Tag: Quantum

Mushy stuff

On By rolcottIn Crazy Theories, Physics: Black Holes and Relativity, UncategorizedLeave a comment

“Amanda! Amanda! Amanda!”

“All right, everyone, settle down for our final Crazy Theorist. Jim, you’re up.”

“Thanks, Cathleen. To be honest I’m a little uncomfortable because what I’ve prepared looks like a follow-on to Newt’s idea but we didn’t plan it that way. This is about something I’ve been puzzling over. Like Newt said, black holes have mass, which is what everyone pays attention to, and charge, which is mostly unimportant, and spin. Spin’s what I’ve been pondering. We’ve all got this picture of a perfect black sphere, so how do we know it’s spinning?”

Voice from the back of the room — “Maybe it’s got lumps or something on it.”

“Nope. The No-hair Theorem says the event horizon is mathematically smooth, no distinguishing marks or tattoos. Question, Jeremy?”

“Yessir. Suppose an asteroid or something falls in. Time dilation makes it look like it’s going slower and slower as it gets close to the event horizon, right? Wouldn’t the stuck asteroid be a marker to track the black hole’s rotation?”

“Excellent question.” <Several of Jeremy’s groupies go, “Oooh.”> “Two things to pay attention to here. First, if we can see the asteroid, it’s not yet inside the horizon so it wouldn’t be a direct marker. Beyond that, the hole’s rotation drags nearby spacetime around with it in the ergosphere, that pumpkin‑shaped region surrounding the event horizon except at the rotational poles. As soon as the asteroid penetrates the ergosphere it gets dragged along. From our perspective the asteroid spirals in instead of dropping straight. What with time dilation, if the hole’s spinning fast enough we could even see multiple images of the same asteroid at different levels approaching the horizon.”

Jeremy and all his groupies go, “Oooh.”

“Anyhow, astronomical observation has given us lots of evidence that black holes do spin. I’ve been pondering what’s spinning in there. Most people seem to think that once an object crosses the event horizon it becomes quantum mush. There’d be this great mass of mush spinning like a ball. In fact, that was Schwarzchild’s model for his non-rotating black hole — a simple sphere of incompressible fluid that has the same density throughout, even at the central singularity.”

VBOR — “Boring!”

“Well yeah, but it might be correct, especially if spaghettification and the Firewall act to grind everything down to subatomic particles on the way in. But I got a different idea when I started thinking about what happened to those two black holes that LIGO heard collide in 2015. It just didn’t seem reasonable that both of those objects, each dozens of solar masses in size, would get mushed in the few seconds it took to collide. Question, Vinnie?”

“Yeah, nice talk so far. Hey, Sy and me, we talked a while ago about you can’t have a black hole inside another black hole, right, Sy?”

“That’s not quite what I said, Vinnie. What I proved was that after two black holes collide they can’t both still be black holes inside the big one. That’s different and I don’t think that’s where Jim’s going with this.”

“Right, Mr Moire. I’m not claiming that our two colliders retain their black hole identities. My crazy theory is that each one persists as a high‑density nubbin in an ocean of mush and the nubbins continue to orbit in there as gravity propels them towards the singularity.”

VBOR —”Orbit? Like they just keep that dance going after the collision?”

“Sure. What we can see of their collision is an interaction between the two event horizons and all the external structures. From the outside, we’d see a large part of each object’s mass eternally inbound, locked into the time dilation just above the joined horizon. From the infalling mass perspective, though, the nubbins are still far apart. They collide farther in and farther into the future. The event horizon collision is in their past, and each nubbin still has a lot of angular momentum to stir into the mush. Spin is stirred-up mush.”

Cathleen’s back at the mic. “Well, there you have it. Amanda’s male-pattern baldness theory, Newt’s hyper‑planetary gear, Kareem’s purple snowball or Jim’s mush. Who wins the Ceremonial Broom?”

The claque responds — “Amanda! Amanda! Amanda!”

~ Rich Olcott

Virial Yang And Yin

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

“But Mr Moire, how does the Virial Equation even work?”

“Sometimes it doesn’t, Jeremy. There’s an ‘if’ buried deep in the derivation. It only works for a system in equilibrium. Sometimes people use the equation as a test for equilibrium.”

“Sorry, what does that mean?”

“Let’s take your problem galaxy cluster as an example. Suppose the galaxies are all alone in the Universe and far apart even by astronomical standards. Gravity’s going to pull them together. Galaxy i and galaxy j are separated by distance Rij. The potential energy in that interaction is Vij = G·mi·mj / Rij. The R‘s are very large numbers in this picture so the V attractions are very small. The Virial is the average of all the V’s so our starting Virial is nearly zero.”

“Nearly but not quite zero, I get that. Wait, if the potential energy starts near zero when things are far apart, and a falling‑in object gives up potential energy, then whatever potential energy it still has must go negative.”

“It does. The total energy doesn’t change when potential energy converts to kinetic energy so yes, we say potential energy decreases even though the negative number’s magnitude gets larger. It’d be less confusing if we measured potential energy going positive from an everything-all-together situation. However, it makes other things in Physics much simpler if we simply write (change in potential energy)+(change in kinetic energy)=0 so that’s the convention.”

“The distances do eventually get smaller, though.”

“Sure, and as the objects move closer they gain momentum and kinetic energy. Gaining momentum is gaining kinetic energy. You’re used to writing kinetic energy as T=m·v²/2, but momentum is p=m·v so it’s just as correct to write T=p²/2m. The two are different ways of expressing the same quantity. When a system is in equilibrium, individual objects may be gaining or losing potential energy, but the total potential energy across the system has reached its minimum. For a system held together by gravity or electrostatic forces, that’s when the Virial is twice the average kinetic energy. As an equation, V+2T=0.”

“So what you’re saying is, one galaxy might fall so far into the gravity well that its potential energy goes more negative than –2T. But if the cluster’s in equilibrium, galaxy‑galaxy interactions during the fall‑in process speed up other galaxies just enough to make up the difference. On the flip side, if a galaxy’s already in deep, other galaxies will give up a little T to pull it outward to a less negative V.”

“Well stated.”

“But why 2? Why not 3½ or some other number?”

“The 2 comes from the kinetic energy expression’s ½. The multiplier could change depending on how the potential energy varies with distance. For both gravity and electrostatic interactions the potential energy varies the same way and 2 is fine the way it is. In a system with a different rule, say Hooke’s Law for springs and rubber bands, the 2 gets multiplied by something other than unity.”

“All that’s nice and I see how the Virial Equation lets astronomers calculate cluster‑average masses or distances from velocity measurements. I suppose if you also have the masses and distances you can test whether or not a collection of galaxies is in equilibrium. What else can we do with it?”

“People analyze collections of stars the same way, but Professor Hanneken’s a physicist, not an astronomer. He wouldn’t have used class time on the Virial if it weren’t good for a broad list of phenomena in and outside of astronomy. Quantum mechanics, for instance. I’ll give you an important example — the Sun.”

“One star, all by itself? Pretty trivial to take its average.”

“Not averaging the Sun as an object, averaging its plasma contents — hydrogen nuclei and their electrons, buffeted by intense heat all the way down to the nuclear reactions that run near the Sun’s core. It’s gravitational potential energy versus kinetic energy all over again, but at the atomic level this time. The Virial Theorem still holds, even though turbulence and electromagnetic effects generate a complicated situation.”

“I’m glad he didn’t assign that as a homework problem.”

“The semester’s not over yet.”

~~ 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

…With Imaginary Answers

On By rolcottIn Mathematics, UncategorizedLeave a comment

“I’ve got another solution to Larry’s challenge for you, Vinnie.”

“Will I like this one any better, Sy?”

“Probably not but here it is. The a‑line has unit length, right?”

“That’s what the picture says.”

“And the b‑line also has unit length, along the i‑axis.”

“Might as well call it that.”

“And we’ve agreed it’s a right triangle because all three vertices touch the semicircle. A right triangle with two one‑unit sides makes it isosceles, right, so how big is the ac angle?”

“Isosceles right triangle … 45°. So’s the bc angle even though it don’t look like that in the picture.”

“Perfect. Now let me split the triangle in two by drawing in this vertical line.”

“That’s its altitude. I told you I remember Geometry.”

“So you did. Describe the ac triangle.”

“Mm, it’s a right triangle because altitudes work that way. We said the ac angle is 45° which means the top angle is also 45°.”

“What about length ca?”

“Gimme a sec, that’s buried deep. … If the hypotenuse is 1 unit long then each side is √½ units.”

“Vinnie, I’m impressed. So what’s the area of the right half of the semicircle?”

“That’s a quarter‑circle with radius ca which is √½ so the area would be ¼πca² which comes to … π/8.’

“Now play the same game with the b‑side.”

“Aw, geez, I see where you’re going. Everything’s the same except cb has an i in it which is gonna get squared. The area’s ¼πcb² which is … ¼π(i√½)² which is –π/8. Add ’em both together and the total area comes to zero again. … Wait!”

“Yes?”

“You said the i‑axis is perpendicular to everything else.” <sketching on the back of the card> “That’s not really a semicircle, it’s two arcs in different planes that happen to have the same center and match up at one point. The c‑line’s bent. Bo‑o‑o‑gus!”

“I said you wouldn’t like it. But I didn’t quite say ‘perpendicular to everything else‘ because it’s not. The problem is that the puzzle depends on a misleadingly ambiguous diagram. The only perpendicular to i that I worked with was the a‑line. You’ve also got it perpendicular to part of the c‑line.”

“Sy, if these i-number things are ambiguous like that, why do you physics and math guys spend so much time with them? And do they really call them i‑numbers?”

“Ya got me. No, they’re actually called ‘complex numbers,’ because they are complex. The problem is with what we call their components.”

“Components plural?”

Figure by “HB” under CCSA 4.0 license

“Mm‑hm. Mathematicians put numbers in different categories. There’s the natural numbers — one, two, three on up. Extend that through zero on down and you have the integers. Roll in the integer/integer fractions and you’ve got the rational numbers. Throw in all the irrationals like pi and √2 and you’re got the full number line. They call that category the reals.”

“That’s everything.”

“That’s everything along the real number axis. Complex numbers have two components, one along the real number line, the other along the perpendicular pure i‑number line. A complex number is one number but it has both real and i components.”

“If it’s not real, it’s imaginary, HAW!”

“More correct than you think. That’s exactly what the ‘i‘ stands for. If Descarte had only called the imaginaries something more respectable they wouldn’t have that mysterious woo‑quality and people wouldn’t have as much trouble figuring them out. Complex numbers aren’t mysterious or ambiguous, they’re just different from the numbers you’re used to. The ‘why’ is because they’re useful.”

“What good’s a number with two components?”

“It’s two for the price of one. The real number line is good for displaying a single variable, but suppose one quantity links two variables, x and y. You can plot them using Descartes’ real‑valued planar coordinates. Or you could use complex numbers, z=x+iy and plot them on the complex plane. Interesting things happen in the complex plane. Look at the difference between ex and eix.”

“One grows, the other cycles.”

“Quantum mechanics depends on that cycling.”

~ 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

A Spherical Bandstand

On By rolcottIn Astronomy: Planetary Science, Mathematics, Mathematics: Spherical Harmonics, UncategorizedLeave a comment
Radial harmonics Rn = Ln e-nr

“Whoa, Sy, something’s not right. Your zonal harmonics — I can see how latitudes go from pole to pole and that’s all there are. Your sectorial harmonic longitudes start over when they get to 360°, fine. But this chart you showed us says that the radius basically disappears crazy close to zero. The radius should keep going forever, just like x, y and z do.”

“Ah, I see the confusion, Susan. The coordinate system and the harmonic systems and the waves are three different things, um, groups of things. You can think of a coordinate system as a multilevel stage where chords of harmonic musicians can interact to play a composition of wave signals. The spherical system has latitude and longitude levels for the brass and woodwind players, plus one in back for the linear percussion section. Whichever direction the brass and woodwinds point, that’s where the signals go out, but it’s the percussion that determines how far they get. Sure, radius lines extend to infinity but except for R0 radial harmonics damp out pretty quickly.”

“Signals… Like Kaski’s team interpreted Juno‘s orbital twitches as a signal about Jupiter’s gravitational unevenness. Good thing Juno got close enough to be inside the active range for those radial harmonics. How’d they figure that?”

“They probably didn’t, Cathleen, because radial harmonics don’t fit easily into real situations. First problem is scale — what units do you measure r in? There’s an easy answer if the system you’re working with is a solid ball, not so easy if it’s blurry like a protein blob or galaxy cluster.”

“What makes a ball easy?”

“Its rigid surface that doesn’t move so it’s always a node. Useful radial harmonics must have a node there, another node at zero and an integer number of nodes between. Better yet, with the ball’s radius as a natural length unit the r coordinate runs linearly between zero at the center and 1.0 at the surface. Simplifies computation and analysis. In contrast, blurries usually don’t have convenient natural radial units so we scrabble around for derived metrics like optical depth or mixing length. If we’re forced into doing that, though, we probably have worse challenges.”

“Like what?”

“Most real-world spherical systems aren’t the same all the way through. Jupiter, for instance, has separate layers of stratosphere, troposphere, several chemically distinct cloud‑phases, down to helium raining on layers of hydrogen in liquid, maybe slushy or even solid form. Each layer has its own suite of physical properties that put kinks into a radial harmonic’s smooth curve. Same problem with the Sun.”

“How about my atoms? The whole Periodic Table is based on atoms having a shell structure. What about the energy level diagrams for atomic spectra? They show shells.”

“Well, they do and they don’t, Susan. Around the turn of the last Century, Lyman, Balmer, Paschen, Brackett and Pfund—”

“Sounds like a law firm.”

“<ironically> Ha, ha. No, they were experimental physicists who gave the theoreticians an important puzzle. Over a 40‑year period first Balmer and then the others, one series at a time, measured the wavelengths of dozens of lines in hydrogen’s spectrum. ’Okay, smarties, explain those!‘ So the theoreticians invented quantum mechanics. The first shot did a pretty good job for hydrogen. It explained the lines as transitions between discrete states with different energy levels. It then explained the energy levels in terms of charge being concentrated at different distances from the nucleus. That’s where the shell idea came from. Unfortunately, the theory ran into problems for atoms with more than one electron.”

“Give us a second… Ah, I get why. If one electron avoids a node, another one dives in there and that radius isn’t a node any more.”

“Got it in one, Cathleen. Although I prefer to think of electrons as charge clouds rather than particles. Anyhow, when an atom has multiple charge concentrations their behavior is correlated. That opens the door to a flood of transitions between states that simply aren’t options for a single‑electron system. That’s why the visible spectrum of helium, with just one additional electron, has three times more lines than hydrogen does.”

“So do we walk away from spherical harmonics for atoms?”

“Oh, no, Susan, your familiar latitude and longitude harmonics fit well into the quantum framework. These days, though, we mostly use combinations of radial fade‑aways like my Sn00 example.”

~~ 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

Zoning Out over Jupiter

On By rolcottIn Astronomy: Planetary Science, Mathematics: Spherical Harmonics, Physics: Waves, 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. “Morning, ladies. Cathleen, prepare to be even more smug.”

“Ooo, what should I be smug about?”

“Your Jupiter suggestion. Grab some coffee and a couple of chairs.” <screen‑tapping on Old Reliable> “Ready? First step — purple and violet. You’ll never see violet or purple light coming from a standard video screen.”

“He’s going spectrum‑y on us, right, Cathleen?”

“More like anti‑spectrum‑y, Susan. Purple light doesn’t exist in the spectrum. We only perceive that color when we see red mixed with blue like that second band on Sy’s display. Violet light is a thing in nature, we can see it in flowers and dyes and rainbows beyond blue. Standard screens can’t show violet because their LEDs just emit red, blue and green wavelengths. Old Reliable uses mixtures of those three to fake all its colors. Where are you going with this, Sy?””

“Deeper into Physics. Cast your eyes upon the squiggles to the right. The one in the middle represents the lightwave coming from purple‑in‑the‑middle. The waveform’s jaggedy, but if you compare peaks and troughs you can see its shape is the sum of the red and blue shapes. I scaled the graphs up from 700 nanometers for red and 450 for blue.”

“Straightforward spectroscopy, Sy, Fourier analysis of a complicated linear waveform. Some astronomers make their living using that principle. So do audio engineers and lots of other people.”

“Patience, Cathleen, I’m going beyond linear. Fourier’s work applies to variation along a line. Legendre and Poisson extended the analysis to—”

“Aah, spherical harmonics! I remember them from Physical Chemistry class. They’re what gives shapes to atoms. They’ve got electron shells arranged around the nucleus. Electron charge stays as close to the nucleus as quantum will let it. Atoms absorb light energy by moving charge away from there. If the atom’s in a magnetic field or near other atoms that gives it a z-axis direction then the shells split into wavey lumps going to the poles and different directions and that’s your p-, d– and f-orbitals. Bigger shells have more room and they make weird forms but only the transition metals care about that.”

The angular portion of the lowest-energy spherical harmonics
Credit: Inigo.quilez, under CCA SA 3.0 license
Adapted from Wieczorek and Meschede under CC BY-NC-ND 4.0

“Considering you left out all the math, Susan, that’s a reasonable summary. I prefer to think of spherical harmonics as combinations of wave shapes at right angles. Imagine a spherical blob of water floating in space. If you tap it on top, waves ripple down to the bottom and back up again and maybe back down again. Those are zonal waves. A zonal harmonic averages over all E‑W longitudes at each N‑S latitude. Or you could stroke the blob on the side and set up a sectorial wave pattern that averages latitudes.”

“How about center‑out radial waves?”

“Susan’s shells do that job. My point was going to be that what sine waves do for characterizing linear things like sound and light, spherical harmonics do for central‑force systems. We describe charge in atoms, yes, but also sound coming from an explosion, heat circulating in a star, gravity shaping a planet. Specifically, Jupiter. Kaspi’s paper you gave me, Cathleen, I read it all the way to the Results table at the tail end. That was the rabbit‑hole.”

“Oh? What’s in the table?”

“Jupiter’s zonal harmonics — J‑names in the first column, J‑intensities in the second. Jn‘s shape resembles a sine wave and has n zeroes. Jupiter’s never‑zero central field is J0. Jn increases or decreases J0‘s strength wherever it’s non‑zero. For Jupiter that’s mostly by parts per million. What’s cool is the pattern you see when you total the dominating Jeven contributions.”

Data from Kaspi, et al.

Cathleen’s squinting in thought. “Hmm… green zone A would be excess gravity from Jupiter’s equatorial bulge. B‘s excess is right where Kaspi proposed the heavy downflow. Ah‑HAH! C‘s pink deficit zone’s right on top of the Great Red Spot’s buoyant updraft. Perfect! Okay, I’m smug.”

~ Rich Olcott

Not Silly-Season Stuff, Maybe

On By rolcottIn Chemistry, Physics: Electromagnetism, UncategorizedLeave a comment

“Keep up the pace, Mr Feder, air conditioning is just up ahead.”

“Gotta stop to breathe, Moire, but I got just one more question.”

“A brief pause, then. What’s your question?”

“What’s all this about LK99 being a superconductor? Except it ain’t? Except maybe it is? What is LK99, anyway, and how do superconductors work? <puffing>”

“So many question marks for just one question. Are you done?”

“And why do news editors care?”

“There’s lots of ways we’d put superconductivity to work if it didn’t need liquid‑helium temperatures. Efficient electric power transmission, portable MRI machines, maglev trains, all kinds of advances, maybe even Star Trek tricorders.”

“Okay, I get how zero‑resistance superconductive wires would be great for power transmission, but how do all those other things have anything to do with it?”

“They depend on superconductivity’s conjoined twin, diamagnetism.”

“Dia—?”

“Means ‘against.’ It’s sort of an application of Newton’s Third Law.”

“That’s the one says, ‘If you push on the Universe it pushes back,’ right?”

“Very good, Mr Feder. In electromagnetism that’s called Lenz’ Law. Suppose you bring a magnet towards some active conductor, say a moving sheet of copper. Or maybe it’s already carrying an electric current. Either way, the magnet’s field makes charge carriers in the sheet move perpendicular to the field and to the prevailing motion. That’s an eddy current.”

“How come?”

“Because quantum and I’m not about to get into that in this heat. Emil Lenz didn’t propose a mechanism when he discovered his Law in 1834 but it works. What’s interesting is what happens next. The eddy current generates its own magnetic field that opposes your magnet’s field. There’s your push‑back and it’s called diamagnetism.”

“I see where you’re going, Moire. With a superconductor there’s zero resistance and those eddy currents get big, right?”

“In theory they could be infinite. In practice they’re exactly strong enough to cancel out any external magnetic field, up to a limit that depends on the material. A maglev train’s superconducting pads would float above its superconducting track until someone loads it too heavily.”

“What about portable MRI you said? It’s not like someone’s gonna stand on one.”

“A portable MRI would require a really strong magnet that doesn’t need plugging in. Take that superconducting sheet and bend it into a doughnut. Run your magnet through the hole a few times to start a current. That current will run forever and so will the magnetic field it generates, no additional power required. You can make the field as strong as you like, again within a limit that depends on the material.”

“Speaking of materials, what’s the limit for that LK99 stuff?”

“Ah, just in time! Ahoy, Susan! Out for a walk yourself, I see. We’re on our way to Al’s for coffee and air conditioning. Mr Feder’s got a question that’s more up your Chemistry alley than my Physics.”

“LK99, right? It’s so newsy.”

“Yeah. What is it? Does it superconduct or not?”

“Those answers have been changing by the week. Chemically, it’s basically lead phosphate but with copper ions replacing some of the lead ions.”

“They can do that?”

“Oh yes, but not as neatly as we’d like. Structurally, LK99’s an oxide framework in the apatite class — a lattice of oxygens with phosphorus ions sitting in most of the holes in the lattice, lead ions in some of the others. Natural apatite minerals also have a sprinkling of hydroxides, fluorides or chlorides, but the reported synthesis doesn’t include a source for any of those.”

“Synthesis — so the stuff is hand‑made?”

“Mm‑hm, from a series of sold‑state reactions. Those can be tricky — you grind each of your reactants to a fine powder, mix the powders, seal them in a tube and bake at high temperature for hours. The heat scrambles the lattices. The atoms can settle wherever they want, mostly. I think that’s part of the problem.”

“Like maybe they don’t?”

“Maybe. There are uncontrollable variables — grinding precision, grain size distribution, mixing details, reaction tube material, undetected but critical impurities — so many. That’s probably why other labs haven’t been able to duplicate the results. Superconductivity might be so structure‑sensitive that you have to prepare your sample j‑u‑s‑t right.”

~~ Rich Olcott