The Limits of Knowledge, Part II: Precise Uncertainty

(Second in a series on the limits of knowledge; see the first post here.)

Of all branches of modern science, quantum mechanics is most seen as magic–either a nihilistic, quasi-Voldemortesque dark magic that needs to be overthrown, or else a wonderful wand that can be waved to justify anything, and I mean anything.

To be sure, Einstein’s relativity disquiets many people.  Without trustworthy, absolute clocks, who can boast about trains running on time?

But quantum mechanics is an order of magnitude stranger. The quantum world is fundamentally uncertain and fuzzy, with slippery wavefunctions leaping from one state to another. Even Einstein himself, who helped to father the field, hated it.

As I’ve written in an earlier post, many SF authors choose either to rebel and literally write quantum mechanics out of the equation, or to use quantum mechanics as a convenient justification for neato pseudo-scientific wish-fulfillment.

All of this is because of fundamental misunderstandings about quantum mechanics.

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Quantum mechanics suffers from a complicated, disjointed origin. Unlike the origin story of relativity–where a shy young German lad is bitten by a radioactive Swiss watch and goes on to revolutionize  physics during a schnitzel break–quantum mechanics evolved slowly and painfully over a period of three decades with half a dozen fathers (including Einstein himself).

Einstein derived both special and general relativity starting from fairly innocuous statements (“you cannot do an experiment to determine absolute motion” and “you cannot do an experiment to distinguish between gravity and uniform acceleration,” respectively) and then, with a few turns of the axiomatic crank and a bit of mathematical perseverance, voila! one has the theory.

But quantum mechanics is not easily derived from such a simply stated premise. Instead, it is more Frankenstein than Einstein, stitched together from the corpses of earlier theories. Even today the scars are still visible, most prominently in the theory of how quantum systems change with time.

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Humans are creatures of the clock, obsessed by time, especially the murky future. Physicists are no different.

After all, it’s prediction physical theories strive for. Newton’s laws predict the flight of a baseball over a green summer field; relativity the path of a photon as it careens near a black hole.  Quantum mechanics, too, plays this game, and very well, as Schrodinger’s time evolution equation describes electrons ricocheting off atoms.

Except that Schrodinger’s equation is not the only rule for change over time. When an “observer” shoves a measuring device into the system, the wavefunction is said to abruptly and instantaneously collapse to a new wavefunction. (The “instantaneous” collapse, along with entanglement, leads to the idea that quantum mechanics violates relativity. But you can’t use either one to transmit information faster than the speed of light.)

Unsurprisingly, a mash-up of two rules for time evolution is unpopular. In response people have formulated alternate “interpretations” of quantum mechanics using the same mathematics and predicting the same experimental outcomes, but visualizing different paradigms (internal pictures of the  physical processes). Unfortunately the alternatives are even more clumsy and impractical, especially for everyday use around the home cyclotron.

Even with the “user friendly” version, it’s not easy to make quantum mechanical predictions.  In classical, Newtonian mechanics, objects have a precisely defined position at all times, and the velocity is a precise measure of how position changes with time. (In practice, all measurements have a small error, which will have big consequences, as I’ll write about in my next post on chaos.)

But this is no longer true in quantum mechanics. An object does not have a set location but instead has a set of probabilities smeared out in space. A ten percent chance you’ll find your keys on the dresser, a two percent you find them in the fridge, only with electrons it’s not forgetting where you left one, it’s fundamental.

Newton postulated a clockwork universe, ticking predictably and precisely.  But quantum mechanics demolished Newton’s cogs and gears. There wasn’t a single fixed answer. You could do the same experiment over and over and get different answers. Einstein hated this.

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In magic, anything can happen. But although quantum mechanics is weird, it’s not that weird.

Repeat the same experiment and you get a different answer. But the range of possible answers is fixed. The probabilities of each possible answer are also fixed and calculable from a wavefunction that pops out of Schrodinger’s equation.

Think of quantum mechanics as a kind of subatomic poll. While there is no single outcome, there is an average outcome. And there is a standard deviation, which as you recall hazily from statistics class is how widely the individual outcomes can deviate from the average. In quantum mechanics, the standard deviation is renamed the uncertainty.

It is, in fact, possible for a particle to have very small uncertainty in position. What Werner Heisenberg infamously showed is that as the uncertainty in position decreases, the uncertainty in velocity must increase.

It’s very important to keep in mind, however, that this is not just “throw up your hands in despair” uncertainty. From the wavefunction you can calculate both the average and the uncertainty as many decimal places as you can measure. So we have a kind of precise uncertainty. You can’t predict the outcome of any one experiment, but you can calculate, with great precision and reliability, the range of possible outcomes.

And, importantly, these predictions have been tested and verified experimentally.

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Yes, yes, I see you holding your hand up, and I’m pretty sure I know what your question will be: How do we know that, deep down, the particles don’t really have precise positions and velocities with zero uncertainty, only we have big clumsy instruments with which to measure them?

You aren’t the first to think of this idea. It’s called a “hidden variable” theory and there have been numerous attempts to construct them.

Many alternatives to quantum mechanics are useless: they may be philosophically more palatable, but make the exact same experimental predictions (and usually are, in practical terms, much more clunky; an example is Bohm’s pilot wave theory). And experiment is the referee that science abides by.

Hidden variable theories, however, do make experimental predictions distinct from quantum mechanics. And, in experiment after experiment, starting in 1982 with Alain Aspect, orthodox quantum mechanics has won, hands down, over broad classes of hidden variable theories. (Lawyer-like fine print: there are still viable hidden variable theories, but they have other, distasteful properties, and for them no one has yet devised a suitable experimental test.)

These experiments test Bell’s inequality, which is far too mathematically complicated to explain here, but which is a measure of quantum entanglement: the results of one experiment are correlated in surprising ways with those of another far, far away. Einstein called entanglement “spooky action at a distance,” but now it has become in the popular imagination “we are all one” magic (suitable for lazy New Age thinking or lazy science fantasy tech). This is in fact not true; entanglement is incredibly fragile and after one measurement the wavefunction collapses and there is no more entanglement, no more magic.

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The real magic is the discovery that the universe, stubbornly, refuses to conform to our everyday prejudices.  Not only that the universe is weirder than we imagine, but that we can, just barely, come to understand it.

We’re stuck with quantum mechanics and its precisely fuzzy uncertainties. It would not surprise me that we eventually find a successor to quantum mechanics. But I am sure that whatever theory we find, it will be even weirder and more unsettling than quantum mechanics. Now there’s a science fiction challenge for you!

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