All this would be clearer if photons were bigger. Say you want to get a good look at a billiard ball on a billiard table. You want to know where it is and how fast it’s going.

You can’t see. “Seeing” is our metaphor for hitting things with photons and then observing the ricochets. So, you’ve got a BB gun and two kinds of BB’s and a bunch of bells lined up in a grid. If one gets hit, it rings and signals your computer that it got rung. You can tell which kind of BB hit by the sound of the ring.

You have little blue BBs. They’re heavy and they hit hard, but when you get one of those ricocheting off from where you aimed, you know from the fact that it bounced off that the thing it hit was where you were aiming. (Aiming blind, but you know which direction your gun was pointing.) You also have big red balloons. They’re light and they don’t hit hard, but with no atmosphere to slow them down, they too shoot straight. When a red balloon ricochets, you know from the fact that it bounced that the thing it hit was somewhere in the way of that broad cross section of your balloon.

You can get a good fix on a billiard ball by shooting it with blue BB’s, because you can tell from the pattern of which ones come back at you and which ones just go downrange and don’t come back, where the billiard was. But in shooting the billiard with all those BB’s you change its motion.

You can get a bit of a fix on a billiard ball by bouncing balloons off it, and you won’t much change its velocity. There’s your tradeoff.

But then it really gets weird. Thing is, the uncertainty in your measurements isn’t just because the billiard ball HAS a definite position and a definite velocity, only you can’t measure both simultaneously with unlimited accuracy. The is-ness of where the ball IS and what its velocity IS is uncertain. There’s this inherent uncertainty built into reality. And worse yet, it’s not just dice rolling. What one particle will do here can be correlated with what another particle does over there, even when there is no way at all to know what either particle will do until you just run the experiment and see. How do they communicate? They don’t. But their destinies are entangled.

And then it gets weirder. We can actually use this to do computations that examine simultaneously a vast number of cases. We can, in principle [and as demonstrated laboratory fact on a small scale] run calculations that would be physically impossible in a Newtonian clockwork universe.

e=mc^2 is simple in comparison. There really is a world in a grain of sand.