  Dirac's Poisson-bracket-to-commutator analogy for the transition from
classical to quantum mechanics assures that for many systems, the classical and
quantum systems share the same algebraic structure. The quantum side of the
analogy (involving operators on Hilbert space with commutators scaled by
Planck's constant $\hbar$) not only gives the algebraic structure but also
dictates the average values of physical quantities in the quantum ground state.
On the other hand, the Poisson brackets of nonrelativistic mechanics, which
give only the classical canonical transformations, do not give any values for
physical quantities. Rather, one must go outside nonrelativistic classical
mechanics in order to obtain a fundamental phase space distribution for
classical physics. We assume that the values of physical quantities in
classical theory at any temperature depend on the phase space probability
distribution which arises from thermal radiation equilibrium including
classical zero-point radiation with the scale set by Planck's constant $\hbar$.
All mechanical systems in thermal radiation will inherit the constant $\hbar$
from thermal radiation. Here we note the connections between classical and
quantum theories (agreement and contrasts) at all temperatures for the harmonic
oscillator in one and three spatial dimensions.
