On meditation and the birth of the universe.

On meditation and the birth of the universe.

This is part of a series of essays prepared to discuss in jail.

Our bodies are chaos engines. 

In our nearby environment, we produce order.  We form new memories.  We build things.  We might have sex and create new life.  From chaos, structure.

As we create local order, though, we radiate disorder into the universe. 

The laws of physics work equally well whether time is moving forward or backward.  The only reason we experience time as flowing forward is that the universe is progressing from order into chaos.

In the beginning, everything was homogeneous.  The same stuff was present everywhere.  Now, some regions of the universe are different from others.  One location contains our star; another location, our planet.  Each of our bodies is very different from the space around us.

This current arrangement is more disorderly than the early universe, but less so than what our universe will one day become.  Life is only possible during this intermediate time, when we are able to eat structure and excrete chaos. 

Hubble peers into a stellar nursery. Image courtesy of NASA Marshall Space Flight on Flickr.

Sunlight shines on our planet – a steady stream of high-energy photons all pointed in the same direction.  Sunshine is orderly.  But then plants eat sunshine and carbon dioxide to grow.  Animals eat the plants.  As we live, we radiate heat – low-energy photons that spill from our bodies in all directions.

The planet Earth, with all its life, acts like one big chaos engine.  We absorb photons from the sun, lower their energy, increase their number, and scatter them.

We’ll continue until we can’t.

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Our universe is mostly filled with empty space. 

But empty space does not stay empty.  Einstein’s famous equation, E equals M C squared, describes the chance that stuff will suddenly pop into existence.  This happens whenever a region of space gathers too much energy.

Empty space typically has a “vacuum energy” of one billionth of a joule per cubic meter.  An empty void the size of our planet would have about as much energy as a teaspoon of sugar.  Which doesn’t seem like much.  But even a billionth of a joule is thousands of times higher than the energy needed to summon electrons into being.

And there are times when a particular patch of vacuum has even more energy than that.

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According to the Heisenberg Uncertainty Principle, time and energy can’t be defined simultaneously.  Precision in time causes energy to spread – the energy becomes both lower and higher than you expected.

In practice, the vacuum energy of a particular region of space will seem to waver.  Energy is blurry, shimmering over time.

There are moments when even the smallest spaces have more than enough energy to create new particles.

Objects usually appear in pairs: a particle and its anti-particle.  Anti-matter is exactly like regular matter except that each particle has an opposite charge.  In our world, protons are positive and electrons are negative, but an anti-proton is negative and an anti-electron is positive.

If a particle and its anti-particle find each other, they explode.

When pairs of particles appear, they suck up energy.  Vacuum energy is stored inside them.  Then the particles waffle through space until they find and destroy each other.  Energy is returned to the void.

This constant exchange is like the universe breathing.  Inhale: the universe dims, a particle and anti-particle appear.  Exhale: they explode.

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Our universe is expanding.  Not only are stars and galaxies flying away from each other in space, but also empty space itself is growing.  The larger a patch of nothingness, the faster it will grow.  In a stroke of blandness, astronomers named the force powering this growth “dark energy.”

Long ago, our universe grew even faster than it does today.  Within each small fraction of a second, our universe doubled in size.  Tiny regions of space careened apart billions of times faster than the speed of light.

This sudden growth was extremely improbable.  For this process to begin, the energy of a small space had to be very, very large.  But the Heisenberg Uncertainty Principle claims that – if we wait long enough – energy can take on any possible value.  Before the big bang, our universe had a nearly infinite time to wait.

After that blip, our universe expanded so quickly because the vacuum of space was perched temporarily in a high-energy “metastable” state.  Technically balanced, but warily.  Like a pencil standing on its tip.  Left alone, it might stay there forever, but the smallest breath of air would cause this pencil to teeter and fall.

Similarly, a tiny nudge caused our universe to tumble back to its expected energy.  A truly stable vacuum.  The world we know today was born – still growing, but slowly.

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During the time of rapid expansion, empty vacuum had so much energy that particles stampeded into existence.  The world churned with particles, all so hot that they zipped through space at nearly the speed of light. 

For some inexplicable reason, for every billion pairs of matter and anti-matter, one extra particle of matter appeared.  When matter and anti-matter began to find each other and explode, this billionth extra bit remained.

This small surplus formed all of stars in the sky.  The planets.  Ourselves.

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Meditation is like blinking.  You close your eyes, time passes, then you open your eyes again.  Meditation is like a blink where more time passes.

But more is different.

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Our early universe was filled with the smallest possible particles.  Quarks, electrons, and photons.  Because their energy was so high, they moved too fast to join together.  Their brilliant glow filled the sky, obscuring our view of anything that had happened before.

As our universe expanded, it cooled.  Particles slowed down.  Three quarks and an electron can join to form an atom of hydrogen.  Two hydrogen atoms can join to form hydrogen gas.  And as you combine more and more particles together, your creations can be very different from a hot glowing gas.  You can form molecules, cells, animals, societies.

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When a cloud of gas is big enough, its own gravity can pull everything inward.  The cloud becomes more and more dense until nuclear fusion begins, releasing energy just like a nuclear bomb.  These explosions keep the cloud from shrinking further.

The cloud has become a star.

Nuclear fusion occurs because atoms in the center of the cloud are squooshed too close together.  They merge: a few small atoms become one big atom.  If you compared their weights – four hydrogens at the start, one helium at the finish – you’d find that a tiny speck of matter had disappeared.  And so, according to E equals M C squared, it released a blinding burst of energy.

The largest hydrogen bomb detonated on Earth was 50 megatons – the Kuz’kina Mat tested in Russia in October, 1961.  It produced a mushroom cloud ten times the height of Mount Everest.  This test explosion destroyed houses hundreds of miles away.

The fireball of Tsar Bomba, the Kuz’kina Mat.

Every second, our sun produces twenty billion times more energy than this largest Earth-side blast.

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Eventually, our sun will run out of fuel.  Our sun shines because it turns hydrogen into helium, but it is too light to compress helium into any heavier atoms.  Our sun has burned for about four billion years, and it will probably survive for another five billion more.  Then the steady inferno of nuclear explosions will end.

When a star exhausts its fuel, gravity finally overcomes the resistance of the internal explosions.  The star shrinks.  It might crumple into nothingness, becoming a black hole.  Or it might go supernova – recoiling like a compressed spring that slips from your hand – and scatter its heavy atoms across the universe.

Planets are formed from the stray viscera of early stars.

Supernova remains. Image by NASA’s Chandra X-Ray Observatory and the European Space Agency’s XMM-Newton.

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Our universe began with only hydrogen gas.  Every type of heavier atom – carbon, oxygen, iron, plutonium – was made by nuclear explosions inside the early stars.

When a condensing cloud contains both hydrogen gas and particulates of heavy atoms, the heavy atoms create clumps that sweep through the cloud far from its center.  Satellites, orbiting the star.  Planets.

Nothing more complicated than atoms can form inside stars.  It’s too hot – the belly of our sun is over twenty million degrees.  Molecules would be instantly torn apart.  But planets – even broiling, meteor-bombarded planets – are peaceful places compared to stars.

Molecules are long chains of atoms.  Like atoms, molecules are made from combinations of quarks and electrons.  The material is the same – but there’s more of it.

More is different.

Some atoms have an effect on our bodies.  If you inhale high concentrations of oxygen – an atom with eight protons – you’ll feel euphoric and dizzy.  If you drink water laced with lithium – an atom with three protons – your brain might become more stable.

But the physiological effects of atoms are crude compared to molecules.  String fifty-three atoms together in just the right shape – a combination of two oxygens, twenty-one carbons, and thirty hydrogens – and you’ll have tetrahydrocannibol.  String forty-nine atoms together in just the right shape – one oxygen, three nitrogens, twenty carbons, and twenty-five hydrogens – and you’ll have lysergic acid diethylamide.

The effects of these molecules are very different from the effects of their constituent parts.  You’d never predict what THC feels like after inhaling a mix of oxygen, carbon, and hydrogen gas.

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An amino acid is comparable in scale to THC or LSD, but our bodies aren’t really made of amino acids.  We’re built from proteins – anywhere from a few dozen to tens of thousands of amino acids linked together.  Proteins are so large that they fold into complex three-dimensional shapes.  THC has its effect because some proteins in your brain are shaped like catcher’s mitts, and the cannibinoid nestles snuggly in the pocket of the glove.

Molecules the size of proteins can make copies of themselves.  The first life-like molecules on Earth were long strands of ribonucleic acid – RNA.  A strand of RNA can replicate as it floats through water.  RNA acts as a catalyst – it speeds up the reactions that form other molecules, including more RNA.

Eventually, some strands of RNA isolated themselves inside bubbles of soap.  Then the RNA could horde – when a particular sequence of RNA catalyzed reactions, no other RNA would benefit from the molecules it made.  The earliest cells were bubbles that could make more bubbles.

Cells can swim.  They eat.  They live and die.  Even single-celled bacteria have sex: they glom together, build small channels linking their insides to each other, and swap DNA.

But with more cells, you can make creatures like us.

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Consciousness is an emergent property.  With a sufficient number of neuron cells connected to each other, a brain is able to think and plan and feel.  In humans, 90 billion neuron cells direct the movements of a 30-trillion-cell meat machine.

Humans are such dexterous clever creatures that we were able to discover the origin of our universe.  We’ve dissected ourselves so thoroughly that we’ve seen the workings of cells, molecules, atoms, and subatomic particles.

But a single human animal, in isolation, never could have learned that much.

Individual humans are clever, but to form a culture complex enough to study particle physics, you need more humans.  Grouped together, we are qualitatively different.  The wooden technologies of Robinson Crusoe, trapped on a desert island, bear little resemblance to the vaulted core of a particle accelerator.

English writing uses just 26 letters, but these can be combined to form several hundred thousand different words, and these can be combined to form an infinite number of different ideas.

More is different.  The alphabet alone couldn’t give anyone insight into the story of your life.

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Meditation is like a blink where more time passes, but the effect is very different.

Many religions praise the value of meditation, especially in their origin stories.  Before Jesus began his ministry, he meditated for 40 days in the Judaean Desert – his mind’s eye saw all the world’s kingdoms prostrate before him, but he rejected that power in order to spread a philosophy of love and charity. 

Before Buddha began his ministry, he meditated for 49 days beneath the Bodhi tree – he saw a path unfurl, a journey that would let travelers escape our world’s cycle of suffering. 

Before Odin began his ministry, he meditated for 9 days while hanging from a branch of Yggdrasil, the world tree – Odin felt that he died, was reborn, and could see the secret language of the universe shimmering beneath him. 

The god Shiva meditated in graveyards, smearing himself with crematory ash.

At its extreme, meditation is purportedly psychedelic.  Meditation can induce brain states that are indistinguishable from LSD trips when visualized by MRI.  Meditation isolates the brain from its surroundings, and isolation can trigger hallucination.

Researchers have found that meditation can boost our moods, attentiveness, cognitive flexibility, and creativity.  Our brains are plastic – changeable.  We can alter the way we experience the world.  Many of our thoughts are the result of habit.  Meditation helps us change those habits.  Any condition that is rooted in our brain – like depression, insomnia, chronic pain, or addiction – can be helped with meditation.

To meditate, we have to sit, close our eyes, and attempt not to think.  This is strikingly difficult.  Our brains want to be engaged.  After a few minutes, most people experience a nagging sense that we’re wasting time.

But meditation gives our minds a chance to re-organize.  To structure ourselves.  And structure is the property that allows more of something to become different.  Squirrels don’t form complex societies – a population of a hundred squirrels will behave similarly to a population of a million or a billion.  Humans form complex webs of social interactions – as our numbers grew through history, societies changed in dramatic ways.

Before there was structure, our entire universe was a hot soup of quarks and electrons, screaming through the sky.  Here on Earth, these same particles can be organized into rocks, or chemicals, or squirrels, or us.  How we compose ourselves is everything.

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The easiest form of meditation uses mantras – this is sometimes called “transcendental meditation” by self-appointed gurus who charge people thousands of dollars to participate in retreats.  Each attendee is given a “personalized” mantra, a short word or phrase to intone silently with every breath.  The instructors dole mantras based on a chart, and each is Sanskrit.  They’re meaningless syllables to anyone who doesn’t speak the language.

Any two-syllable word or phrase should work equally well, but you’re best off carving something uplifting into your brain.  “Make peace” or “all one” sound trite but are probably more beneficial than “more hate.”  The Sanskrit phrase “sat nam” is a popular choice, which translates as “truth name” or more colloquially as “to know the true nature of things.”

The particular mantra you choose matters less than the habit – whichever phrase you choose, you should use it for every practice.  Because meditation involves sitting motionless for longer than we’re typically accustomed, most people begin by briefly stretching.  Then sit comfortably.  Close your eyes.  As you breathe in, silently think the first syllable of your chosen phrase.  As you breathe out, think the second.

Repeating a mantra helps to crowd out other thoughts, as well as distractions from your environment.  Your mind might wander – if you catch yourself, just try to get back to repeating your chosen phrase.  No one does it perfectly, but practice makes better.  When a meditation instructor’s students worried that their practice wasn’t good enough, he told them that “even on a shallow dive, you still get wet.”

In a quiet space, you might take a breath every three to six seconds.  In a noisy room, you might need to breathe every second, thinking the mantra faster to block out external sound.  The phrase is a tool to temporarily isolate your mind from the world.

Most scientific studies recommend you meditate for twenty minutes at a time, once or twice a day, each and every day.  It’s not easy to carve out this much time from our daily routines.  Still, some is better than nothing.  Glance at a clock before you close your eyes, and again after you open them.  Eventually, your mind will begin to recognize the passage of time.  After a few weeks of practice, your body might adopt the approximate rhythm of twenty minutes.

Although meditation often feels pointless during the first week of practice, there’s a difference between dabbling and a habit.  Routine meditation leads to benefits that a single experience won’t.

More is different.

On uncertainty (with cartoon ending).

The whole cartoon is at the end.
See this monstrosity, in its entirety, at the end of this essay.

Reading about the uncertainty principle in popular literature almost always sets my teeth on edge.

CaptureI assume most people have a few qualms like that, things they often see done incorrectly that infuriate them.  After a few pointed interactions with our thesis advisor, a friend of mine started going berserk whenever he saw “it’s” and “its” misused on signs.  My middle school algebra teacher fumed whenever he saw store prices marked “.25% off!” when they meant you’d pay three quarters of the standard price, not 99.75%.  A violinist friend with perfect pitch called me (much too early) on a Sunday morning to complain that the birds on her windowsill were out of tune… how could she sleep when they couldn’t hit an F#??

“Ha,” I say.  “That’s silly… they should just let it go.”  But then I start frowning and sputtering when I read about the uncertainty principle.  Anytime somebody writes a line to the effect of, we’ve learned from quantum mechanics that measurement obscures the world, so we will always be uncertain what reality might have been had we not measured it.

My ire is risible in part because the idea isn’t so bad.  It even holds in some fields.  Like social psychology, I’d say.  If a research group identifies a peculiarity of the human mind and then widely publicizes their findings, that particularity might go away.  There was a study published shortly before I got my first driver’s license concluding that the rightmost lanes of toll booths were almost always fastest.  Now that’s no longer true.  Humans can correct their mistakes, but first they have to realize they’re mistaken.

That’s not the uncertainty principle, though.

CaptureAnd, silly me, I’d always thought that this misconception was due to liberal arts professors wanting to cite some fancy-sounding physics they didn’t understand.  I didn’t realize the original misconception was due to Heisenberg himself.  In The Physical Principles of Quantum Theory. he wrote (and please note that this is not the correct explanation for the uncertainty principle):

Thus suppose that the velocity of a free electron is precisely known, while the position is completely unknown.  Then the principle states that every subsequent observation of the position will alter the momentum by an unknown and undeterminable amount such that after carrying out the experiment our knowledge of the electronic motion is restricted by the uncertainty relation.  This may be expressed in concise and general terms by saying that every experiment destroys some of the knowledge of the system which was obtained by previous experiments.

Most of this isn’t so bad, despite not being the uncertainty principle.  The next line is worse, if what you’re hoping for is an accurate translation of quantum mechanics into English.

This formulation makes it clear that the uncertainty relation does not refer to the past; if the velocity of the electron is at first known and the position then exactly measured, the position for times previous to the measurement may be calculated.  Then for these past times ∆p∆q [“p” stands for momentum and “q” stands for position in most mathematical expressions of quantum mechanics] is smaller than the usual limiting value, but this knowledge of the past is of a purely speculative character, since it can never (because of the unknown change in momentum caused by the position measurement) be used as an initial condition in any calculation of the future progress of the electron and thus cannot be subjected to experimental verification.

That’s not correct.  Because the uncertainty principle is not about measurement, it’s about the world and what states the world itself can possibly adopt.  We can’t trace the position & momentum both backward through time to know where & how fast an electron was earlier because the interactions that define a measurement create discrete properties, i.e. they are not revealing crisp properties that pre-existed the measurement.

Heisenberg was a brilliant man, but he made two major mistakes (that I know of, at least.  Maybe he had his own running tally of things he wished he’d done differently).  One mistake may have saved us all, as was depicted beautifully in Michael Frayn’s Copenhagen (also… they made a film of this?  I was lucky enough to see the play in person, but I’ll have to watch it again!) — who knows what would’ve happened if Germany had the bomb?

Heisenberg’s other big mistake was his word-based interpretation of the uncertainty principle he discovered.

CaptureHis misconception is understandable, though.  It’s very hard to translate from mathematics into words.  I’ll try my best with this essay, but I might botch it too — it’s going to be extra-hard for me because my math is so rusty.  I studied quantum mechanics from 2003 to 2007 but since then haven’t had professional reasons to work through any of the equations.  Eight years of lassitude is a long time, long enough to forget a lot, especially because my mathematical grounding was never very good.  I skipped several prerequisite math courses because I had good intuition for numbers, but this meant that when my study groups solved problem sets together we often divided the labor such that I’d write down the correct answer then they’d work backwards from it and teach me why it was correct.

I solved equations Robert Johnson crossroads style, except I had a Texas Instruments graphing calculator instead of a guitar.

The other major impediment Heisenberg was up against is that the uncertainty principle is most intuitive when expressed in matrix mechanics… and Heisenberg had no formal training in linear algebra.  I hadn’t realized this until I read Jagdish Mehra’s The Formulation of Matrix Mechanics and Its Modifications from his Historical Development of Quantum Theory.  A charming book, citing many of the letters the researchers sent to one another, providing mini-biographies of everyone who contributed to the theory.  The chapter describing Heisenberg’s rush to learn matrices in order to collaborate with Max Born and Pascual Jordan before the former left for a lecture series in the United States has a surprising amount of action for a history book about mathematics… but the outcome seems to be that Heisenberg’s rushed autodidacticism left him with some misconceptions.

Which is too bad.  The key idea was Heisenberg’s, the idea that non-commuting variables might underlie quantum behavior.

Commuting? I should probably explain that, at least briefly.  My algebra teacher, the same one who turned apoplectic when he saw miswritten grocery store discount signs, taught the subject like it was gym class (which I mean as a compliment, despite hating gym class).  Each operation was its own sport with a set of rules.  Multiplication, for instance, had rules that let you commute, and distribute, and associate.  When you commute, you get to shuffle your players around.  7 • 5 will give you the same answer as 5 • 7.

CaptureBut just because kicks to the head are legal in MMA doesn’t mean you can do ’em in soccer.  You’re allowed to commute when you’re playing multiplication, but you can’t do it in quantum mechanics.  You can’t commute matrices either, which was why Born realized that they might be the best way to express quantum phenomena algebraically.  If you have a matrix A and another matrix B, then A • B will often not be the same as B • A.

That difference underlies the uncertainty principle.

So, here’s the part of the essay wherein I will try my very best to make the math both comprehensible and accurate.  But I might fail at one or the other or both… if so, my apologies!

A matrix is an array of numbers that represents an operation.  I think the easiest way to understand matrices is to start by imagining operators that work in two dimensions.

Just like surgeons all dressed up in their scrubs and carrying a gleaming scalpel and peering down the corridors searching for a next victim, every operator needs something to operate on.  In the case of surgeons, it’s moneyed sick people.  In the case of matrices, it’s “vectors.”

As a first approximation, you can imagine vectors are just coordinate pairs.  Dots on a graph.  Typically the term “vector” implies something with a starting point, a direction, and a length… but it’s not a big deal to imagine a whole bunch of vectors that all start from the origin, so then all you need to know is the point at which the tip of an arrow might end.

It’ll be easiest to show you some operations if we have a bunch of vectors.  So here’s a list of them, always with the x coordinate written above the y coordinate.

3        4        5        2        6        1         7         3          5

0 ,      0 ,      0 ,      1 ,      1 ,      2 ,       2 ,       5 ,        5

That set of points makes a crude smiley face.

graph-1

And we can operate on that set points with a matrix in order to change the image in a predictable way.  I’ve always thought the way the math works here is cute… you have to imagine a vector leaping out of the water like a dolphin or killer whale and then splashing down horizontally onto the matrix.  Then the vector sinks down through the rows.

It won’t be as fun when I depict it statically, but the math works like this:

Picture 2

Does it make sense why I imagine the vector, the (x,y) thing, flopping over sideways?

The simplest matrix is something called an “identity” matrix.  It looks like this:

Picture 4

When we multiply a vector by the identity matrix, it isn’t changed.  The zeros mean the y term of our initial vector won’t affect the x term of our result, and the x term of our initial vector won’t affect the y term of our result.  Here:

Picture 5

And there are a couple other simple matrices we might consider (you’ll only need to learn a little more before I get back to that “matrices don’t commute” idea).

If we want to make our smiling face twice as big, we can use this operator:

2   0

0   2

Hopefully that matrix makes a little bit of sense.  The x and y terms still do not affect each other, which is why we have the zeros on the upward diagonal, and every coordinate must become twice as large to scoot everything farther from the origin, making the entire picture bigger.

We could instead make a mirror image of our picture by reflecting across the y axis:

-1   0

0    1

Or rotate our picture 90º counterclockwise:

0  -1

1   0

The rotation matrix has those terms because the previous Y axis spins down to align with the negative X axis, and the X axis rotates up to become the positive Y axis.

And those last two operators, mirror reflection and rotation, will let us see why the commutative property does not hold in linear algebra.  Why A • B is not necessarily equal to B • A if both A & B are matrices.

Here are some nifty pictures showing what happens when we first reflect our smile then rotate, versus first rotating then reflecting.  If the matrices did commute, if A • B = B • A, the outcome of the pair of operations would be the same no matter what order they were applied in.  And they aren’t! The top row of the image below shows reflection then rotation; the bottom row shows rotating our smile then reflecting it.

graph-2

And that, in essence, is where the uncertainty principle comes from.  Although there is one more mathematical concept that I should tell you about, the other rationale for using matrices to understand quantum mechanics in the first place.

You can write a matrix that would represent any operation or any set of forces.  One important class of matrices are those that use the positions of each relevant object, like the locations of each electron around a nucleus, in order to calculate the total energy of a system.  The electrons have kinetic energy based on their momentum (the derivative of their position with respect to time) and potential energy related to their position itself, due to interaction with the protons in the nucleus and, if there are multiple electrons, repulsive forces between each other…

Elliptic_orbit(I assume you’ve heard the term “two-body problem” before, used by couples who are trying to find a pair of jobs in the same city so they can move there together.  It’s a big issue in science and medicine, double matching for residencies, internships, post-docs, etc.  Well, it turns out that nobody thinks it’s funny to make a math joke out of this and say, “At least two-body problems are solvable.  Three-body problems have to be approximated numerically.”)

…but once you have a wavefunction (which is basically just a fancy vector, now with a stack of functions instead of a stack of numbers), you can imagine acting upon it with any matrix you want.  Any measurement you make, for instance, can be represented by a matrix.  And the cute thing about quantum mechanics, the thing that makes it quantized, is that only a discrete set of answers can come out of most measurements.  This is because a measurement causes the system to adopt an eigenfunction of the matrix representing that measurement.

An eigenfunction is a vector that still looks the same after it’s been operated upon by a particular matrix (from the German word “eigen,” which means something like “own” or “self”).  If we consider the operator for reflection that I jotted out above, you can see that a vector pointing straight up will still resemble itself after it’s been acted upon.

And a neat property of quantum mechanics is that every operator has a set of eigenfunctions that spans whatever space you’re working with.  For instance, the X & Y axes together span all of two-dimensional space… but so do any pair of non-parallel lines.  You could pick any pair of lines that cross and use them as a basis set to describe two-dimensional space.  Any point you want to reach can indeed be arrived at by moving some distance along your first line and then some distance along your second.

This is relevant to quantum mechanics because any measurement collapses the system into an eigenfunction of its representative matrix, and the probability that it will end up in any one state is determined by the amount of that eigenfunction you need to describe its previous wavefunction in your new basis set.

That is one ugly sentence.

Maybe it’s not so surprising that Heisenberg described this incorrectly in words, because this is somewhat arduous…

Here, I’ll draw another nifty picture.  We’ll have to imagine two different operations (you could even get ahead of me and imagine that these represent measuring position and momentum, since that’s the pair of famous variables that don’t commute), and the eigenvectors for these operations are represented by either the blue arrows or the red arrows below.

graph-3

If we make a measurement with the blue matrix, it’ll collapse the system into one of the two blue eigenvectors.  If we decide to measure the same property again, i.e. act upon the system with the blue matrix again, we’re sure to see that same blue eigenvector.  We’ll know what we’ll be getting.

But once the system has collapsed into a blue arrow, if we measure with the red matrix the system has to shift to align with one of the red arrows.  And our probability of getting each red answer depends upon how similar each red arrow is to the blue arrows… the one that looks more like our current state is more likely to occur, but because neither red arrow matches a blue arrow perfectly, there’s a chance we’ll end up with either answer.

And if we want to make a blue measurement, then red, then blue… the two blue measurements won’t necessarily be the same.  After we’re in a state that matches a red eigenvector, we have some probability to flop back to either blue eigenvector, depending, again, on how similar each is to the red eigenvector we land in.

That’s the uncertainty principle.  That position is simply not well-defined when momentum is precisely known, and vice versa.  The eigenfunctions for one type of measurement do not resemble the eigenfunctions for the other measurement.  Which means that the type of measurement you have to make in order to know one or the other property invariably changes the system and gives you an unpredictable result… it’s like you’re rolling dice every time you switch which flavor of measurement you’re making.

But the measurement isn’t causing error.  It’s revealing an underlying probability distribution.  That is, there is no conceivable “gentle” way of measuring that will give a predictable answer, because the phenomenon itself is probabilistic.  Because the mechanics are quantized, because there are no in-between states, the system flops like a landbound fish from eigenvectors of one measurement to eigenvectors of the other.

Which is why it bothers me so much to see the uncertainty principle described as measurement obscuring reality when the idea crops up in philosophy or literature.  Those allusions also tend to place too much import on the idea of “observers,” like the old adage about a tree making or not making sound when it falls in an empty forest.  Perhaps I did a bad job of this too by writing “measurement” so often.  Maybe that word makes it sound as though quantum collapse requires intentional human involvement.  It doesn’t.  Any interaction between quantum mechanics and a semi-classical system will couple them and can cause the probabilistic distribution of wavefunctions to condense into particle-like behavior.

And I think the biggest difference between the uncertainty principle and the way it’s often portrayed in literature is that, rather than measurements obscuring reality, you could almost say that measurements create reality.  There wasn’t a discrete state until the measurement was made.  It’s like asking an inebriated collegiate friend who just learned something troubling about his romantic partner, “Well, what are you going to do?”  He’ll probably answer.  While you’re talking about it, it’ll seem like he’s going to stick to that answer.  But if you hadn’t asked he probably would’ve continued to mull things over, continued to exist in that seemingly in-between state where there’s both a chance that he’ll break up or try to work things out.  By asking, you learn his plan… but you also forced him to come up with a plan.

And it’s important that our collegian be drunk in this analogy… because making a different measurement has to re-randomize behavior.  Even after he resolves to break up, if you ask “Where should we go for our midnight snack,” mulling that over would make him forget what he’d planned to do about the whole dating situation.  The next time you ask, he might decide to ride it out.  It’s only when allowed to keep the one answer in the forefront of his mind that the answer stays consistent.

The uncertainty principle says that position and momentum can’t both be known precisely not because measurement is difficult, but because elementary particles are too drunk to remember where they are when you ask how fast they’re moving.

And, here, a treat!  As a reward for wading through all this, I’ve drawn a cartoon version of Heisenberg’s misconception.  Note that this is not, in fact, the correct explanation for the uncertainty principle… but do you really need me to sketch a bunch of besotted electrons?

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