JohnH said:
I read a without-math description online
Where? Please give a specific reference. We can't comment on something we can't read for ourselves.
JohnH said:
the uncertainty relation via delta mc multiplied by delta x is greater than or equal to Planck's constant
For a single quantum particle, yes, that is the uncertainty relation (although momentum would usually be written ##m v##, not ##m c##, since we're not talking about light). But we're not talking about a single quantum particle. We're talking about a macroscopic object with perhaps ##10^{25}## atoms. So you can't just wave your hands and say "position" and "momentum". The object doesn't have a single "position" or "momentum"; it has ##10^{25}## particles, each of which has their own position and momentum.
You can talk about the
center of mass position and momentum, but that just means you have rearranged your description of the object slightly so you have 1 degree of freedom that you are explicitly dealing with (the center of mass), and ##10^{25} - 1## other degrees of freedom (the position/momentum of all but one of the object's particles, since the last one is fully determined by the center of mass plus all the others) that you ignore. You can write down an uncertainty relation for the center of mass, but that only applies to the center of mass degree of freedom; it doesn't mean you've pinned down the exact position and momentum of every particle in the ball to that level of accuracy.
JohnH said:
for a long time I've been puzzling over what a more mathematical description of that looks like
It looks like the classical equations for the center of mass position and momentum, derived under the assumption that whatever quantum uncertainty is involved with the ##10^{25} - 1## other degrees of freedom doesn't affect the classical behavior of the center of mass equations. The center of mass equations themselves are classical because, if you just look at the center of mass degree of freedom by itself, you can (at least heuristically) apply the uncertainty principle using the overall mass of the macroscopic object, so you can indeed show that the uncertainty in the
center of mass position and momentum is so small that the center of mass behaves classically to a good enough approximation for all practical purposes.
Depending on what source you are reading, this approach might be simply asserted without argument, given a hand-wavy heuristic justification, or justified by something like the Ehrenfest theorem based on some reasonable attempt at writing down actual mathematical expressions for the center of mass position and momentum observables.
JohnH said:
I thought that meant that each particle in a Hilbert space is definitionally orthogonal to every other
This doesn't even make sense.
JohnH said:
such that the Hup essentially applied to each particle individually
It does. But you don't need to deal with this if all you're interested in is the center of mass behavior.
JohnH said:
you can think about this in one degree of freedom such that you are thinking of all the particles collectively contributing to a very low uncertainty
Sort of. See above for a better description.