Gin And The Art of Quantum Mechanics

On May 30, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, Uncategorized2 Comments

“Fancy a card game, Johnny?”
“Sure, Jennie, deal me in. Wot’re we playin’?”
“Gin rummy sound good?”

Great idea, and it fits right in with our current Entanglement theme. The aspect of Entanglement that so bothered Einstein, “spooky action at a distance,” can be just as spooky close-up. Check out this magic example — go ahead, it’s a fun trick to figure out.

Spooky, hey? And it all has to do with cards being two-dimensional. I know, as objects they’ve got three dimensions same as anyone (four, if you count time), but functionally they have only two dimensions — rank and suit.

When you’re looking at a gin rummy hand you need to consider each dimension separately. The queens in this hand form a set — three cards of the same rank. So do the three nines. In the suit dimension, the 4-5-6-7 run is a sequence of ranks all in the same suit.

A physicist might say that evaluating a gin rummy hand is a separable problem, because you can consider each dimension on its own. <Hmm … three queens, that’s a set, and three nines, another set. The rest are hearts. Hey, the hearts are in sequence, woo-hoo!>

“Gin!”

If you chart the hand, the run and sets and their separated dimensions show up clearly even if you don’t know cards.

A standard strategy for working a complex physics problem is to look for a way to split one kind of motion out from what else is going on. If the whole shebang is moving in the z-direction, you can address the z-positions, z-velocities and z-forces as an isolated sub-problem and treat the x and y stuff separately. Then, if everything is rotating in the xy plane you may be able to separate the angular motion from the in-and-out (radial) motion.

But sometimes things don’t break out so readily. One nasty example would be several massive stars flying toward each other at odd angles as they all dive into a black hole. Each of the stars is moving in the black hole’s weirdly twisted space, but it’s also tugged at by every other star. An astrophysicist would call the problem non-separable and probably try simulating it in a computer instead of setting up a series of ugly calculus problems.

The card trick video uses a little sleight-of-eye to fake a non-separable situation. Here’s the chart, with green dots for the original set of cards and purple dots for the final hand after “I’ve removed the card you thought of.” The kings are different, and so are the queens and jacks. As you see, the reason the trick works is that the performer removed all the cards from the original hand.

The goal of the illusion is to confuse you by muddling ranks with suits. What had been a king of diamonds in the first position became a king of spades, whereas the other king became a queen. You were left with an entangled perception of each card’s two dimensions.

In quantum mechanics that kind of entanglement crops up any time you’ve got two particles with a common history. It’s built into the math — the two particles evolve together and the model gives you no way to tell which is which.

Suppose for instance that an electron pair has zero net spin (spin direction is a dimension in QM like suit is a dimension in cards). If the electron going to the left is spinning clockwise, the other one must be spinning counterclockwise. Or the clockwise one may be the one going to the right — we just can’t tell from the math which is which until we test one of them. The single test settles the matter for both.

Einstein didn’t like that ambiguity. His intuition told him that QM’s statistics only summarize deeper happenings. Bohr opposed that idea, holding that QM tells us all we can know about a system and that it’s nonsense to even speak of properties that cannot be measured. Einstein called the deeper phenomena “elements of reality” though they’re currently referred to as “hidden variables.” Bohr won the battle but maybe not the war — Einstein had such good intuition.

~~ Rich Olcott

Oh, what an entangled wave we weave

On May 23, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Here’s the poly bag wiff our meals, Johnny. ‘S got two boxes innit, but no labels which is which.”
“I ordered the mutton pasty, Jennie, anna fish’n’chips for you.”
“You c’n have this box, Johnny. I’ll take the other one t’ my place to watch telly.”
…
<ring>
” ‘Ullo, Jennie? This is Johnny. The box over ‘ere ‘as the fish. You’ve got mine!”

In a sense their supper order arrived in an entangled state. Our friends knew what was in both boxes together, but they didn’t know what was in either box separately. Kind of a Schrödinger’s Cat situation — they had to treat each box as 50% baked pasty and 50% fried codfish.

But as soon as Johnny opened one box, he knew what was in the other one even though it was somewhere else. Jennie could have been in the next room or the next town or the next planet — Johnnie would have known, instantly, which box had his meal no matter how far away that other box was.

By the way, Jennie was free to open her box on the way home but that’d make no difference to Johnnie — the box at his place would have stayed a mystery to him until either he opened it or he talked to her.

Information transfer at infinite speed? Of course not, because neither hungry person knows what’s in either box until they open one or until they exchange information. Even Skype operates at light-speed (or slower).

But that’s not quite quantum entanglement, because there’s definite content (meat pie or batter-fried cod) in each box. In the quantum world, neither box holds something definite until at least one box is opened. At that point, ambiguity flees from both boxes in an act of global correlation.

There’s strong experimental evidence that entangled particles literally don’t know which way is up until one of them is observed. The paired particle instantaneously gets that message no matter how far away it is.

Niels Bohr’s Principle of Complementarity is involved here. He held that because it’s impossible to measure both wave and particle properties at the same time, a quantized entity acts as a wave globally and only becomes local when it stops somewhere.

Here’s how extreme the wave/particle global/local thing can get. Consider some nebula a million light-years away. A million years ago an electron wobbled in the nebular cloud, generating a spherical electromagnetic wave that expanded at light-speed throughout the Universe.

cats-eye nebula — The Cat’s Eye Nebula (NGC 6543)
courtesy of NASA’s Hubble Space Telescope

Last night you got a glimpse of the nebula when that lightwave encountered a retinal cell in your eye. Instantly, all of the wave’s energy, acting as a photon, energized a single electron in your retina. That particular lightwave ceased to be active elsewhere in your eye or anywhere else on that million-light-year spherical shell.

Surely there was at least one other being, on Earth or somewhere else, that was looking towards the nebula when that wave passed by. They wouldn’t have seen your photon nor could you have seen any of theirs. Somehow your wave’s entire spherical shell, all 10¹² square lightyears of it, instantaneously “knew” that your eye’s electron had extracted the wave’s energy.

But that directly contradicts a bedrock of Einstein’s Special Theory of Relativity. His fundamental assumption was that nothing (energy, matter or information) can go faster than the speed of light in vacuum. STR says it’s impossible for two distant points on that spherical wave to communicate in the way that quantum theory demands they must.

Want some irony? Back in 1906, Einstein himself “invented” the photon in one of his four “Annus mirabilis” papers. (The word “photon” didn’t come into use for another decade, but Einstein demonstrated the need for it.) Building on Planck’s work, Einstein showed that light must be emitted and absorbed as quantized packets of energy.

It must have taken a lot of courage to write that paper, because Maxwell’s wave theory of light had been firmly established for forty years prior and there’s a lot of evidence for it. Bottom line, though, is that Einstein is responsible for both sides of the wave/particle global/local puzzle that has bedeviled Physics for a century.

~~ Rich Olcott

Think globally, act locally. Electrons do.

On May 16, 2016November 30, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“Watcha, Johnnie, you sure ‘at particle’s inna box?”
“O’course ’tis, Jennie! Why wouldn’t it be?”
“Me Mam sez particles can tunnel outta boxes ’cause they’re waves.”
“Can’t be both, Jessie.”

Maybe it can.

Nobel-winning (1965) physicist Richard Feynman said the double-slit experiment (diagrammed here) embodies the “central mystery” of Quantum Mechanics.

When the bottom slit is covered the display screen shows just what you’d expect — a bright area opposite the top slit.

When both slits are open, the screen shows a banded pattern you see with waves. Where a peak in a top-slit wave meets a peak in the bottom-slit wave, the screen shines brightly. Where a peak meets a trough the two waves cancel and the screen is dark. Overall there’s a series of stripes. So electrons are waves, right?

But wait. If we throttle the beam current way down, the display shows individual speckles where each electron hits. So the electrons are particles, right?

Now for the spooky part. If both slits are open to a throttled beam those singleton speckles don’t cluster behind the slits as you’d expect particles to do. A speckle may appear anywhere on the screen, even in an apparently blocked-off region. What’s more, when you send out many electrons one-by-one their individual hits cluster exactly where the bright stripes were when the beam was running full-on.

It’s as though each electron becomes a wave that goes through both slits, interferes with itself, and then goes back to being a particle!

By the way, this experiment isn’t a freak observation. It’s been repeated with the same results many times, not just with electrons but also with light (photons), atoms, and even massive molecules like buckyballs (fullerene spheres that contain 60 carbon atoms). In each case, the results indicate that the whatevers have a dual character — as a localized particle AND as a wave that reacts to the global environment.

Physicists have been arguing the “Which is it?” question ever since Louis-Victor-Pierre-Raymond, the 7th Duc de Broglie, raised it in his 1924 PhD Thesis (for which he received a Nobel Prize in 1929 — not bad for a beginner). He showed that any moving “particle” comes along with a “wave” whose peak-to-peak wavelength is inversely proportional to the particle’s mass times its velocity. The longer the wavelength, the less well you know where the thing is.

I just had to put numbers to de Broglie’s equation. With Newton in mind, I measured one of the apples in my kitchen. To scale everything, I assumed each object moved by one of its diameters per second. (OK, I cheated for the electron — modern physics says it’s just a point, so I used a not-really-valid classical calculation to get something to work with.)

“Particle”	Mass, kilograms	Diameter, meters	Wavelength, meters	Wavelength, diameters
Apple	0.2	0.07	7.1×10^-33	1.0×10^-31
Buckyball	1.2×10^-24	1.0×10^-9	0.083	8.3×10⁺⁷
Hydrogen atom	1.7×10^-27	1.0×10^-10	600	6.0×10⁺¹²
Electron	9.1×10^-31	3.0×10^-17	3.7×10⁺¹²	1.2×10⁺²⁹

That apple has a wave far smaller than any of its hydrogen atoms so I’ll have no trouble grabbing it for a bite. Anything tinier than a small virus is spread way out unless it’s moving pretty fast, as in a beam apparatus. For instance, an electron going at 1% of light-speed has a wavelength only a nanometer wide.

Different physicists have taken different positions on the “particle or wave?” question. Duc de Broglie claimed that both exist — particles are real and they travel where their waves tell them to. Bohr and Heisenberg went the opposite route, saying that the wave’s not real, it’s only a mathematical device for calculating relative probabilities for measuring this or that value. Furthermore, the particle doesn’t exist as such until a measurement determines its location or momentum. Einstein and Schrödinger liked particles. Feynman and Dirac just threw up their hands and calculated.

Which brings us to the other kind of quantum spookiness — “entanglement.” In fact, Einstein actually used the word spukhafte (German for “spooky”) in a discussion of the notion. He really didn’t like it and for good reason — entanglement rudely collides with his own Theory of Relativity. But that’s another story.

~~ Rich Olcott

Location, Location, Location

On May 9, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“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.

.
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

Particles and Poetry

On May 2, 2016November 28, 2025 By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

“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

Hard Science Ain't Hard

Month: May 2016

Gin And The Art of Quantum Mechanics

Oh, what an entangled wave we weave

Think globally, act locally. Electrons do.

Location, Location, Location

Particles and Poetry