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How do we know at exactly what point the theories of classical physics break down, because there must be a smooth transistion between the two; so how do we decide at exacly what scale in spacetime to leave the theories of classical physics and apply our knowledge of quantum physics? Is the line between the two blurry or is it sharp?

Thank you.

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    $\begingroup$ See also: physics.stackexchange.com/q/343462/125 $\endgroup$ – Mateen Ulhaq Nov 23 '17 at 7:27
  • $\begingroup$ The classical description is never exact, but the error made using it can be quantified explicitly in some situations. I would say that the classical description is acceptable as long as the error is small enough to make predictions sufficiently accurate. So I would say that there is not a sharp "line", and it strongly depends from the context. $\endgroup$ – yuggib Nov 23 '17 at 7:42
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A useful approximation for size scales at which classical mechanics and the quantum description of particles "switch over" is as follows: for an object the size of an orange, quantum effects are too small to detect. Now we go down in scale to an object the size of an atom: quantum effects are important at that scale; for example, they give rise to the discrete energy levels which the electrons belonging to that atom must occupy. Now go down further in scale to the realm of the nucleus of that atom: here, quantum effects are fully dominant.

The switchover from classical to quantum is gradual and blurry. Note that any model we might propose to account for what goes on at the size scale of an atomic nucleus is required to gradually revert to the classical picture as we scale things up.

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