Category: Physics

Heisenberg’s trade-offs

On By rolcottIn Physics: Quantum Mechanics, UncategorizedLeave a comment

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

On By rolcottIn Physics: Quantum Mechanics, Uncategorized3 Comments

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.

The 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.

Now 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

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

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