödinger mechanics or Heisenberg mechanics, you can do quantum physics with either.
