I Planck-Einstein: how does it separate classical from Quantum?

It's assumed everything obeys quantum laws. As scale goes up, quantum effects are less noticeable. Things begin to behave more classically. There is no sharp dividing line, however.

 

Planck's and Einstein's (and also Bohr's) "old quantum theory" is dissatisfactory, because it consists of a lot if intrinsically inconsistent ad-hoc assumptions with quite strange (if not esoteric) notions like wave-particle duality. Thus very soon, in 1925/26, modern non-relativistic quantum theory has been discovered already in three equivalent versions (Heisenberg+Born+Jordan+Pauli: "matrix mechanics", Schrödinger ("wave mechanics"), Dirac ("transformation theory")) and brought to a rigorous mathematical form in terms of Hilbert-space theory by von Neumann.

In modern quantum theory the classical physics occurs as an emergent phenomenon derivable from quantum theory by appropriate coarse-graining to effectively describe the relevant macroscopic degrees of freedom.

 

And later Dirac's transformation theory was bought into rigorous mathematical form thanks to some of the greatest mathematicians of the later part of the 20th century - Gelfland, Grothendieck and Schwartz.

Often mathematicians preempt the mathematics physicists and other applied mathematicians need, but here it was reversed and led to some very beautiful, deep and highly applicable math.

Thanks
Bill