Physicist: We’re definitely missing something, but we’re always missing something. One of the most famous quotes about quantum physics, often used in lieu of a shrug, is due to St. Feynman: “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” Which is fair, but it applies equally well to bicycle mechanics. Or paper airplane design. Not even cake science has a hope of ever being completely understood.
The classical world (the world we experience) has one set of rules and the quantum world has another set of rules. But of course, they’re describing the same, one-and-only universe. The “correspondence principle” says that the quantum laws should, on a large and noisy enough scale, reproduce the classical laws. So far that seems to be exactly the case; as weird as they are, the laws of quantum mechanics are always compatible with the world we see around us. In other words, the very particular laws we consider “normal” are a special case of quantum laws. How that works is unfortunately a case-by-case thing. There’s no clean “quantum-to-classical translation technique”. The reason a thrown rock follows a particular path has a very different explanation than the cause of rainbows. Even the distinction itself, between classical and quantum, is often impossible to nail down.
Quantum mechanical laws are rules for the universe, and in that sense they’re no weirder than gravity or anything that Newton did. And we explore them in the same way: follow clues, come up with models, test them out, find out that practically all of them are wrong, etc. When it comes to the actual doing-of-the-math and the exploration-of-the-physics, there’s nothing unusual about quantum physics. After all, it wasn’t obvious that “for a set of isolated objects, the sum of their masses times the derivative of their positions with respect to time is invariant.” The math and experiments took a while to develop, but then we gave it a name and teach it to kids as “conservation of momentum” (an object in motion stays in motion…).
When you get into the nuts and bolts of physics (the math), no subject is terribly intuitive. If you study classical mechanics (how stuff moves) you run headfirst into Lagrangians and principles of least time or action, and suddenly conversations about balls bouncing and spinning tops become… abstract. Honestly, the big difference isn’t in the difficulty of the physics, it’s in the implications. If you learn about how to predict orbital trajectories, you’ll walk away with the sense that orbits are complicated. If you learn about how to predict quantum tunneling, you’ll walk away with the distinct sense that either the universe is messing with you or that someone, somewhere made a huge mistake and nobody’s bothered to double check their work. The difficulty is comparable, the unease is not.
So why is quantum mechanics so unintuitive? Because we’re not used to it. It would be difficult to get used to moving around in the world if you were a tiny insect, and drops of water stood up like boulders, or if you were a bird, and the air had a “landscape” of movement. We think of the world we live in as intuitive because we not only grew up and live in it, but evolved for it over millions of generations. Quantum effects seem strange to our minds because our brains have never had to deal with them before.
On the other hand (literally), our bodies have had to deal with quantum effects since before they were bodies. While a human mind is really good at maneuvering a human body through the world, dealing with moving objects, weather, other humans, and making up dirty limericks, there hasn’t been much call (over evolutionary timescales) for us to worry about quantum effects. But on the scale of biochemistry, quantum effects are incredibly pervasive, and the tiny chemical mechanisms in every one of our cells take advantage of them all the time. On a chemical level, life has been dealing with quantum phenomena since it began, and (even though we don’t have to think about it) it’s gotten really good at it. For example, photosynthesis involves maintaining the coherence of incoming light (it acts like a wave, not a particle) until it can be directed to a set of molecules that can actually use the energy to make food. If plants didn’t take advantage of superposition and coherence (inherently quantum things), they’d need light-as-a-particle to bullseye the tiny receptors; a huge waste of the vast majority of photons that would be off-target.
At a basic level, every chemical process is inherently quantum mechanical. Chemistry and biochemistry are just applied quantum physics, so if you want to see some ridiculously fancy quantum physics at work, go no farther than your mirror. Or your dinner for that matter.
That said, your conscious mind isn’t there to understand every detail of how your body works down to the atoms, it’s there to react to information from your senses and direct your physical body in such a way that you live through the day (also fall in love and explore and appreciate beauty or whatever). So we’re good at that and we’re used to it. Atoms and quantum mechanics, although not inherently more weird than the “classical world”, take some getting used to.
Fortunately for us, we seem to be pretty adaptable. For example, the Earth intuitively seems to be flat, motionless, and not a brief exception to an infinite nothingness, when in fact it’s round, spinning, unimaginably old, and hurtling through the void at 30 km/s. And yet, most people don’t seem to mind. The nature of the planet we’re on is totally unintuitive, but it’s something you get used to with experience.
The classical world is already incredibly complex and weird and, as “intuitive” as it seems, it still takes a lot of work to understand it (as much as anyone can). But we get used to it. Quantum mechanics is weird too and involves plenty of ideas that seem totally bizarre to our squishy hominid brains. But we get used to it.