Maybe even smaller?

There’s a sofa in my office. Sometimes it’s used to seat some clients for a consultation, sometimes I use it for a nap. This evening Anne and I are sitting on it, close together, after a meal of Eddie’s Pizza d’amore.

“I’ve been thinking, Sy. I don’t want to use my grow-shrink superpower very much.”

“Fine with me, I like the size you are. Why’d you decide that?”

“I remember Alice saying, ‘Three inches is such a wretched height to be.’ She was thinking about what her cat would do to her at that height. I’m thinking about what an amoeba might do to me if I were down to bacteria-size and I wouldn’t be able to see it coming because I’d be too small to see light. It would be even messier further down.”

“Well, mess is the point of quantum mechanics — all we get is the averages because it’s all chaos at the quantum level. Bohr would say we can’t even talk about what’s down there, but you’d be in the thick of it.”

She shudders delicately, leans in tighter. <long, very friendly pause> “Where’d that weird number come from, Sy?”

“What weird number?”

“Ten-to-the-minus-thirty-fifth. You mentioned it as a possible bottom to the size range.”

“I’ve got this new superpower, I need to think about stuff.  Besides, we’ve finished the pizza.”

<sigh> “This conversation reminds me of our elephant adventure.  Oh well.  Umm. It may have started on a cold, wet afternoon. You know, when your head’s just not up to real work so you grab a scratchpad and start doodling? I’ll bet Max Planck was in that state when he started fiddling with universal constants, like the speed of light and his own personal contribution ħ, the quantum of action.”

“He could change their values?”

“No, of course not. But he could combine them in different ways to see what came out. Being a proper physicist he’d make sure the units always came out right. I’m not sure which unit-system he worked in so I’ll just stick with SI units, OK?”

“Why should I argue?”

“No good reason to. So… c is a velocity so its units are meters per second. Planck’s constant ħ is energy times time, which you can write either as joule-seconds or kilogram-meter² per second. He couldn’t just add the numbers together because the units are different. However, he could divide the one by the other so the per-seconds canceled out. That gave him kilogram-meters, which wasn’t particularly interesting. The important step was the next one.”

“Don’t keep me in suspense.”

“He threw Newton’s gravitational constant G into the mix. Its units are meter³ per kilogram per second². ‘Ach, vut a mess,’ he thought, ‘but maybe now ve getting somevere. If I multiply ħ by G the kilograms cancel out und I get meter5 per second³. Now … Ah! Divide by c³ vich is equal to multiplying by second³/meter³ to cancel out all the seconds and ve are left mit chust meter² vich I can take the square root uff. Wunderbar, it is simply a length! How ’bout that?‘”

“Surely he didn’t think ‘how ’bout that?‘”

“Maybe the German equivalent. Anyway, doodling like that is one of the ways researchers get inspirations. This one was so good that (Għ/c³)=1.6×10-35 meter is now known as the Planck length. That’s where your ten-to-the-minus-thirty-fifth comes from.”

“That’s pretty small. But is it really the bottom?”

“Almost certainly not, for a couple of different reasons. First, although the Planck formula looks like a fundamental limit, it’s not. In the same report Planck re-juggled his constants to define the Planck mass (ħc/G)=2.2×10-8 kilograms or 22 micrograms. Grains of sand weight less than that. If Planck’s mass isn’t a limit, Planck’s length probably isn’t either. Before you ask, the other reason has to do with relativity and this is not the time for that.”

“Mmm … so if space is quantized, which is where we started, the little bits probably aren’t Planck-sized?”

“Who knows? But my guess is, no, probably much smaller.”

“So I wouldn’t accidentally go out altogether like a candle then. That’s comforting to know.”

My turn to shudder. <another long, friendly pause>.

~~Rich Olcott

Beautiful Realization

“Whaddaya mean, Sy, ‘charge and resistance and voltage all playing beautiful together‘? How’s that beautiful?”

“It is when they play together in a Kibble Balance, Vinnie. That beautifully-designed device solved the realization problem for two of the revised fundamental standards of measurement. Here’s the one for electricity.”

“That’s odd. It says ‘electric current’ but the number’s about charge. And I don’t see anything in there about voltage or resistance.”

“True. The electronic charge e is one of our universal constants. It and the speed of light and Planck’s constant h are the same on Mars as they are here on Earth. Take a cesium-based laser from Earth to Mars and its frequency doesn’t change. That’s why the revisions are measure-anywhere standards, no need to carry something to Paris to compare it to a physical object.”

“This is another one of those definition tricks, isn’t it? Like the cesium frequency — we defined the second by saying it’s the time required for so-and-so many waves of that light beam. Here, it’s not like someone measured the charge in coulombs, it’s we’re gonna make the coulomb exactly big enough so when we do measure an electron it’ll match up.”

“You’re not wrong, Vinnie, but it’s not quite that arbitrary. Lots of people did measure the electron against the old standard. This number represents the most accurate estimate across all the measurements. The standards board just froze it there. It’s the same strategy they took with the other six fundamental standards — each of them sits on top of a well-known constant.”

“Like Newton’s gravitational constant?”

“Sorry, Al, not that one. It’s universal, alright, but it’s only known to four significant figures, nowhere near the 8-or-better level the metrologists demand.”

“So tell us about the beauty part, Sy.”

I grab some paper napkins from the dispenser at our table. Al gives me a look. In his opinion Vinnie uses way too many of those and he doesn’t want it to spread. “Just using what I need to make a point, Al. Vinnie, I know you like pictures better than algebra but bear with me.”

“Yeah, you went through the kilogram thing pretty quick what with the garlic and all.”

“Oooh, yeah.” <scribbling on the first napkin> “Well anyway, here’s a sketch of the Kibble Balance rigged for weighing but let’s just pay attention to the parts in the dark blue oval. That zig-zag line labeled RK is a resistor that exploits the quantum Hall effect. Quantum math says its resistance is given by RK=h/e2. That’s exactly 25812.80756 ohms.”

“That a lot more digits than gravity.”

“Nice catch, Al. Now the second component in the oval is a quantum voltmeter. If you put a voltage V across it, the Josephson junction inside passes an alternating current whose frequency is f=V/CJ, where CJ=h/2e.” <scribbling on the second napkin> “Put another way. the frequency tells you the voltage from V=f×CJ and that’s the same as V=f×h/2e.” <scribbling on the third napkin> “The current iW going through RK is V/RK and that’s going to be iW=(f×CJ)/(RK)=f×(CJ/RK)=f×(h/2e)/(h/e2)=(f/2)×e. You with me?”

“Gimme a minute… You’re saying that the current is going to be half some frequency, which we can measure, times the charge on an electron. Yeah, that makes sense, ’cause the current is electrons and you got us counting electrons. Hey, wait, what happened to the h?”

“Canceled out in the fraction, just the way that e canceled out in the fraction for the kilogram.”

“Cute.”

“Better than cute, it’s beautiful. The same equipment, the Kibble Balance plus a gravimeter, gives you the realization of a kilogram depending only on h, AND the realization of the ampere depending only on e. Once you know the standards for time, which depends only on that cesium frequency, and for length, which depends only on time and the speed of light, you can get standards for mass and electric current in the NIST lab here on Earth or up on Mars or anywhere.”

“Almost anywhere.”

“In space, where there’s no gravity.”

“Einstein covered that with his Equivalence Principle.”

~~ Rich Olcott

Revenge of The Garlic Calzone

“So what’s the next two steps?” Vinnie asks.

“I’m thinking a dose of the pink stuff and a glass of milk. That garlic calzone’s just not giving up.”

“Nah, we were talking about the new mass standard and how it uses a Kibble Balance protocol you said had three steps but you only got to the gravity-measuring step. You wanna talk to take your mind off your gut, do some more of that.”

“<burp-sigh> OK, assume we did an accurate measurement of gravity’s acceleration g right next to the Balance.” <pulling Old Reliable from its holster...> “Here’s the device in the protocol’s second step, ‘weighing mode’. Bottom to top we’ve got a permanent magnet A and a coil of wire B that’s hooked up to some electronics. The coil floats in the magnetic field because it’s carrying an electric current from that adjustable power source C. The balance’s test pan D rides on the coil and supports our target mass E. Up top, laser interferometer F keeps track of the test pan’s position. Got all that?”

“Mass goes in the pan, got it.”

“Good. You adjust the current through the coil until the interferometer tells you the test pan is floating motionless. Here’s where the electronics come into play. The same current goes through resistor RK, a quantum Hall effect device chilling in a magnetic field and a bath of liquid helium. Quantum math says its resistance is h/e², where e is the charge on an electron and h is Planck’s constant. Those’re both universals like Einstein’s lightspeed c. RK comes to 25812.807557 ohms. You remember the V-I-R diagram?”

“Once Ms Kendall drills it into your brain it’s there forever. Volts equals current in amps times resistance in ohms.”

“Yep. In the Kibble Balance we evaluate the coil’s balancing current by measuring the voltage drop across RK. The voltmeter uses a Josephson junction, another quantum thingie. At a voltage V the junction passes a small alternating current whose frequency is f=V/CJ, where CJ=h/2e. Count the frequency and you can calculate the voltage. DivideV by RK to get the current iW going through the resistor. Everything here meets the count-based, stable, reproducible-anywhere standard.”

“I suppose the w suffixes mean ‘weigh mode’ and m in the bottom equation is the mass. Makes sense that heavier masses need more current to float them. What’s G?”

“Hold on, I got another burp coming … <bo-o-o-O-O-ORP!>”

“Impressive.”

“Thanks, I suppose. G rolls up all those geometry factors — size, shape and power of the magnet and so forth — that you complained about when I said ‘motor-generator.’ We take care of that in the third step. Here’s the diagram for that.”

“Looks pretty much the same.”

“We took out the target mass and the power source, and see, there’s v-subscripts for ‘velocity mode.’ We move the coil vertically while
the atomic clock ticks and the interferometer tracks the pan’s position. That lets us calculate speed s. The coil moving through the magnetic field generates a voltage V=fvCj=sG. Because the geometry factor G is identical between modes, the linkage between coil speed and output power is exactly the same as the linkage between current and input power. That’s the middle equation — velocity-mode voltage divided by speed equals weighing-mode force divided by current.”

“That’s weird.”

“But true, and so elegant. Every variable in that equation save the mass comes from a high-accuracy, high-precision reproducible standard. That makes mass a measure-anywhere dimension, too. But wait, there’s more.”

“Just a little more. Plug all these equations together and you get the bottom one. That’s exciting.”

“Doesn’t look exciting to me.”

“It goes back to the universal constants thing. The first factor in th middle is a ratio of count-derived quantities. Plug the quantum definitions into the second factor and you get CJ²/RK=(h²/4e²)(e²/h)=h/4. What that says is mass is expressible in units of Planck’s constant. That’s deep stuff! What’s exciting is that the standards people used that result in defining the kilogram.”

“Well, blow me down. And one more of your garlic burps or any more math just might.”

~~ Rich Olcott

Virtualosity

No knock, the door just opened suddenly.

“Hello, Jeremy.  Rule of Three?”

“Huh?  No, I was down the hall just now when I saw you go into your office so I knew you hadn’t gotten busy with something yet.  Sir.  What’s the Rule of Three?”

“Never mind.  You’re up here about virtual particles, I guess.”

“Yessir.  You said they’re ‘now you might see them, now you probably don’t.’  What’s that about and what do they have to do with abstraction and Einstein’s ‘underlying reality’?”

“What have you heard about Heisenberg’s Uncertainty Principle?”

“Ms Plenum says you can’t know where you are and how fast you’re going.”

“Ms Plenum’s got part of the usual notion but she’s missing the idea of simultaneous precision and a few other things.  Turns out you CAN know approximately where you are AND approximately how fast you’re going at a particular moment, but you can’t know both things precisely.  There’s going to be some imprecision in both measurements.  Think about Coach using a radar gun to track a thrown baseball.  How does radar work?”

“It bounces a light beam off of something and measures the light’s round-trip travel time.  I suppose it multiplies by the speed of light to convert time to distance.”

“Good.  Now how does it get the ball’s speed?”

“Uhh… probably uses two light pulses a certain time apart and calculates the speed as distance difference divided by time difference.”

“Got it in one.  Now, suppose that a second after the ball’s thrown the radar says the ball is 61 feet away from the plate and traveling at 92 mph.  Air resistance acts to slow the ball’s flight so that 92 is really an average.   Maybe it was going 92.1 mph at the first radar pulse and 91.9 mph at the second pulse.  So that reported speed has an 0.2 mph range of uncertainty.”

“Oh, and neither of the two pulses caught the ball at exactly 61 feet so that’s uncertain, too, right?”

“There you go.  We know the two averages, but each of them has a range.  The Uncertainty Principle says that the product of those two ranges has to be greater than Planck’s constant, 10-34 Joule·second.  Plugging that Joule-fraction and the mass of an electron into Einstein’s E=mc², we restate the constant as about 10-21 of an electron-second.  Those are both teeny numbers — but they’re not zero.”

“So speed and location make an uncertainty pair.  Are there others?”“A few.  The most important for this discussion is energy and time.”

“Wait a minute, those two can’t be linked that way.”

“Why not?”

“Well, because … umm … speed is change of location so those two go together, but energy isn’t change of time.  Time just … goes, and adding energy won’t make it go faster.”

“As a matter of fact, there are situations where adding energy makes time go slower, but that’s a couple of stories for another day.  What we’re talking about here is uncertainty ranges and how they combine.  Quantum theory says that if a given particle has a certain energy, give or take an energy range, and it retains that energy for a certain duration, give or take a time range, then the product of the two ranges has to be larger than that same Planck constant.   Think about a 1-meter cube of empty space out there somewhere.  Got it?

“Sure.”

“Suppose a particle appeared and then vanished somewhere in that cube sometime during a 1-second interval.  What’s the longest time that particle could have existed?”

“Easy — one second.”

“Zero.  Wait, it’d be the smallest possible non-zero time, wouldn’t it?”

“Good catch.  So what’s the time uncertainty?”

“One second minus that tiniest bit of time.”

“And what’s the corresponding energy range?”

“That constant number that I forget.”

“10-21 electron-second’s worth.  Now let’s pick a shorter interval.  What’s the mass range for a particle that appears and disappears sometime during the 10-19 second it takes a photon to cross a hydrogen atom?”

“That’s 10-21 electron-second divided by 10-19 second, so it’d be, like, 0.01 electron.”

“How about 1% of that 10-19 second?”

“Wow — that’d be a whole electron.”

“A whole electron’s worth of uncertainty.  But is the electron really there?”

“Probably not, huh?”

“Like I said, ‘Now you probably don’t’.”

~~ Rich Olcott

A kite floating on the breeze.  Optimal work-life balance.  Smoothly functioning free markets.  The Heisenberg Uncertainty Principle.  Why would an alien from another planet recognize the last one but maybe not the others?

The kite is a physical object, intentionally built by humans to human scale.  The next two are idealized theoretical constructs, goals to be approached but rarely achieved.  The Heisenberg Uncertainty Principle (HUP) is fundamental to how the Universe works.

The first three are each in a dynamic equilibrium that is constantly buffeted by competing forces.  The HUP comes straight out of the deep math for where those forces come from.  Kites and work stress and markets may be peculiar to Earth, but the HUP is in play on every planet and star.

In the last post we saw that thanks to the HUP we can precisely identify an oboe’s pitch if it plays forever.  We can know precisely when a pitchless cymbal crashed.  But it’s mathematically impossible to get both exact pitch and exact time for the same sound.  Thank goodness, we can have imprecise knowledge of both quantities and actually play some music.

We determine a pitch (cycles per second) by counting sound waves passing during a given duration — and that limits our knowledge.  We can’t know that a wave has passed unless we see at least two peaks.  Our observation period must be at least long enough to see two peaks.  To put it the other way, the pitch must be high enough to give us at least two peaks during the time we’re watching.  This isn’t quantum mechanics, it’s just arithmetic, but it’s basic to physics.

Mathematically the HUP is as simple as Einstein’s E=mc2 equation, except the HUP is an inequality:

[A-uncertainty] x [B-uncertainty] ≥ h / 4π

where A and B are two paired quantities like pitch and duration.

(That h is Planck’s constant, “the quantum of action,” 6.6×10-34 joule-sec.  That’s a very small number indeed but it shows up everywhere in quantum physics.  To put h in scale, one gram of TNT packs 4184 joules of explosive energy.  TNT has a detonation velocity of 6900 meters/sec and density of 1.60 gram/cm3, so we can figure a 1-gram cube of the stuff would burn for 1.2 microseconds and generate a total action of about 5×10-3 joule-sec.  Divide that by Avagadro’s number to get that one molecule of TNT is good for 10-26 joule-sec.  That’s about 10 million times h.  So, yeah, h is small.)

Back to the HUP inequality.  A and B are our paired quantities.  The standard examples that everyone’s heard of are position and momentum, as in the old physicist joke, “I haven’t a clue where I’m going, but I know how fast I’m getting there.”  For things that are tied to a central attractor like an atomic nucleus, A and B would be angular position and angular momentum.  If you’re into solid-state physics you may have run into another example — the number of electrons in a superconducting current is paired with a metric that reflects the degree of order in the conducting medium.  One more pair is energy and time, but that’s a story for another week.

But what’s in the HUP inequality isn’t A and B, but rather our uncertainty about each.  A billiard ball might be on the lip of the near cup or it can be all the way across the table — HUP won’t care.  What’s important to HUP is whether the ball is here plus/minus one inch, or here plus/minus a millionth of an inch.  Similarly, HUP doesn’t care how fast the ball is going, but it does care whether the speed is plus/minus one inch per second or plus/minus one millionth of an inch per second.  HUP tells us that we can know one of the pair precisely and the other not at all, or that we can know both imprecisely.  Furthermore, even the imprecision has a limit.

We can’t simultaneously know both A and B more precisely than that little teeny h, but some physicists believe h may have been big enough to launch our Universe.

Next week — HUP, two, three, four

~~ Rich Olcott