Location, Location, Location

“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.” PIB stack

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.

.PIB sum
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.

Inverse tennisSuppose 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


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