particle in a box – Hard Science Ain't Hard

“Hoy, Johnny, still got that particle inna box?”
“Sure do, Jessie.”
“So where’s hit in there?”
“Me Pap says hit’s spread-out like but hit’s mostly inna middle.”
“Why’s hit spread then?”
“The more I taps the box, the wider hit spreads. Sommat to do wiff energy.”

Newton would have answered Jessie’s question by saying, sort of, “Pick a point anywhere in the box. The probability that the particle is at that point is equal to the probability that it’s at any other point.”

Quantum physicists take a different approach. They start by saying, “We know there’s zero probability that the particle is anywhere outside of the box, so there must be zero probability that it’s exactly at any wall.”

Now for a trick that we’re actually quite used to. When you listen to an orchestra, you can usually pick out the notes being played by a particular instrument. Someone blessed/cursed with perfect pitch can tell when a note is just a leetle bit flat, say an A being played at 438 cycles instead of 440. You can create any sound by mixing together the right frequencies in the right proportion. That’s how an MP3 recorder does it.

QM solutions use that strategy the other way round. They calculate probabilities by adding together sets of symmetric elementary shapes, all of which are zero at certain places, like the box walls. For instance, on average Johnnie’s particle will be near the middle of his box, so we start a set with an orange mound of probability right there. That mound is like our base frequency — it has no nodes, no non-wall places where the probability is zero.

Then we add a first overtone, the one-node yellow shape that represents equal probability on either side of a plane of zero probability.

Two nodal planes at right angles give us the four-peaked green shape. Further steps up have more and more nodal planes (cyan then blue, and so on). The video shows the running total up to 46 nodes.

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As we add more nodes, the cumulative shape gets smoother and broader. After a huge number of steps, the sum will look pretty much like Newton’s (except for right at the walls, of course).

So if the classical and QM boxes wind up looking the same, why go to all that trouble? Because those nodes don’t come for free.

Suppose you’re playing goalie in an inverse tennis game. There’s a player in each service box. Your job is to run the net line using your rackets to prevent either player from getting a ball into the opposing half-court. Basically, you want the ball’s locations to look like the single-node yellow shape up above. You’ll have to work hard to do that.

Now suppose they give you a second, crosswise net (the green shape). You’re going to have to work twice as hard. Now add a third net, and so on … each additional nodal plane is going to be harder (cost more energy) to keep empty. Not a problem if you have an infinite amount of energy.

Enter Planck and Einstein. They showed there’s a limit for small systems like atoms and molecules. Electrons dash about in atom- or molecule-shaped boxes, but the principle is the same. The total probability distribution is still the sum of bounded elementary shapes. However, you can’t use an infinite number of them. Rather, you start with the cheapest shapes (the fewest nodes) and build upward.

Tally two electrons for each shape you use. Why two? Because that’s the rule, no arguments.

It’s important to realize that QM does NOT say that two specific electrons occupy one shape. All the charge is spread out over all the shapes — we’re just keeping count.

When you run out of electrons the accumulated model shows everything we can know about the electronic configuration. You won’t know where any particular electron is, but you’ll know where some electron spends some time. For a chemist that’s the important thing — the peaks and nodes, the centers of negative and positive charge, are the most likely regions for chemical reactions to happen.

Johnnie’s energetic taps make his particle boldly go where no particle has gone before.

~~ Rich Olcott

“Hoy, Johnny, wotcher got inna box?”
“Hit’s a particle, Jessie.”
“Ooo, lovely for you. Umm… wot’s a particle then?”
“Me Pap says hit’s sommat you calc’late about wiffout knowin’ wot ’tis.”

Pap’s right. Newton was a particle guy all the way (he was a strong supporter of the idea that light is composed of particles). One of his most important insights was that he could simplify gravitational calculations if he replaced an object with an equally massive “particle” located at the object’s center of mass. Could be a planet, or a moon, or that apple — he could treat each of them as a “particle.” That worked fine for his purposes, because the distances between his object centers were vastly larger than the object sizes.

Fleas — “Great fleas have little fleas upon their backs to bite ’em / And little fleas have lesser fleas and so on infinitum.” ~~ Augustus De Morgan

It took Roche to work out what happens when the distances get small. Gravitational forces break the original “particles” into littler particles. And when two of the little ones approach closely enough they break up, and then those break up… You get the idea. Take the process far enough and you get Saturn’s Rings, for instance.

But the analysis can keep going. Consider one “particle” in Saturn’s A-ring. It’s probably about 3″ across, made of ice, and contains something like 10²⁴ particles that happen to be molecules of H₂O. Each molecule contains 3 nuclei (2 protons and one oxygen nucleus) and 10 electrons, all 13 of which merit “particle” status if you’re calculating molecules. They’re all held together by a blizzard of photons carrying the electromagnetic forces between them. The oxygen nucleus contains 16 nuclear particles, each of which contains 3 quarks. The quark structures would fly apart except for a host of gluons that pass back and forth transmitting the nuclear strong force. Hooboy, do we got particles.

“Particle” is a slippery word. For Newton’s purposes, if an object is small relative to its distance from other objects, that was all he needed to know to treat it as a particle.

One dictionary specifies “a small localized object which has identifiable physical or chemical properties such as volume or mass.” However, there are theoretical grounds to believe that the classic “particle of light,” the photon, has neither mass nor volume. Physicists have had long arguments trying to devise a good working definition. Nobelist (1999) Gerard ‘t Hooft ended one such discussion by saying, “A particle is fundamental when it’s useful to think of it as fundamental.”

It may seem a little strange for a physicist to argue for imprecision. In fact, ‘t Hooft was arguing for a broad, even poetic but still precise understanding of the word.

Poets use metaphor to help us understand the world. Part of their art is to pack as much meaning as they can into the minimum number of words. In the same way, scientists use mathematics to pack observed relationships into a simile called an equation — a brief bit of math may connect and illuminate many disparate phenomena.

Think of physics as metaphor, with numbers.

Newton’s Law of Gravity works for for galaxies roving through a cluster and for basketball-sized satellites orbiting Earth and for stars circling a black hole (if they don’t get too close). Maxwell’s Equations, just 30 symbols including parentheses and equal signs, give the speed of light and describe the operation of electric motors. The particle physicists’ Standard Model makes predictions that match experimental results to more than a dozen decimal places.

Good equations are so successful that Nobelist (1963) Eugene Wigner wrote an influential paper entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

We sometimes get into trouble by confusing metaphor with reality. Poetic metaphors can be carried too far — Hamlet’s lungs were not in fact filling with water from his “sea of troubles.”

Mathematical models can also be carried too far. Popular (and practitioner) discussion of quantum mechanics is rife with over-extended metaphors. QM calculations yield only statistical results — an average position, say, plus or minus so much. It’s an average, but of what? The “many worlds” hypothesis is an unnecessarily long jump. There are simpler, less extravagant ways to account for statistical uncertainty.

~~ Rich Olcott

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