Category: Physics: Quantum Mechanics

Flipping An Edge Case

On By rolcottIn Chemistry, Physics: Quantum Mechanics, UncategorizedLeave a comment

“Why’s the Ag box look weird in your chart, Susan?”

“That’s silver, Eddie. It’s an edge case. The pure metal’s diamagnetic. If you alloy silver with even a small amount of iron, the mixture is paramagnetic. How that works isn’t my field. Sy, it’s your turn to bet and explain.”

I match Eddie’s bet (the hand’s not over). “It’s magnetism and angular momentum and how atoms work, and there are parts I can’t explain. Even Feynman couldn’t explain some of it. Vinnie, what do you remember about electromagnetic waves?”

“Electric part pushes electrons up and down, magnetic part twists ’em sideways.”

“Good enough, but as Newton said, action begets reaction. Two centuries ago, Ørsted discovered that electrons moving along a wire create a magnetic field. Moving charges always do that. The effect doesn’t even depend on wires — auroras, fusion reactor and solar plasmas display all sorts of magnetic phenomena.”

“You said it’s about how atoms work.”

“Yes, I did. Atoms don’t follow Newton’s rules because electrons aren’t bouncing balls like those school‑book pictures show. An electron’s only a particle when it hits something and stops; otherwise it’s a wave. The moving wave carries charge so it generates a magnetic field proportional to the wave’s momentum. With me?”

“Keep going.”

“That picture’s fine for a wave traveling through space, but in an atom all the charge waves circle the nucleus. Linear momentum in open space becomes angular momentum around the core. If every wave in an atom went in the same direction it’d look like an electron donut generating a good strong dipolar magnetic field coming up through the hole.”

“You said ‘if’.”

“Yes, because they don’t do that. I’m way over‑simplifying here but you can think of the waves pairing up, two single‑electron waves going in opposite directions.”

“If they do that, the magnetism cancels.”

“Mm‑hm. Paired‑up configurations are almost always the energy‑preferred ones. An external magnetic field has trouble penetrating those structures. They push the field away so we classify them as diamagnetic. The gray elements in Susan’s chart are almost exclusively in paired‑up configurations, whether as pure elements or in compounds.”

“Okay, so what about all those paramagnetic elements?”

The angular portion of the lowest-energy spherical harmonic modes
Credit: Inigo.quilez, under CCA SA 3.0 license

“Here’s where we get into atom structure. An atom’s electron cloud is described by spherical harmonic modes we call orbitals, with different energy levels and different amounts of angular momentum — more complex shapes have more momentum. Any orbital hosting an unpaired charge has uncanceled angular momentum. Two kinds of angular momentum, actually — orbital momentum and spin momentum.”

“Wait, how can a wave spin?”

“Hard to visualize, right? Experiments show that an electron carries a dipolar magnetic field just like a spinning charge nubbin would. That’s the part that Feynman couldn’t explain without math. A charge wave with spin and orbital angular momentum is charge in motion; it generates a magnetic field just like current through a wire does. The math makes good predictions but it’s not something that everyday experience prepares us for. Anyway, the green and yellow‑orange‑ish elements feature unpaired electrons in high‑momentum orbitals buried deep in the atom’s charge cloud.”

“So what?”

“So when an external magnetic field comes along, the atom’s unpaired electrons join the party. They orient their fields parallel to the external field, in effect allowing it to penetrate. That qualifies the atom as paramagnetic. More unpaired electrons means stronger interaction, which is why iron goes beyond paramagnetic to ferromagnetic.”

“How does iron have so many?”

“Iron’s halfway across its row of ten transition metals—”

“I know where you’re going with this, Sy. It’ll help to say that these elements tend to lose their outer electrons. Scandium over on the left ionizes to Sc3+ and has zero d‑electrons. Then you add one electron in a d orbital for each move to the right.”

“Thanks, Susan. Count ’em off, Vinnie. Five steps over to iron, five added d‑electrons, all unpaired. Gadolinium, down in the lanthanides, beats that with seven half‑filled f‑orbitals. That’s where the strength in rare earth magnets arises.”

“So unpaired electrons from iron flip alloyed silver paramagnetic?”

“Vinnie wins this pot.”

~ Rich Olcott

A Carefully Plotted Tale

On By rolcottIn Cosmology, Physics: Quantum Mechanics, UncategorizedLeave a comment

<chirp, chirp> “Moire here.”

“Hello, Mr Moire. Remember me?”

“Yes, I do, Walt. I hope your people were satisfied with what you brought them from our last meeting.”

“They were, which is why I’m calling. Buy you pizza at Eddie’s, fifteen minutes?”

“Make it twenty.”

We’re at the rear‑corner table, Walt facing both doors, naturally. “So, what’s the mysterious question this time?”

“Word on the street is that the CPT Law’s being violated. We want to know who’s involved, and what’s their connection with ChatGPT.”

Good thing I’ve just bit into my pizza so I can muffle my chuckle in my chewing. “What do you know about anti‑matter?”

“Inside‑out atoms — protons outside whizzing around electrons in the nucleus.”

“Common misconception. One proton has the mass of 1800 electrons. An atom built as you described would be unstable — the thing would fly apart. You’ve got anti‑matter’s charges arranged right but not the particles. Anti‑matter has negative anti‑protons in the nucleus and positrons, positive electrons, on the outside.”

<writing rapidly in his notebook> “You can do that? Just flip the sign on a particle?”

“No, positrons and such are respectable particles in their own right, distinct from their anti‑partners. Electric charge comes built into the identity. What’s important is, an anti‑atom behaves exactly like a normal atom does. Maxwell’s Equations and everything derived from them, including quantum mechanics, work equally well for either charge structure.”

“There’s a bit of Zen there — change but no‑change.”

“Nice. Physicists call that sort of thing a symmetry. In this case it’s charge symmetry, often written as C.”

“The C in CPT?”

“Exactly.”

“What about the P and T?”

“When someone says something is symmetrical, what do you think of first?”

“Right side’s a reflection of left side. Symmetrical faces look better but they’re usually less memorable.”

“Interesting choice of example. Anyway, reflection symmetry is important in common physical systems.”

“Classical Greek and Cambodian architecture; the Baroque aesthetic without the decorative frills.”

“I suppose so. Anyway, we call reflection symmetry Parity, or P for short.”

“And T?”

“Time.”

“Time’s not symmetrical. It’s always past‑to‑future.”

“Maybe, maybe not. In all our physical laws that deal with a small number of particles, you can replace t for time with –t and get the same results except for maybe a flipped sign. Newton’s Laws would run the Solar System in reverse just as well as they do forward.”

“But … Ah, ‘small number of particles,’ that’s your out. If your system has a large number of particles, you’re in chaos territory where randomness and entropy have to increase. Entropy increase is the arrow for one‑way time.”

“Good quote.”

“I’ve been in some interesting conversations. You’re not my only Physics source. So CPT is about Charge AND Parity AND Time symmetries. But you can’t simply add them together.”

“You multiply them. Technically, each of them is represented by a mathematical operator—”

“Step away from the technically.”

“Understood. This’ll be simpler. If a system’s atoms have positive nuclei, set C=1, otherwise set C=–1. If the system’s naturally‑driven motion is counterclockwise set P=1, otherwise P=–1. If time is increasing, set T=1, otherwise set T=–1. Okay?”

“Go on.”

“You can summarize any system’s CPT state by multiplying the prevailing symmetry values. The product will be either +1 or –1. The CPT Law says that in any universe where quantum mechanics and relativity work, one CPT state must hold universe‑wide.”

“Make it real for me.”

“You know the Right-hand Rule for electromagnetism?”

“Grab the wire with your right hand, thumb pointing along the current. Your fingers wrap in the direction of the spiraling magnetic field.”

“Perfect. Suppose C*P*T=+1 for this case. Now reverse the charge, making C=–1. What happens?”

“Ssss… The magnetic spin flips orientation. That’s a reflection operation so P=–1. The C*P*T calculation is (+1)*(–1)*(–1)=+1, no change.”

“The CPT Law in action. The CPT violation you’ve heard about is only observed in rare weak‑force‑mediated radioactive decays of a carefully prepared nucleus. That was a 1956 Nobel‑winning discovery, though the right person didn’t win it.”

“1956. Decades before A.I.”

“Yup, ChatGPT is off the hook. For that.”

“Bye.”

“Don’t mention it.”

~ Rich Olcott

  • Thanks to Caitlin, the hand model.

It’s in The Book

On By rolcottIn Chemistry, Physics: Entropy and Thermodynamics, Physics: Quantum Mechanics, UncategorizedLeave a comment

A young man’s knock, eager yet a bit hesitant. “Door’s open, Jeremy, c’mon in.”

“Hi, Mr Moire, I’ve got something to show you. It’s from my acheii, my grandfather. He said he didn’t need it any more now he’s retired so he gave it to me. What do you think?”

“Wow, the CRC Handbook of Chemistry And Physics, in the old format, not the 8½×11″ monster. An achievement award, too — my congratulations to your grandfather. Let’s see … over 3000 pages, and that real thin paper you can read through. It’s still got the math tables in front — they moved those to an Appendix by the time I bought my copy. Oooh yeah, lots of data in here, probably represents millions of grad student lab hours. Tech staff, too. And then their bosses spent time checking the work before publishing.”

“Acheii said I’d have to learn a lot before I could use it properly. I see lots of words in there I don’t recognize.” <opens book to a random page> “See, five- and six‑figure values for, what’re Specific Heat and Enthalpy?”

“Your grandfather’s absolutely correct. Much of the data’s extremely specialized. Most techs, including me, have a few personal‑favorite sections they use a lot, never touch the rest of the book. These particular pages, for instance, would be gold for a someone who designs or operates steam‑driven equipment.”

“But what do these numbers mean?”

“Specific Heat is the amount of heat energy you need to put into a certain mass of something in order to raise its temperature by a certain amount. In the early days the Brits, the Scots really, defined the British Thermal Unit as the amount of energy it took to raise the temperature of one pound of liquid water by one degree Fahrenheit. You’d calculate a fuel purchase according to how many BTUs you’d need. Science work these days is metric so these pages tabulate Specific Heat for a substance in joules per gram per °C. Tech in the field moves slow so BTUs are still popular inside the USA and outside the lab.”

“But these tables show different numbers for different temperatures and they’re all for water. Why water? Why isn’t the Specific Heat the same number for every temperature?”

“Water’s important because most power systems use steam or liquid water as the working fluid or coolant. Explaining why heat capacity varies with temperature was one of the triumphs of 19th‑century science. Turns out it’s all about how atomic motion but atoms were a controversial topic at the time. Ostwald, for instance—”

“Who?”

“Wilhelm Ostwald, one of science’s Big Names in the late 1800s. Chemistry back then was mostly about natural product analysis and seeing what reacted with what. Ostwald put his resources into studying chemical processes themselves, things like crystallization and catalysis. He’s regarded as the founder of Physical Chemistry. Even though he invented the mole he steadfastly maintained that atoms and molecules were nothing more than diffraction‑generated illusions. He liked a different theory but that one didn’t work out.”

“Too bad for him.”

“Oh, he won the first Nobel Prize in Chemistry so no problem. Anyway, back to Specific Heat. In terms of its molecules, how do you raise something’s temperature?”

“Um, temperature’s average kinetic energy, so I’d just make the molecules move faster.”

“Well said, except in the quantum world there’s another option. The molecules can’t just waggle any which way. There are rules. Different molecules do different waggles. Some kinds of motion take more energy to excite than others do. Rule 1 is that the high‑energy waggles don’t get to play until the low‑energy ones are engaged. Raising the temperature is a matter of activating more of the high‑energy waggles. Make sense?”

“Like electron shells in an atom, right? Filling the lowest‑energy shells first unless a photon supplies more energy?”

“Exactly, except we’re talking atoms moving within a molecule. Smaller energies, by a factor of 100 or more. My point is, the heat capacity of a substance depends on which waggles activate as the temperature rises. We didn’t understand heat capacity until we applied quantum thinking to the waggles.”

“What about ‘Enthalpy’ then?”

Crystal photo by JJ Harrison

~ Rich Olcott

The Not-so-dangerous Banana

On By rolcottIn Astronomy: Techniques, Physics: Quantum Mechanics, UncategorizedLeave a comment

“Y’know, Cathleen, both our ladders boil down to time. Your Astronomy ladder connects objects at different times in the history of the Universe. My Geology ladder looks back into the Solar System’s history.”

“As an astronomer I normally think of parsecs or lightyear distances but you have a point, Kareem. Edwin Hubble linked astronomical space with time. Come to think of it, my cosmologist colleagues work almost exclusively in the time domain, like ‘T=0 plus a few lumptiseconds.’ Billions of years down to that teeny time interval — how does your time ladder compare?”

“Lumptiseconds out to a hundred trillion times the age of the Universe. I win.”

“C’mon, Kareem.”

“No, really, Sy. My ladder uses isotopes. Every carbon atom has 6 protons in the nucleus, right? Carbon‑12 adds 6 neutrons and it’s stable but another isotope, carbon‑14, has 8 neutrons. It’s radioactive — spits out an electron and becomes stable nitrogen‑14 with 7 and 7. Really heavy isotopes like uranium‑238 spit out alpha particles.”

“Wait, if carbon‑14 spits out an electron doesn’t that make it a carbon ion?”

“Uh‑uh, Cathleen, the electron comes out of the nucleus, not the electron cloud. It’s got a hundred thousand times more energy than a chemical kick could give it. Sy could explain—”

“Nice try, Kareem, this is your geologic time story. Let’s stay with that.”

“If I must. So, the stable isotopes last forever, pretty much, but the radioactive ones are ticking bombs with random detonation times.”

“What’s doing the ticking? Surely there’s no springs or pendulums in there.”

“Quantum, Cathleen. Sy’s trying to stay out of this so I’ll give you my outsider answer. I picture every kind of subatomic particle constantly trying to leave every nucleus, butting their little heads bazillions of times a second against walls set up by the weak and strong nuclear forces. Nearly every try is a bounce‑back, but one success is enough to break the nucleus. Every isotope has its own personal set of parameters for each kind of particle — wall height, wall thickness, something like an internal temperature ruling how hard the particles hit the walls. The ticking is those head‑butts; the randomness comes from quantum’s goofy rules somehow. How’s that, Sy?”

Chart by Sjlegg, licensed under CC S-A 3.0 unported

“Good enough for jazz, Kareem. Carry on.”

“Right. So every kind of radioisotope is characterized by what kinds of particle it emits, how much energy each kind has after busting through a wall, and how often that happens in a given sample size. And the isotope’s chemistry, of course, which is the same as every other isotope that has the same number of protons. The general rule is that the stable isotopes have maybe a few more neutrons than protons but nearly every element has some unstable isotopes. The ones with too many neutrons, like carbon‑14, emit electrons as beta particles. They go up a square in the Periodic Table. Too few and they drop down by emitting a positron.”

“All those radioactive stand‑ins for normal atoms. Sounds ghastly. Why are we still here and not all burnt up?”

“First, when one of these atoms decays by itself it’s a lot of energy for that one atom, but the energy spreads out as heat across many atoms. Unless a bunch of atoms crumble at about the same time, there’s only a tiny bit of general heating. The major biological danger from radioactivity comes from spit‑out particles breaking protein or DNA molecules.”

“Mutated, not burnt.”

“Mm‑hm. Second, the radioactives are generally rare relative to their stable siblings. In many cases that’s because the bad guys, like aluminum‑26, have had time to decay to near‑zero. That banana you’re eating has about half a gram of potassium atoms but only 0.012% are unstable potassium‑40. Third, an isotope with a long half‑life doesn’t lose many atoms per unit time. A kilogram of tellurium‑128, for instance, loses 2000 atoms per year. The potassium‑40 in your banana has a half‑life of nearly 2 million years. Overall, it releases only about 1300 beta particles per second producing less than a nanowatt of heat‑you‑up power. Not to worry.”

“Two million years? How do you measure something that slow?”

~~ Rich Olcott

Why Physics Is Complex

On By rolcottIn Mathematics, Physics: Light and Relativity, Physics: Quantum Mechanics, UncategorizedLeave a comment

“I guess I’m not surprised, Sy.”

“At what, Vinnie?”

“That quantum uses these imaginary numbers — sorry, you’d prefer we call them i‑numbers.”

“Makes no difference to me, Vinnie. Descartes’ pejorative term has been around for three centuries so that’s what the literature uses. It’s just that most people pick up the basic idea more quickly without the woo baggage that the real/imaginary nomenclature carries along. So, yes, it’s true that both i‑numbers and quantum mechanics appear mystical, but really quantum mechanics is the weird one. And relativity.”

“Wait, relativity too? That’s hard to imagine, HAW!”

“Were you in the room for Jim’s Open Mic session where he talked about Minkowski’s geometry?”

“Nope, missed that.”

“Ah, okay. Do you remember the formula for the diagonal of a rectangle?”

“That’d be the hypotenuse formula, c²=a²+b². Told you I was good at Geometry.”

“Let’s use ‘d‘ for distance, because we’re going to need ‘c‘ for the speed of light. While we’re at it, let’s replace your ‘a and ‘b‘ with ‘x‘ and ‘y,’ okay?”

“Sure, why not?”

<casting image onto office monitor> “So the formula for the body diagonal of this box is…”

“Umm … That blue line across the bottom’s still √(x²+y²) and it’s part of another right triangle. d‘s gotta be the square root of x²+y²+z².”

“Great. Now for a fourth dimension, time, so call it ‘t.’ Say we’re going for light’s path between A at one moment and B some time t later.”

“Easy. Square root of x²+y²+z²+t².”

“That’s almost a good answer.”

“Almost?”

“The x, y and z are distance but t is a duration. The units are different so you can’t just add the numbers together. It’d be like adding apples to bicycles.”

“Distance is time times speed, so we multiply time by lightspeed to make distance traveled. The formula’s x²+y²+z²+(ct)². Better?”

“In Euclid’s or Newton’s world that’d be just fine. Not so much in our Universe where Einstein’s General Relativity sets the rules. Einstein or Minkowski, no‑one knows which one, realized that time is fundamentally perpendicular to space so it works by i‑numbers. You need to multiply t by ic.”

“But i²=–1 so that makes the formula x²+y²+z²–(ct)².”

“Which is Minkowski’s ‘interval‘ between an event at A and another event at B. Can’t do relativity work without using intervals and complex numbers.”

“Well that’s nice but we started talking about quantum. Where do your i‑numbers come into play there?”

“It goes back to the wave equation— no, I know you hate equations. Visualize an ocean wave and think about describing its surface curvature.”

Adapted from “The Great Wave off Kanagawa”
by Katsushika Hokusai, in public domain

“Curvature?”

“How abruptly the slope changes. If the surface is flat the slope is zero everywhere and the curvature is zero. Up near the peak the slope changes drastically within a short distance and we say the surface is highly curved. With me?”

“So far.”

“Good. Now, visualize the wave moving past you at some convenient speed. Does it make sense that the slope change per unit time is proportional to the curvature?”

“The pointier the wave segment, the faster its slope has to change. Yeah, makes sense.”

“Which is what the classical wave equation says — ‘time‑change is proportional to space‑change’. The quantum wave equation is fundamental to QM and has exactly the same form, except there’s an i in the proportionality constant and that changes how the waves work.” <casting a video> “The equation’s general solution has a complex exponential factor eix. At any point its value is a single complex number with two components. From the x‑direction, the circle looks like a sine wave. From the i‑direction it also looks like a sine wave, but out of phase with the x‑wave, okay?”

A traveling wave packet.
Image by “Xcodexif” under CCS-A 4.0 license

“Out of phase?”

“When one wave peaks, the other’s at zero and vice‑versa. The point is, rotation’s built into the quantum waves because of that i‑component.” <another video> “Here’s a lovely example — that black dot emits a photon that twists and releases the electromagnetic field as it moves along.”

~ Rich Olcott

Marconi Would Be Proud

On By rolcottIn General: Fun Stuff, Physics: Quantum Mechanics, UncategorizedLeave a comment

A warmish Spring day.  I’m under a shady tree by the lake, waiting for the eclipse and doing some math on Old Reliable.  Suddenly there’s a text‑message window on its screen.  The header bar says 710‑555‑1701 . Old Reliable has never held a messaging app, that’s not what I use it for, but the set-up is familiar. I type in, Hello?

Hello, Mr Moire. Remember me?

Of course I do.  That sultry knowing stare, those pointed ears.  Hello, Lieutenant Baird.  It’s been a year.  What can I do for you?

Not Lieutenant any more, I’m back up to Commander, Provisional.

Congratulations. Did you invent something again?

Yes, but I can’t discuss it on this channel. I owe you for the promotion. I got the idea from one of your Crazy Theories posts. You and your friends have no clue but you come up with interesting stuff anyway.

You’re welcome, I suppose. Mind you, your science is four centuries ahead of ours but we do the best we can.

I know that, Mr Moire. Which is why I’m sending you this private chuckle.

Private like with Ralphie’s anti‑gravity gadget? I suggested he add another monitoring device in between two of his components. That changed the configuration you warned me about. He’s still with us, no anti‑gravity, but now he blames me.

Good ploy. Sorry about the blaming. Now it’s your guy Vinnie who’s getting close to something.

Vinnie? He’s not the inventor type, except for those maps he’s done with his buddy Larry. What’s he hit on?

His speculation from your Quantum Field Theory discussion that entanglement is somehow involved with ripples in a QFT field, ripples that are too weak to register as a particle peak. He’s completely backwards on entanglement, but those ripples—

Wait, what’s that about entanglement?

Entanglement is the normal state for quantized particles. Our 24th‑Century science says every real and virtual particle in the Universe is entangled with every other particle that shares the same fields. It’s an all‑embracing quantum state. Forget your reductionist 20th‑Century‑style quantum states, this is something … different. Your Hugh Everett and his mentor John Archibald Wheeler had an inking of that fact a century before your time, though of course they didn’t properly understand the implications and drew a ridiculous conclusion. Anyway, when your experimenting physicists say they’ve created an entangled particle pair, they’ve simply extracted two particles from the common state. When they claim to transmit one of the particles somewhere they’re really damping out the local field peak linked to their particle’s anti‑particle’s anti‑peak at the distant location and that puts an anti‑anti‑particle‑particle peak there. Naturally, that happens nearly instantaneously.

I don’t follow the anti‑particle‑anti‑peak part. Or why it’s naturally instantaneous.

I didn’t expect you to or else I wouldn’t have told you about it. The Prime Directive, you know. Which is why the chuckle has to be private, understand?

I won’t tell. I live in “the city that knows how to keep its secrets,” remember?

Wouldn’t do you any good if you did tell and besides, Vinnie wouldn’t think it’s funny. Here’s the thing. As Vinnie guessed, there are indeed sub‑threshold ripples in all of the fundamental fields that support subatomic particles and the forces that work between them. And no, I won’t tell you how many fields, your Standard Model has quite enough complexity to <heh> perturb your physicists. A couple hundred years in your future, humanity’s going to learn how to manipulate the quarks that inhabit the protons and neutrons that make up a certain kind of atom. You’ll jiggle their fields and that’ll jiggle other fields. Pick the right fields and you get ripples that travel far away in space but very little in time, almost horizontal in Minkowski space. It won’t take long for you to start exploiting some of your purposely jiggled fields for communication purposes. Guess what a lovely anachronism you’ll use to name that capability.

‘Jiggled fields’ sounds like communications tech we use today based on the electromagnetic field — light waves traveling through glass fibers, microwave relays for voice and data—

You’re getting there. Go for the next longer wavelength range.

Radio? You’ll call it radio?

Subspace radio. Isn’t that wonderful?

~~ Rich Olcott

Fields of Dreams

On By rolcottIn Cosmology, Physics: Quantum Mechanics, Physics: Waves, Uncategorized1 Comment

Vinnie takes a slug of his coffee. “So the gravitational field carries the gravitational wave. I suppose Einstein would blame sound waves on some kind of field?”

I take a slug of mine. “Mm-hm. We techies call it a pressure field. Can’t do solar physics without it. The weather maps you use when plotting up a flight plan — they lay three fields on top of the geography.”

“Lessee — temp, wind … and barometer reading. In the old days I’d use that one to calibrate my altimeter. You say those are fields?”

“In general, if a variable has a value at every point in the region of interest, the complete set of values is a field. Temperature and pressure are the simplest type. Their values are just numbers. Each point in a wind field has both speed and direction, two numbers treated as a single value, so you’ve got a field of vector values.”

“Oh, I know vectors, Sy. I’m a pilot, remember? So you’re saying instead of looking at molecules back‑and‑forthing to make sound waves we step back and look at just the pressure no matter the molecules. Makes things simpler, I can see that. Okay, how about the idea those other guys had?”

“Hm? Oh, the other wave carrier idea. Einstein’s gravitational waves are just fine, but the Quantum Field Theorists added a collection of other fields, one for each of the 17 boxes in the Standard Model.”

“Boxes?”

<displaying Old Reliable’s screen> “Here’s the usual graphic. It’s like the chemist’s Periodic Table but goes way below the atomic level. There’s a box each for six kinds of quarks; another half‑dozen for electrons, neutrinos and their kin; four more boxes for mediating electromagnetism and the weak and strong nuclear forces. Finally and at last there’s a box for the famous Higgs boson which isn’t about gravity despite what the pop‑sci press says.”

“What’s in the boxes?”

“Each box holds a list of properties — rest mass, spin, different kinds of charge, and a batch of rules for how to interact with the other thingies.”

“Thingies, Sy? I wouldn’t expect that word from you.”

“I would have said ‘particle‘ but that would violate QFT’s tenets despite the graphic’s headline.”

“Tenets? That word sounds more like you. What’s the problem?”

“That the word ‘particle‘ as we normally use it doesn’t really apply at the quantum field level. Each box names a distinct field that spreads its values and waves all across the Universe. There’s an electron field, a photon field, an up‑quark field, a down‑quark field, and so on. According to QFT, what we’d call a particle is nothing more than a localized peak in its underlying field. Where you find a peak you’ll find all the properties listed in its box. Wherever the field’s value is below its threshold, you find none of them.”

“All or none, huh? I guess that’s where quantum comes in. Wait, that means there could be gazillions of one of them popping up wherever, like maybe a big lump of one kind all right next to each other.”

“No, the rules prevent that. Quarks, for instance, only travel in twos or threes of assorted kinds. The whole job of the gluons is to enforce QFT rules so that, for instance, two up‑quarks and a down‑quark make a proton but only if they have different color charges.”

“Wait, color charge?”

“Not real colors, just quantum values that could as easily have been labeled 1, 2, 3 or A, B, C. There’s also an anti- for each value so the physicists could have used ±1, ±2, ±3, but they didn’t, they used ‘red’ and ‘anti‑red’ and so forth. And ‘color charge‘ is a different property from electric charge. Gluons only interact with color charge, photons only interact with electric charge. The rules are complicated.”

“You said ‘waves.’ Each of these fields can have waves like gravity waves?”

“Absolutely. We can’t draw good pictures of them because they’re 3‑dimensional. And they’re constantly in motion, of course.”

“How fast can those waves travel?”

“The particles are limited to lightspeed or slower; the waves, who knows?”

“Ripples zipping around underneath the quantum threshold could account for entanglement, ya’ know?”

“Maybe.”

~~ Rich Olcott

Shapes And Numbers

On By rolcottIn Mathematics: Spherical Harmonics, Physics: Quantum Mechanics, UncategorizedLeave a comment

I’m nursing my usual mug of eye‑opener in Cal’s Coffee Shop when astronomer Cathleen and chemist Susan chatter in and head for my table. Susan fires the first volley.

“Sy, those spherical harmonics you’ve posted don’t look anything like the atomic orbitals in my Chem text. Shouldn’t they?”
 ”How do you add and multiply shapes together?”
  ”What does the result even mean?”
   ”And what was that about solar seismology?”

The angular portion of the lowest‑energy spherical harmonics.
Adapted from work by Inigo.quilez, under CCA SA 3.0 license

“Whoa, have you guys taken interrogation lessons from Mr Feder? One at a time, please. Let’s start with the basics.” <sketching on a paper napkin> “For example, the J2 zonal harmonic depends only on the latitude, not on the longitude or the distance from the center, so whatever it does encircles the axis. Starting at the north pole and swinging down to the south pole, the blue line shows how J2 varies from 1.0 down to some negative decimal. At any latitude, whatever else is going on will be multiplied by the local value of J2.”

“The maximum is 1.0, huh? Something multiplied by a number less than 1 becomes even smaller. But what happens where J2 is zero? Or goes negative?”

“Wherever Jn‘s zero you’re multiplying by zero which makes that location a node. Furthermore, the zero extends along its latitude all around the sphere so the node’s a ring. J2‘s negative value range does just what you’d expect it to — multiply by the magnitude but flip the product’s sign. No real problem with that, but can you see the problem in drawing a polar graph of it?”

“Sure. The radius in a polar graph starts from zero at the center. A negative radius wouldn’t make sense mixed in with positives in the opposite direction.”

“Well it can, Cathleen, but you need to label it properly, make the negative region a different color or something. There are other ways to handle the problem. The most common is to square everything.” <another paper napkin> “That makes all the values positive.”

“But squaring a magnitude less than 1 makes it an even smaller multiplier.”

“That does distort the shapes a bit but it has absolutely no effect on where the nodes are. ’Nothin’ times nothin’ is nothin’,’ like the song says. Many of the Chem text orbital illustrations I’ve seen emphasize the peaks and nodes. That’s exactly what you’d get from a square‑everything approach. Makes sense in a quantum context, because the squared functions model electron charge distributions.”

“Thanks for the nod, Sy. We chemists care about charge peaks and nodes around atoms because they control molecular structures. Chemical bonds and reactions tend to localize near those places.”

“I aim for fairness, Susan. There is another way to handle a negative radius but it needs more context to look reasonable. Meanwhile, we’ve established that at any given latitude each Jn is just a number so let’s look at longitudes.” <a third paper napkin> “Here’s the first two sectorial harmonics plotted out in linear coordinates.”

“Looks familiar.”

“Mm‑hm. Similar principles, except that we’re looking at a full circle and the value at 360° must match the value at 0°. That’s why Cm always has an even node count — with an odd number you’d have -1 facing +1 and that’s not stable. In polar coordinates,” <the fourth paper napkin> “it’s like you’re looking down at the north pole. C0 says ‘no directional dependence,’ but C1 plays favorites. By the way, see how C1‘s negative radii in the 90°‑270° range flip direction to cover up the positives?”

“Ah, I see where you’re going, Sy. Each of these harmonics has a numeric value at each angle around the center. You’re going to tell us that we can multiply the shapes by multiplying their values point by point, one for each latitude for a J and each longitude for a C.”

“You’re way ahead of me as usual, Cathleen. You with us, Susan?”

“Oh, yes. In my head I multiplied your J2 by C0 and got a pz orbital.”

“I’m impressed.”
 ”Me, too.”

“Oh, I didn’t do it numerically. I just followed the nodes. J2 has two latitude nodes, C0 has no longitude nodes. There it is, easy‑peasy.”

~~ Rich Olcott

Little Strings And Big Ones

On By rolcottIn Cosmology, Physics: Quantum Mechanics, Physics: Waves, UncategorizedLeave a comment

It’ll be another hot day so I’m walking the park early. No geese in the lake — they’ve either flown north or else they’re attacking a farmer’s alfalfa field. A familiar voice shatters the quiet. “Wait up, Moire, I got questions.”

“Good morning, Mr Feder. First question, but please pick up your pace, I want to get back to the air conditioning.”

“I thought string theory was about little teeny stuff but this guy said about cosmic strings. How can they be teeny and cosmic?”

“They can’t. Totally different things, probably. Next question.”

“C’mon, Moire, that wasn’t even an answer, just opened up a bunch more questions.”

“It’s a tangled path but the track mostly started in the late 18th Century. Joseph Fourier derived the equation for how heat progresses along a uniform metal bar. Turned out the equation’s general solution was the sum of an infinite series of sine waves.”

“Sign waves? Like a protest rally?”

“Haha. No, s‑i‑n‑e, a mathematical function where something regularly and smoothly deviates about some central value. Anyhow, mathematicians soon realized that Fourier’s cute trick for his heat equation could be applied to equations for everything from sound waves to signal processing to pretty much all of Physics. Economics, even. Any time you use the word ‘frequency‘ you owe something to Fourier.”

“If he ain’t got it in writing from the Patent Office, I ain’t paying nothing.”

“It’s not the kind of thing you can patent, and besides, he lived in France and died almost two centuries ago. Be generous with your gratitude, at least. Let’s move on. Fourier’s Big Idea was already <ahem> in the air early in the 20th Century when Bohr and the Physics gang were looking at atoms. No surprise, they extended the notion to describe how electronic charge worked in there.”

“I’m waiting for the strings.”

“The key is that an atom’s a confined system like a guitar string that only vibrates between the bridge and whatever fret you’re pressing on. Sound waves traveling in open space can have any wavelength, but if you pluck a confined guitar string the only wavelengths you can excite are whole number multiples of its active length. No funny fractions like π/73 of the length no matter how hard or soft you pluck the string. Atoms work the same way — charge is confined around the nucleus so only certain wave sizes and shapes are allowed.”

“You said ‘strings.’ We getting somewhere finally?”

“Closing in on it. String theory strings aren’t just teeny. If your body were suddenly made as large as the Observable Universe, string theory is about what might happen inside a box a billion times smaller than your size now.”

“Really tight quarters, got it, so only certain vibrations are allowed.”

“Mm-hm, except it’s not really vibration, it would be something that acts mathematically like vibration. Go back to your guitar string. Plucking gives it up‑down motion, strumming moves it side‑to‑side. Two degrees of freedom. The math says whatever’s going on in a string theory box needs 8 or 11 or maybe 25 degrees of freedom, depending on the theory. At the box‑size scale if there’s structure at all it looks nothing like a string.”

“Then how about the big cosmic strings? What’s confining them?”

“Nothing, and I mean that literally. If they exist they’re bounded by different flavors of empty space. It goes back to what we think happened right after the Big Bang during rapid space expansion. Whatever forces drove the process were probably limited by lightspeed. Local acceleration in one region wouldn’t immediately affect events in regions lightyears away. Nearly adjacent parts of the Universe could have been evolving at very different rates. Have you ever watched the whirlpools that form when a fast‑moving stream of water meets a slower‑moving one?”

“Fort Lee had a storm‑sewer pipe that let into the Hudson River. You got crazy whirlpools there after a hard rain.”

“Whirlpools are one kind of topological defect. They die away in water because friction dissipates the angular momentum. Hiding behind a whole stack of ifs and maybes, some theorists think collisions between differently‑evolving spacetime structures might generate long‑lived defects like cosmic strings or sheets.”

~~ Rich Olcott

Noodles or A Sandwich?

On By rolcottIn Cosmology, Physics: Quantum Mechanics, UncategorizedLeave a comment

“Wait, Sy, your anti-Universe idea says there are exactly two um, sub‑Universes. Even the word ‘multiverse‘ suggests more than that.”

“You’re right, Susan, most of the multiverse proposals go to the other extreme. Maybe the most extreme version grew in reaction to one popular interpretation of quantum theory. Do you know about the ‘Many Worlds‘ notion?”

“Many Worlds? Is that the one about when I decide between noodles for lunch or a sandwich, the Universe splits and there’s one of me enjoying each one?”

“That’s the popular idea. The physics idea is way smaller, far bigger and even harder to swallow. Physicists have been arguing about it for a half‑century.”

“Come again? Smaller AND bigger?”

“Smaller because it’s a quantum‑based idea about microscopic phenomena. Doesn’t say anything about things big enough to touch. Remember how quantum calculations predict statistics, not exact values? They can’t give you anything but averages and spreads. Einstein and Bohr had a couple of marquee debates about that back in the 1930s. Bohr maintained that our only path to understanding observations at the micro‑scale was to accept that events there are random and there’s no point discussing anything deeper than statistics. Einstein’s position was that the very fact that we’re successfully using an average‑based strategy says that there must be finer‑grained phenomena to average over. He called it ‘the underlying reality.’ The string theory folks have chased that possibility all the way down to the Planck‑length scale. They’ve found lots of lovely math but not much else. Hugh Everett had a different concept.”

“With that build‑up, it’d better have something to do with Many Worlds.”

“Oh, it does. Pieces of the idea have been lying around for centuries, but Everett pulled them all together and dressed them up in a quantum suit. Put simply, in his PhD thesis he showed how QM’s statistics can result from averaging over Universes. Well, one Universe per observation, but you experience a sequence of Universes and that’s what you average over.”

“How can you show something like that?”

“By going down the rabbit hole step by step and staying strictly within the formal QM framework. First step was to abstractify the operation of observing. He said it’s a matter of two separate systems, an observer A and a subject B. The A could be a person or electronics or whatever. What’s important is that A has the ability to assess and record B‘s states and how they change. Given all that, the next step is to say that both A and B are quantized, in the sense that each has a quantum state.”

“Wait, EACH has a quantum state? Even if A is a human or a massive NMR machine?”

“That’s one of the hard‑to‑swallows, but formally speaking he’s okay. If a micro‑system can have a quantum state then so can a macro‑system made up of micro‑systems. You just multiply the micro‑states together to get the macro‑state. Which gets us to the next step — when A interrogates B, the two become entangled. We then can only talk about the combined quantum state of the A+B system. Everett referred to an Einstein quote when he wrote that a mouse doesn’t change the Moon by looking at it, but the Moon changes the mouse. The next step’s a doozy so take a deep breath.”

“Ready, I suppose.”

“B could have been in any of its quantum states, suppose it’s . After the observation, A+B must be an entangled mixture of whatever A was, combined with each of B‘s possible final states. Suppose B might switch to . Now we can have A+B(#42), separate from a persisting A+B(#10), plus many other possibles. As time goes by, A+B(#42) moves along its worldline independent of whatever happens to A+B(#10).”

“If they’re independent than each is in its own Universe. That’s the Many Worlds thing.”

“Now consider just how many worlds. We’re talking every potential observing macro‑system of any size, entangled with all possible quantum states of every existing micro‑system anywhere in our Observable Universe. We’re a long way from your noodles or sandwich decision.”

“An infinity of infinities.”

“Each in its own massive world.”

“Hard to swallow.”

~~ Rich Olcott