The Universe and Werner H.

Heisenberg’s Area ( about 10-34 Joule-second) is small, one ten-millionth of the explosive action in a single molecule of TNT.  OK, that’s maybe important for sub-atomic physics, but it’s way too small to have any implications for anything bigger, right?  Well, it could be responsible for shaping our Universe.

Quick recap: The Heisenberg Uncertainty Principle (HUP) says that certain quantities (for instance, position and momentum) are linked in a remarkable way.  We can’t measure either of them perfectly accurately, but we can make repeated more-or-less sloppy measurements that give us average values.  The linkage is in that sloppiness.  Each repeated measurement lands somewhere in a range of values around the average.  HUP says that even with very careful measurement the product of those two spans must be greater than Heisenberg’s Area.

So now let’s head out to empty space, shall we?  I mean, really empty space, out there between the galaxies, where there’s only about one hydrogen atom per cubic meter.

Here’s a good cubic meter … sure enough, it’s got exactly one hydrogen atom in it.

g25For practice using Heisenberg’s Area, what can we say about the atom? (If you’re checking my math it’ll help to know that the Area, h/4π, can also be expressed as 0.5×10-34 kg m2/s; the mass of one hydrogen atom is 1.7×10-27 kg; and the speed of light is 3×108 m/s.)  On average the atom’s position is at the cube’s center.  Its position range is one meter wide.  Whatever the atom’s average momentum might be, our measurements would be somewhere within a momentum range of (h/4π kg m2/s) / (1 m) = 0.5×10-34 kg m/s. A moving particle’s momentum is its mass times its velocity, so the velocity range is (0.5×10-34 kg m/s) / (1.7×10-27 kg) = 0.3×10-7 m/s.

With really good tools we could determine the atom’s velocity within plus or minus 0.000 000 03 m/s.  Pretty good.

Now zoom in.  Dial that one-meter cube down a billion-fold to a nanometer (10-9 meters, which is still about ten times the atom’s width).  Yeah, the atom’s still in the box, but now its velocity range is 300 m/s.  The atom could be just hanging out at the center, or it could zoom out of the cube a microsecond after we looked — we just can’t tell which.

All of which illuminates the contrast between physics Newton-style and the physics that has bloomed since Einstein’s 1905 “miracle year.”  If Newton were in charge of the Universe, Heisenberg’s Area would be zero.  We could determine that atom’s position and momentum with complete accuracy.  In fact in principle we could accurately determine everything’s position and momentum and then calculate where everything would be at any time in the future.  But he isn’t and it’s not and we can’t.

Theorists and experimenters use the word “measurement” in different ways. A measurement done by a theoretician is generally based on fundamental constants and an Valueselaborate mathematical structure. If the measurement is a quantum mechanical result, part of that structure is our familiar bell-shaped curve.  It’s an explicit recognition that way down in the world of the very small, we can’t know what’s really going on.  Most calculations have to be statistical, predicting an average and an expected range about that average. That prediction may or may not pan out, depending on what the experimentalists find.

By contrast, when experimenters measure something, even as an average of multiple tests, it’s an estimate of the real distribution.  The research group (usually it’s a group these days) reports a distribution that they claim overlaps well with a real one out there in the Universe.  Then another group dives in to prove they or the theoreticians or both are wrong.  That’s how Science works.

You are hereSo there could be a collection of bell-curves gathered about the experimental result. Remember those extra dimensions we discussed earlier?  One theory that’s been floated is that along those extra dimensions the fundamental constants like h might take on different values.  Maybe further along “Dimension W” the value of h is bigger than it is in our Universe, and quantum effects are even more important than they are here.

Now how can we test that?

BTW, Heisenberg will be 114 on Dec 5.  Alles Gute zum Geburtstag, Werner!

~~ Rich Olcott

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Heisenberg’s Area

Unlike politicians, scientists want to know what they’re talking about when they use a technical word like  “Uncertainty.”  When Heisenberg laid out his Uncertainty Principle, he wasn’t talking about doubt.  He was talking about how closely experimental results can cluster together, and he was putting that in numbers.

ArrowsThink of Robin Hood competing for the Golden Arrow.  For the showmanship of the thing, Robin wasn’t just trying to hit the target, he wanted his arrow to split the Sheriff’s.  If the Sheriff’s shot was in the second ring (moderate accuracy, from the target’s point of view), then Robin’s had to hit exactly the same off-center location (still moderate accuracy but great precision).  The Heisenberg Uncertainty Principle (HUP) is all about precision (a.k.a, range of variation).

We’ve all encountered exams that were graded “on the curve.”  But what curve is that?  I can say from personal experience that it’s extraordinarily difficult to create an exam where  the average grade is 75.  I want to give everyone the chance to show what they’ve learned.  Each student probably learned only part of what’s in the unit, but I won’t know which part until after the exam is graded.  The only way to be fair is to ask about everything in the unit.  Students complained that my tests were really hard because to get 100 they had to know it all.

Translating test scores to grades for a small class was straightforward.  I would plot how many papers got between 95 and 100, how many got 90-95, etc, and look at the graph.  Nearly always it looked like the top example.  TestsThere’s a few people who clearly have the material down pat; they clearly earned an “A.”  Then there’s a second group who didn’t do as well as the A’s but did significantly better than the rest of the class — they earned a “B.”  As the other end there’s a (hopefully small) group of students who are floundering.  Long-term I tried to give them extra help but short-term I had no choice but to give them an “F.”

With a large class those distinctions get blurred and all I saw (usually) was a single broad range of scores, the well-known “bell-shaped curve.”  If the test was easy the bell was centered around a high score.  If the test was hard that center was much lower.  What’s interesting, though, is that the width of that bell for a given class stayed pretty much the same.  The curve’s width is described by a number called the standard deviation (SD), proportional to the width at half-height.  If a student asked, “What’s my score?” I could look at the curve for that exam and say there’s a 66% chance that the score was within one SD of the average, and a 95% chance that it was within two SD’s.

The same bell-shape also shows up in research situations where a scientist wants to measure some real-world number, be it an asteroid’s weight or elephant gestation time.  He can’t know the true value, so instead he makes many replicate measurements or pays close attention to many pregnant elephants.  He summarizes his results by reporting the average of all the measurements and also the SD calculated from those measurements.  Just as for the exams, there’s a 95% chance that the true value is within two SD’s of the average.  The scientist would say that the SD represents the uncertainty of the measured average.

Which is what Heisenberg’s inequality is about.  Heisenberg area 1He wrote that the product of two paired uncertainties (like position and momentum) must be larger than that teeny “quantum of action,” h.  There’s a trade-off.  We can refine our measurement of one variable but we’ll lose precision on the other.  If we plot results for one member of the pair against results for the other, there’s no linkage between their average values.  However, there will be a rectangle in the middle representing the combined uncertainty.

Heisenberg tells us that the minimum area of that rectangle is a constant.

It’s a very small rectangle, area = h/4π = 0.5×10-34 Joule-sec, but it’s significant on the scale of atoms — and maybe on the scale of the Universe (see next week).

~~ Rich Olcott

Heisenberg’s trade-offs

KiteA 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.

TNT(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.

Balance 1But 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

Don’t blame Heisenberg

There was the time I discovered that a chemical compound I’d made is destroyed by the light of the spectrometer I was using to study it. The NYT just ran an article about how biologists have a new-tech problem studying animals in the field because a camera drone can scare the critters away (or provoke an attack).  A teacher can’t shut down an ongoing bullying campaign because student chatter stops when they see him coming.  What’s the common thread in these situations?

You probably thought “Heisenberg,” but please don’t dis the poor guy for them.  You may have seen the for-real Heisenberg Uncertainty Principle in action, but only if you’re a physicist or a music-reading percussionist.  Rather, the incidents in the first paragraph are all examples of the Observer Effect, which is completely separate from the work of Werner H.

Observer observedThe confusion arises because the Observer Effect is often used in classroom explanations of the Heisenberg Uncertainty Principle (the HUP).  The Observer Effect could well apply pretty much anywhere there’s an observer and an observee (see photo), which is why research psychologists and police interrogators use one-way mirrors.

By contrast, the HUP is in play in only a few circumstances, chiefly audio and physics labs.  The key is that word uncertainty, because the HUP is all about the limits of our knowledge.  It says that there are certain pairs of quantities where we must trade off knowledge of one against knowledge of the other.  The more precisely we know the value of one, the more uncertain we are about the other one’s value.

drum notesLet’s start with sound.  Did you know that sheet music for a drummer doesn’t really use a “proper” staff with keys and all?  Oh, sure, they use a staff, sort of, but the “notes” indicate strokes rather than tones.  Here’s one variant of many notations out there.

Suppose an oboist plays a tone for you, that nice, long “A” that the orchestra tunes to.  (It’s generally the oboe playing that note, by the way, for two reasons.  First, the oboe uses very little air to produce its sound, so the oboist can hold that note much longer than a flautist or trumpeter could.  More important, though, is that the oboe simply isn’t adjustable — everyone else perforce has to re-tune to match up.)  The primary component of that “A” sound should be a wave of 440 cycles per second.

OboeNow suppose the oboist plays that “A” in shorter and shorter bursts — half-note, quarter-note, etc., down to where all that comes out is a blip.  His fingering and embouchure don’t change, so he’s still playing an “A.” However, when the emitted sound wave is very short we can no longer identify the pitch because there aren’t enough cycles there.  We need at least 2 cycles in a known time period to be able to say how many cycles per second the tone has.

Now the oboist switches up an octave (880 cycles per second) with the same burst length.  That gives us twice as many cycles in the blip and we can identify the new pitch.  However, if he cuts the note’s length in half once more, then again we don’t have enough cycles to count.  The shorter the note, the more precisely we know when it sounded, but the less precisely we know what note it was.

A cymbal crash is basically the limiting case.  It has no distinct pitch (or the physicist would say it has a huge number of pitches that all die away after a few cycles).  Rather than tell the percussionist to play an unidentifiably short note, the composer says, “T’heck with it!” and writes an “X” somewhere on the staff.

And vice-versa — at the start of the oboist’s note the sound contained an mixture of other frequencies.  The interlopers eventually died out as the note proceeded.  There will be another mixing when the oboist runs out of breath.  We can only have a really pure tone if the note never starts and never ends — the poor oboist plays that one note forever.

Thank to Heisenberg, we can be confident that even Bach’s well-tempered clavier was imprecise.

Next week — more fun with Heisenberg.

~~ Rich Olcott

Buttered Cats — The QM perspective

You may have heard recently about the “buttered cat paradox,” a proposition that starts from two time-honored claims:

  • Cats always land on their feet.
  • Buttered toast always lands buttered side down.

“The paradox arises when one considers what would happen if one attached a piece of buttered toast (butter side up) to the back of a cat, then dropped the cat from a large height. …
“[There are those who suggest] that the experiment will produce an anti-gravity effect. They propose that as the cat falls towards the ground, it will slow down and start to rotate, eventually reaching a steady state of hovering a short distance from the ground while rotating at high speed as both the buttered side of the toast and the cat’s feet attempt to land on the ground.”

~~ en.wikipedia.org/wiki/Buttered_cat_paradox

After extensive research (I poked around with Google a little), I’ve concluded that no-one has addressed the situation properly from the quantum mechanical perspective. The cat+toast system in flight clearly meets the Schrödinger conditions — we cannot make an a priori prediction one way or the other so we must consider the system to be in a 50:50 mix of both positions (cat-up and cat-down).

In a physical experiment with a live cat it’s probable that cat+toast actually would be rotating. As is the case with unpolarized light, we must consider the system’s state to be a 50:50 mixture of clockwise and counter-clockwise rotation about its roll axis (defined as one running from the cat’s nose to the base of its tail). Poor kitty would be spinning in two opposing directions at the same time.

Online discussions of the problem have alluded to some of the above considerations. Some writers have even suggested that the combined action of the two opposing adages could generate infinite rotational acceleration and even anti-gravity effects. Those are clearly incorrect conclusions – the concurrent counter-rotations would automatically cancel out any externally observable effects. As to the anti-gravity proposal, not even Bustopher Jones is heavy enough to bend space like a black hole. Anyway, he has white spats.

However, the community appears to have completely missed the Heisenbergian implications of the configuration.

The Heisenberg Uncertainty Principle declares that it’s impossible to obtain simultaneous accurate values for two paired variables such as a particle’s position and momentum. The better the measurement of one variable, the less certain you can be of the other, and vice-versa. There’s an old joke about a cop who pulled a physicist to the side of the road and angrily asked her, “Do you have any idea how fast you were going?”  “I’m afraid not, officer, but I know exactly where I am.”

It’s less commonly known that energy and time are another such pair of variables – the stronger the explosion, the harder it is to determine precisely when it started.

Suppose now that our cat+toast system is falling slowly, perhaps in a low-gravity environment. The landing, when it finally occurs, will be gentle and extend over an arbitrarily long period of time. Accordingly, the cat will remain calm and may not even awake from its usual slumberous state.

Tom and toastBy contrast, suppose that cat+toast falls rapidly. The resulting impact will occur over a very small duration. As we would expect from Heisenberg’s formulation, the cat will become really really angry and with strong probability will attack the researcher in a highly energetic manner.

From a theoretical standpoint therefore, we caution experimentalists to take proper precautions in preparing a laboratory system to test the paradox.

Next week – Getting more certain about Heisenberg

~~ Rich Olcott