I was recently invited to contribute a guest post to the blog ribbonfarm, and I chose as my topic the somewhat ambitious question “where do electric forces come from?”

You can read my contributed post here.

As a teaser, I’ll give you the following snippet, along with an illustration from the article:

Empty space, or vacuum, is the stage upon which the pageantry of nature plays out.  Just as the setup of the stage in a theater determines what kind of plays can be performed, so too do the properties of the vacuum determine what kind of natural laws we have.

Coulomb’s law describes the repulsive force between two garden hoses

Magnets are complicated.

This fact is pretty well illustrated by how hard it is to predict whether a given material will make a good magnet or not. For example, if someone tells you the chemical composition of some material and then asks you “will it be magnetic?”, you (along with essentially all physicists) will have a hard time answering.  That’s because whether a material behaves like a magnet usually depends very sensitively on things like the crystal structure of the material, the valence of the different atoms, and what kind of defects are present.  Subtle changes to any of these things can make the difference between having a strong magnet and having an inert block.  Consequently, a whole scientific industry has grown up around the question “will X material be magnetic?”, and it keeps thousands of scientists gainfully employed.

On the other hand, there is a pretty simple answer to the more general question “where does magnetism come from?”  And, in my experience, this answer is not very well-known, despite the obvious public outcry for an explanation.

So here’s the answer:  Magnetism comes from the exchange interaction.

In this post, I want to explain what the “exchange interaction” is, and outline the essential ingredients that make magnets work.  Unless you’re a physicist or a chemist, these ingredients are probably not what you expect.  (And, despite some pretty good work by talented science communicators, I haven’t yet seen a popular explanation that introduces both of them.)

## Ingredient #1: Electrons are little magnets

As I discussed in a recent post, individual electrons are themselves like tiny magnets.  They create magnetic fields around themselves in the same way that a tiny bar magnet would, or that a spinning charged sphere would.  In fact, the “north pole” of an electron points in the same direction as the electron’s “spin”.

(This is not to say that an electron actually is a little spinning charged sphere.  My physics professors always told me not to think about it that way… but sometimes I get away with it anyway.)

The magnetic field created by one electron is relatively strong at very short distances: at a distance of one angstrom, it is as strong as 1 Tesla.  But the strength of that field decays quickly with distance, as $1/(\text{distance})^3$.  At any distance $\gtrsim 3$ nm away the magnetic field produced by one electron is essentially gone — it’s weaker than the Earth’s magnetic field (tens of microTesla).

What this means is that if you want a collection of electrons to act like a magnet then you need to get a large fraction of them to point their spins (their north poles) in the same direction.  For example, if you want a permanent magnet that can create a field of 1 Tesla (as in the internet-famous neodymium magnets), then you need to align about one electron per cubic angstrom (which is more than one electron per atom).

So what is it that makes all those electron align their spins with each other?  This is the essential puzzle of magnetism.

Before I tell you the right answer, let me tell you the wrong answer.  The wrong answer is probably the one that you would first think of.  Namely, that all magnets have a natural pushing/pulling force on each other: they like to align north-to-south.  So too, you might think, will these same electrons push each other into alignment through their north-to-south attraction.  Your elementary/middle school teacher may have even explained magnets to you by showing you a picture that looks like this:

(Never mind the fact that the picture on the right, in addition to its aligned North and South poles, has a bunch of energetically costly North/North and South/South side-by-side pairs.  So it’s not even clear that it has a lower energy than the picture on the left.)

But if you look a little more closely at the magnetic forces between electrons, you quickly see that they are way too weak to matter.  Even at about 1 Angstrom of separation (which is the distance between two neighboring atoms), the energy of magnetic interaction between two electrons is less than 0.0001 electronVolts (or, in units more familiar to chemists, about 0.001 kcal/mol).

The lives of electrons, on the other hand, are played out on the scale of about 10 electronVolts: about 100,000 times stronger than the magnetic interaction.  At this scale, there are really only two kinds of energies that matter: the electric repulsion between electrons (which goes hand-in-hand with the electric attraction to nuclei) and the huge kinetic energy that comes from quantum motion.

In other words, electrons within a solid material are feeling enormous repulsive forces due to their electric charges, and they are flying around at speeds of millions of miles per hour.  They don’t have time to worry about the puny magnetic forces that are being applied.

So, in cases where electrons decide to align their spins with each other, it must be because it helps them reduce their enormous electric repulsion, and not because it has anything to do with the little magnetic forces.

## Ingredient #2: Same-spin electrons avoid each other

The basic idea behind magnetism is like this: since electrons are so strongly repulsive, they want to avoid each other as much as possible.  In principle, the simplest way to do this would be to just stop moving, but this is prohibited by the rules of quantum mechanics.  (If you try to make an electron sit still, then by the Heisenberg uncertainty principle its momentum becomes very uncertain, which means that it acquires a large velocity.)  So, instead of stopping, electrons try to find ways to avoid running into each other.  And one clever method is to take advantage of the Pauli exclusion principle.

In general, the Pauli principle says that no two electrons can have the same state at the same time.  There are lots of ways to describe “electron state”, but one way to state the Pauli principle is this:

No two electrons can simultaneously have the same spin and the same location.

This means that any two electrons with the same spin must avoid each other.  They are prohibited by the most basic laws of quantum mechanics from ever being in the same place at the same time.

Or, from the point of view of the electrons, the trick is like this: if a pair of electrons points their spin in the same direction, then they are guaranteed to never run into each other.

It is this trick that drives magnetism.  In a magnet, electrons point their spins in the same direction, not because of any piddling magnetic field-based interaction, but in order to guarantee that they avoid running into each other.  By not running into each other, the electrons can save a huge amount of repulsive electric energy.  This saving of energy by aligning spins is what we (confusingly) call the “exchange interaction”.

To make this discussion a little more qualitative, one can talk about the probability $P(r)$ that a given pair of electrons will find themselves with a separation $r$.  For electrons with opposite spin (in a metal), this probability distribution looks pretty flat: electrons with opposite spin are free to run over each other, and they do.

But electrons with the same spin must never be at the same location at the same time, and thus the probability distribution $P(r)$ must go to zero at $r = 0$; it has a “hole” in it at small $r$.  The size of that hole is given by the typical wavelength of electron states: the Fermi wavelength $\lambda_F \sim n^{-1/3}$ (where $n$ is the concentration of electrons).  This result makes some sense: after all, the only meaningful way to interpret the statement “two electrons can’t be in the same place at the same time” is to say that “two electrons can’t be within a distance $R$ of each other, where $R$ is the electron size”.  And the only meaningful definition of the electron size is the electron wavelength.

If you plot two probability distributions together, they look something like this:

If you want to know how much energy the electrons save by aligning their spins, then you can integrate the distribution $P(r)$ multiplied by the interaction energy law $V(r)$ over all possible distances $r$, and compare the result you get for the two cases.  The “hole” in the orange curve is sometimes called the “exchange hole”, and it implies means that electrons with the same spin have a weaker average interaction with each other.  This is what drives magnetism.

(In slightly more technical language, the most useful version of the probability $P(r)$ is called the “pair distribution function“, which people spend a lot of time calculating for electron systems.)

## Epilogue: So why isn’t everything magnetic?

To recap, there are two ingredients that produce magnetism:

1. Electrons themselves are tiny magnets.  For a bulk object to be a magnet, a bunch of the electrons have to point their spins in the same direction.
2. Electrons like to point their spins in the same direction, because this guarantees that they will never run into each other, and this saves them a lot of electric repulsion energy.

These two features are more-or-less completely generic.  So now you can go back to the first sentence of this post (“Magnets are complicated”) and ask, “wait, why are they complicated?  If electrons universally save on their repulsive energy by aligning their spins, then why doesn’t everything become a magnet?”

The answer is that there is an additional cost that comes when the electrons align their spins.  Specifically, electrons that align their spins are forced into states with higher kinetic energy.

You can think about this connection between spin and kinetic energy in two ways.  The first is that it is completely analogous to the problem of atomic orbitals (or the simpler quantum particle in a box).  In this problem, every allowable state for an electron can hold only two electrons, one in each spin direction.  But if you start forcing all electrons to have the same spin, then each energy level can only hold one electron, and a bunch of electrons get forced to sit in higher energy levels.

The other way to think about the cost of spin polarization is to notice that when you give electrons the same spin, and thereby force them to avoid each other, you are really confining them a little bit more (by constraining their wavefunctions to not overlap with each other).  This extra bit of confinement means that their momentum has to go up (again, by the Heisenberg uncertainty principle), and so they start moving faster.

Either way, it’s clear that aligning the electron spins means that the electrons have to acquire a larger kinetic energy.  So when you try to figure out whether the electrons actually will align their spins, you have to weigh the benefit (having a lower interaction energy) against the cost (having a higher kinetic energy).  A quantitative weighing of these two factors can be difficult, and that’s why so many of scientific types can make a living by it.

But the basic driver of magnetism is really as simple as this: like-spin electrons do a better job of avoiding each other, and when electrons line up their spins they make a magnet.

So the next time someone asks you “magnets: how do they work?”, you can reply “by the exchange interaction!”  And then you can have a friendly discussion without resorting to profanity or name-calling.

## Footnotes:

1. The simplest quantitative description of the tradeoff between the interaction energy gained by magnetism and the kinetic energy cost is the so-called Stoner model.  I can write a more careful explanation of it some time if anyone is interested.

2. One thing that I didn’t explicitly bring up (but which most popular descriptions of magnetism do bring up) is that electrons also create magnetic fields by virtue of their orbits around atomic nuclei.  This makes the story a bit more complicated, but doesn’t change it in a fundamental way.  In fact, the magnetic fields created by those orbits are nearly equal in magnitude to the ones created by the electron spin itself, so thinking about them doesn’t change any order-of-magnitude estimates.  (But you will definitely need to think about them if you want to predict the exact strength of the magnetic field in a material.)

3. There is a pretty simple version of magnetization that occurs within individual atoms.  This is called Hund’s rule, which says that when you have a partially-filled atomic orbital, the electrons within the orbital will always arrange themselves so as to maximize the amount of spin alignment.  This “magnetization of a single atom” happens for the same reason that I outlined above: when electron spins align, they do a better job of avoiding each other, and their energy is lower.

4. If I were a good popularizer of science, then I would really go out of my way to emphasize the following point.  The existence of magnetism is a visible manifestation of quantum mechanics.  It cannot be understood without the Pauli exclusion principle, or without thinking about the electron spin.  So if magnets feel a little bit like magic, that’s partly because they are a startling manifestation of quantum mechanics on a human-sized scale.

Today, April 16, is the one day in the year when I allow myself to co-opt this blog for very personal purposes.  Usually I use this time as a way to remember Virginia Tech, and my time there.  Today, however, I want to be a little more audacious.

Forgive me, I want to talk about God and godlessness.

The theme and the day have something of a connection, of course.  But, as before, I will steadfastly refuse to make it.

If you’re here for physics-related content, I apologize; a new post should be up within a couple days.

My childhood was a very religious one.

I mean this not just in the sense that I spent a lot of time in church (which I did), or that religious doctrines played an outsized role in shaping and constraining the events of my life (which they did), but in the sense that religion felt very important to me.  From a very early age, the religious ideas to which I was exposed felt intensely valuable and deeply moving.

I was a Christian (a Mormon, in fact), and the doctrines of Christianity fascinated and moved me in a way that few things in my life have.  I felt very personally the Christian call to strive for a particular ideal of Christ-likeness, one defined by patience, charity, loving kindness, and faith.  It is probably not an exaggeration to say that I took some inspiration from this ideal almost every day of my life, starting from my earliest moments of literacy through the first year or two of college.  Christianity, to me, felt challenging and profound; I’m sure I expected that my religion would be the most important thing I would ever study.  Through deep intellectual thought, I supposed, and through the cultivation of very personal emotional experiences, I would become ever closer to this Christ-like person.  And to do so felt not just like a rewarding challenge, but like a moral imperative.

In short: I was a serious-minded kid, and my religion was perhaps the thing I took most seriously.

I was also a pretty analytical-minded kid.  I loved building things, solving puzzles, and learning how things work.  I relished those moments of wonder when you first realize that something seemingly mundane is much bigger or more intricate than you first supposed.

And, during childhood, those analytical proclivities seemed very much compatible with my religious feelings.  In fact, they usually mixed together in a truly beautiful way.  I remember, for example, the first time I looked at a cell.  I had peeled the thin skin off an onion from the fridge, and I lay down on my stomach on the prickly June grass and looked at the onion skin with a little mirror-lit microscope.  That was a moment of real wonder, as I’m sure it is for many a child: the first time you get a visual sense of the impossible intricacy of life.  And for me, those feelings were compounded and deepened by my awe at their creator.

In those years, every piece of learning or exploration of Nature made me feel like I was getting a little bit closer to God, like I was learning to see the universe just a little bit more like He was able to see it.  And that was exhilarating.

In addition to my seriousness and my nerdiness, my childhood was generally defined by happiness.  I had (and continue to have) a wonderful family, and from them I received the gift of feeling safe and feeling loved in a very fundamental way.  These feelings, not surprisingly, also got entangled with my gratitude for God, as the ostensible source from which love and protection flows.  And gratitude is a beautiful feeling when it is felt deeply and recognized for what it is.

In this way my religion provided the frame for the deepest feelings and experiences in my life.  It was the channel through which I experienced wonder and gratitude, and it provided the context through which I interpreted the love and security that I felt in life.

It was a beautiful way to live.

Of course, I realize in retrospect that there was something dangerous about this religious way of experiencing life.  Specifically, there is a particular trick commonly played by religion (if I can phrase it this way without ascribing bad intention to any particular person) for which I fell completely.

The trick goes like this: in most churches one is taught, quite explicitly, and from the youngest possible age, that God loves you and is watching out for you.  Therefore, the lesson continues, you can feel loved and safe in the world.  It’s a comforting lesson, and when felt deeply it is quite moving.  Implicit in this lesson, however, is something like a threat.  Namely, that if for some reason you don’t know that there is a God who loves you and protects you, then you will have no grounds for feeling loved and safe in the world.

Even without being explicitly taught that unhappy converse statement of the “God loves you” lesson, I accepted it.  I doubt that I even realized how deeply I had accepted it; it just seemed natural.  This was, perhaps, the most dangerous idea that I absorbed from my religion: that I was dependent on my religious belief for the most valuable emotions in my life.  (And maybe this is always the case, that the most dangerous religious lessons are the ones that are absorbed through implication rather than through explicit statement.)

I remember, for example, somewhere in my early teenage years, having a gentle creationism argument with an atheist.  The things that this person was saying (basic things, like “there is no God” — this was a very primitive discussion) seemed, to me, impossible to accept.  Not because they were illogical, but because it seemed intolerable to believe that human life was a cosmic accident with no purpose.  Such an idea simply could not (or should not) be accepted, I thought, because it would rob one of the ability to feel things that were essential for happiness.

As I aged through my teenage years, however, I found it increasingly difficult to be a “religious person.”  Maintaining religious belief and religious feeling required a constant struggle, whose gains were made by concentrated acts of studying, prayer, and willing oneself into a state of religious feeling.  Indeed, the very nature of religion was often described explicitly as a struggle, or a battle: a “good fight” that must be won.  But as I got older, this fight got increasingly difficult.  It involved grappling with deeper and more personal questions, and staving off doubts and alternative ideologies that seemed increasingly natural (and which I often kept at bay by equating their acceptance with a kind of moral laziness).

This process was largely terrifying, and intensely stressful.  At the conscious level, I propelled myself forward by appealing to the moral imperative associated with being a Christ-like force for good in the world.  (Again, this type of motivation, and the “battle” imagery that accompanied it, was often made explicit at church.)  But at the unconscious level, I was very afraid that if I lost my religion then I would lose completely the channel through which I experienced happiness.

This struggle for my own soul left me stressed, dour, and judgmental.

Eventually, it all fell apart.  Somewhere during my first years of college I became unable (or, as I would reproachfully describe myself, lazily unwilling) to maintain real religious belief.  My level of certainty about religious ideas shrank and shrank over time, until one day I found that I had nothing at all – no strong belief, and essentially no religious feeling.

That realization was a dark moment of real despair, and I became terribly depressed.  I felt hopeless, and isolated from my own family.  I felt as if I had a shameful secret that I had to keep from them.  Perhaps even worse, I felt isolated from a part of myself that I had loved dearly and that had made me happy.  I felt like a person whom I could no longer like or respect.

Eventually, though, after at least a year in this kind of state, a truly wonderful thing happened.  I remember very clearly: I was in the middle of a long, solo road trip, driving through the staggering mountains of Colorado along I-70 west of Denver.  The radio didn’t work in my car, so I had nothing to do for entertainment but to provoke myself to internal argument.  Suddenly, during the course of one of these arguments, and among the mountains of Colorado beneath a beautiful bright sky, I had an epiphany.

All my life my religion had exhorted me to seek truth, which was to be obtained through that painful “good fight” process, under the premise that real knowledge of deep truths would enable me to be a happy and a good person.  But what I realized, in that flash under the Colorado sky, was that I could be perfectly happy with no knowledge about any deep truths at all.

What a beautiful moment that was — it was one of the happiest instants of my life. (And, ironically, it felt exactly the way that I had always been taught religious revelation would feel.)

What I realized in that moment is that my happiness in life did not come from, or rely upon, any religious idea.  It was not dependent on any particular idea of God or any specific narrative about my place in the universe.  My feelings of being fundamentally safe and loved in the world were gifts that had been given to me by my family; even if I interpreted them in a religious way, they were never inherently religious feelings.  Today I am just as capable as ever of feeling like a good and a happy person, even though I lack any absolute standard for “good” or any plausible ultimate source from whom that happiness flows.  My religion had always taught me to credit its god for those feelings, but I realize now that they exist whether I believe in Him or not.

I can’t tell you how beautifully liberating I find that realization to be.

I should make clear, in closing, that I am not trying to make an anti-religious statement.  In general, I don’t feel qualified to make any large-scale comments about the “value” of religion.  Whether religion (in some particular form) is more “moral” than atheism, whether religion in general does more good than harm in the world, or whether any particular person will profit from adopting a particular set of religious beliefs – these are all hard questions that are best left to someone who has given them much more thought than I have.

But the point of this post is that I do have one comment that I think is worth making, directed to anyone who may find themselves unable to hold on to their religious feelings:

Though it may seem impossible, there is still wonder on the other side of belief.  Even without a God around whom to focus your wonder.  There is still gratitude, even without a God to whom you can direct it.  There is still love and kindness, and the world can still be deeply moving.  It is still possible to be happy and to feel like a good person, even without an ultimate arbiter to give your life meaning or to tell you what “good” means.

To those of you who have never constructed your lives around religion, these statements may sound obvious to the point of being asinine.  But to me they were among the most valuable and difficult ideas that I ever learned.

## Credit:

I was inspired to write this little essay after reading this wonderful blog post, which includes the lines:

If one thinks of creationism as a sequoia with God in the towering trunk and the various aspects of the natural world as branches going outward at every height, then [in contrast] science in general and evolution in particular is a web of unimaginable richness, with connections in every conceivable direction, splitting and rejoining and looping in almost infinite variety.  The strength of the sequoia is its enormous trunk, a monolithic invulnerability; that of the web is its deep interconnectedness, so that even if a few of its strands are found to be flawed (and they surely are, from time to time), the overall structure retains its integrity with room to spare.

In the year 1908, while the great minds of natural philosophy were puzzling over how to understand the structure of atoms in terms of the recently-discovered electron, Vladimir Lenin (yes, that Vladimir Lenin) declared that:

The electron is as inexhaustible as the atom

Lenin was making a philosophical point, but you can view his declaration as something like a scientific prediction.  Just as scientists of his era were discovering that atoms were made from smaller constituents like electrons, so too, he said, would we eventually find that the electron was made from even smaller components.  And on the cycle would go, with each new piece of matter being an “inexhaustible” source of new discovery.

It was a pretty reasonable expectation at the time, given the relentless progress of reductionist science.  But so far, despite more than a century of work by scientists (including many presumably extra-motivated Soviets), we still have no indication that the electron is made from anything else, or that it has any internal structure.  Students of physics in the 21st century are taught that the electron is a point in space with certain properties — mass, charge, and spin — but that it cannot be thought of as a spinning sphere or anything else that has a size and shape.  The electron looks pretty exhausted.

In this post, though, I want to take Lenin’s side, and ask the heretical question: “If the electron actually is a real, physical object with a finite size, then how big is it?”  Not surprisingly, there is no clear answer to this question, but some of the candidate answers turn out to be pretty interesting.

## Option #1: The Bohr radius

If you’re a physicist, and someone asks you “how big is an electron?”, then the most canonically correct thing to say is “There is no concept of an electron size other than the spatial extent of the electron wavefunction.  The size of the electron wavefunction is the electron size.”

This opinion essentially amounts to telling someone that they need to stop trying to think about a quantum electron as a physical object that you could hold in your hand like a baseball if only it was enlarged $10^{10}$ times.  People who profess this opinion are presumably also the ones who get annoyed by the popular declaration that “matter is 99.999% empty space.”

So, if such a person were pressed to give a numerical value for the “size of the electron”, they might say something like “Well, most electrons in the universe are bound into atoms.  So the typical ‘size’ of an electron is about the same as the typical size of an atom.”

As has been discussed here before, the typical size of an atom is given by the Bohr radius:

$a_B \sim 4 \pi \epsilon_0 \hbar^2/me^2$.

[Here, $\hbar$ is Planck’s constant, $\epsilon_0$ is the vacuum permittivity, and $m$ is the electron mass, and $e$ is the electron charge.]

Just to remind you how this length scale appears, you can figure out the rough size of an atom by remembering that an electron bound to an atom exists in a balance between two competing energies.  First, there is the attractive potential energy $U \sim -e^2/4 \pi \epsilon_0 r$ that pulls the electron toward the nucleus, and that gets stronger as the electron size $r$ (which is about the same as the average distance between the electron and the nucleus) gets smaller.  Second, there is the kinetic energy of the electron $KE \sim p^2/2m$.  The typical momentum of the electron gets larger as $r$ gets smaller, as dictated by the Heisenberg uncertainty principle, $p \sim \hbar/r$, which means that the electron kinetic energy gets larger as the atom size $r$ shrinks: $KE \sim \hbar^2/mr^2$.

In the balanced state that is an atom, $PE$ and $KE$ are about the same, which means that $e^2/4 \pi \epsilon_0 r \sim \hbar^2/mr^2$.  Solve for $r$, and you get that the electron size is about equal to the Bohr radius, which numerically works out to about 1 Angstrom, or  $10^{-10}$ m.

So now we have our first candidate answer to the question “how big is an electron?”.  If someone asks you this question, then you can sort of roll your eyes and then say “usually, about $10^{-10}$ meters”.

But let’s keep going, and see if there are any other concepts of an “electron size”.  Perhaps you don’t really like the answer that “the size of the electron wavefunction is the electron size”, because the electron wavefunction can be different in different situations, which means that this definition of “size” isn’t really an immutable property of all electrons.  Maybe you prefer to think about the electron as a tiny little ball, and the wavefunction as a sort of probability distribution that describes where the ball tends to be from time to time.  If you insist on this point of view, then what can you say about the size of the ball?

## Option #2: The classical electron radius

If you’re going to think about the electron as a tiny charged ball, then there is one thing that should bother you: that ball will have a lot of energy.

To see why this is true, imagine the hypothetical process of assembling your tiny charged ball from a bunch of smaller pieces, each with a fraction of the total charge.  Since the pieces have an electric repulsion from each other, and since you are bringing them very close to each other, the ball will be very hard to put together.

In fact, the energy required to “build” the electron is $E \sim e^2/4 \pi \epsilon_0 r$, where $r$ is the electron size.  So the smaller the electron is, the harder it is to build, and the more energy gets stored in the form of electric repulsion between all the pieces.  The large self-energy of the  electron also means that if the electron were to get “broken”, then all the tiny pieces would fly apart from each other, and release a tremendous amount of energy.

But, in fact, by the end of the first decade of the 20th century, people already knew that a single electron stored a lot of energy.  Einstein’s famous equation, $E = mc^2$, suggested that even a very small piece of matter was really an intensely concentrated form of energy.  One electron represents about $10^{-13}$ Joules (500,000 electron volts), which means that it only takes a few cups’ worth of electrons to have enough rest mass energy to equal the energy of an atomic bomb.  (Of course, if you ever actually removed all the electrons from a few cups’ worth of matter, you would create something much worse than an atomic bomb.)

So one of the early attempts to estimate a size for the electron was to equate these two ideas.  Maybe, the logic goes, the large “mass energy” of an electron is actually the same as the energy stored in the electric repulsion of its constituent pieces.  Equating those two expressions for the energy, $E = e^2/4\pi \epsilon_0 r = mc^2$ gives an estimate for $r$ that is called the “classical radius of the electron”:

$r_c = e^2/(4 \pi \epsilon_0 m c^2) \sim 10^{-15}$ m

So that’s our first estimate for the intrinsic electron size: about  $10^{-15}$ m.

This classical radius, by the way, works out to be about the same size as the typical size of an atomic nucleus.  And this is, perhaps, where we get the typical middle school picture of the atom: if you think about the electron using its classical radius, then the picture you get is of a tiny negatively-charged speck orbiting another equally tiny positively-charged speck, with about a factor $10^5$ difference between the speck sizes and the orbit sizes.

## Option #3: The Compton Wavelength

This picture of the electron as a little charged ball with size $10^{-15}$ meters seemed relatively okay up until the 1920s, when it was discovered that the electron had more properties than just its charge and its mass.  The electron also has what we now call a “spin”.

Basically, the concept of spin comes down to the fact that the electron is magnetic: it has a north pole and a south pole, and it creates a magnetic field around itself that is as large as about 1 Tesla at a distance of 1 Angstrom away, and that decays in strength as $1/(\text{distance})^3$.  Another way of saying this same thing is that the electron has a “magnetic moment” whose value is equal to the “Bohr magneton”, $\mu_B = e \hbar/2m$.  (As it turns out, this magnetic moment is essentially the same as the magnetic moment created by the orbit of an electron around a nucleus.)

At first sight, this magnetic-ness of the electron doesn’t seem like a problem.  A charged sphere can create a magnetic field around itself, as long as the sphere is spinning.  Remember, for example, that electrical currents create magnetic fields:

and that current is just moving electric charge.  So a spinning charged sphere also creates magnetic fields by virtue of the movement of its charge-containing surface:

This sort of image is, in fact, why the electron’s magnetic-ness was first called “spin”.  People imagined that the little electron was actually spinning.

A problem arises with this picture pretty quickly, though.  Namely, if the electron is very small, then in order for it to create a noticeable magnetic field it has to be spinning really quickly.  In particular, the magnetic moment of a spinning sphere with charge $e$ is something like $\mu \sim e \omega r^2$, where $\omega$ is the sphere’s rotation frequency.  If the sphere is spinning very quickly, then $\omega$ is large, and the equator of the sphere is moving at a very fast speed $v \sim \omega r$.

We know, however, that nothing can move faster than the speed of light (including, presumably, the waistline of an electron).  This puts a limit on how fast the electron can spin, which translates to a limit on how small the radius of the electron can be if we have any hope of explaining the observed electron magnetic field in terms of a physical rotation.

In particular, if you set $\mu = e \omega r^2 = e v r = \mu_B$, and require that $v < c$, then you get that the size of the electron $r$ must be bigger than

$\lambda_c \sim \hbar/(m c) \approx 10^{-12}$ m.

In other words, if you hope to explain the electron magnetic field as coming from an actual spinning motion, then the size of the electron needs to be at least $10^{-12}$ m.  This is about a thousand times larger than the classical electron radius.

Coincidentally, this value $\lambda_c$ is called the “Compton wavelength”, and it has another important meaning.  The Compton wavelength is more or less the smallest distance to which you can confine an electron.  If you try to squeeze the electron into an even smaller distance, then its momentum will become so large (via the uncertainty principle) that it its kinetic energy will be larger than $mc^2$.  In this case, there will be enough energy to create (from the vacuum) a new electron-positron pair, and the newly-created positron can just annihilate the trapped electron while the newly-created electron flies away.

## Option #4: the empiricist’s view

At this point you may start to feel like none of the above estimates for the electron size seems meaningful.  (Although don’t be too harsh in discarding them: the concepts of the Bohr radius, the classical electron radius, and the Compton wavelength appear over and over again in physics.)

So let’s consider a purely empirical view: what do experiments tell us about how big an electron is?

I know of two types of experiments that qualify as saying something about the electron size.  The first is a measurement of the electric dipole moment of the electron.  The idea is that, perhaps, the electron does not always have its charge arranged in a perfectly spherical way.  Maybe its “shape” can be slightly asymmetric, like this:

An asymmetric electron

In this case the electron would would want to align its “head” with an external electric field.  The strength of the electron asymmetry is quantified by the electron dipole moment $d$, which is defined as something like $d \sim ($charge in the bottom half of the electron $-$charge in the top half of the electron $) \times ($ electron size).

So far, however, experiments looking for a finite value of the electron dipole moment have not found one, and their experiments place an upper limit of $d < 10^{-30} \text{ electron charges} \times \text{meters}$.  This means that either the electron is an extremely symmetric object (for example, a very perfect sphere) or its size is smaller than about $10^{-30}$ meters.

Another set of experiments looks for corrections to the way that the electron interacts with the vacuum.  At a very conceptual level, an electron can absorb and re-emit photons from the vacuum, and this slightly alters its magnetic moment in a way that is mind-bogglingly well-described.  If the electron were to have a finite size, then this would alter its interaction with the vacuum a little bit, and the magnetic moment would very slightly change relative to our theories based on size-less electrons.

So far, however, experiments have seen no evidence of such an effect.  The accuracy of the experimental observations places an apparent upper limit on the electron size of about $10^{-18}$ m.

## Option #5: The Planck length

At this point, seeing the apparent failure of the classical description to produce a coherent picture, and seeing the experimental appearance of such spectacularly small numbers as $10^{-18}$ and $10^{-30}$, you may be willing to abandon Lenin’s hope of an “inexhaustible” electron and simply declare that the electron really is size-less.  From now on, you may resolve, when you draw an electron, you’ll draw it as a single pixel.  But only because you can’t draw it as a half-pixel.

But this brings up the last, and strangest, question: what is the smallest conceivable length that anything can have?  In other words, if the universe has a fundamental “pixel size”, then what is it?

Of course, I don’t know the answer to this question.  Whether there really is a “smallest possible length” is interesting to consider, but at the moment it can only be addressed with speculation.  Nonetheless, we do know that there is a length scale below which our most basic theories of the universe stop making sense.  This is called the “Planck length”, $\ell_p$.  At distances smaller than the Planck length, we are unable to describe even what empty space is like.

As I understand it, the Planck length problem can be viewed like this.  Our modern understanding of the vacuum (i.e., of empty space) is that it contains one photon mode for every possible photon wavelength.  This means that empty space is essentially full of photons of all conceivable energies and with all conceivable wavelengths.

However, photons with very small wavelength $\ell$ have very large energy, $E \sim \hbar c/\ell$.  It should therefore be possible to convert that energy, if only for a brief instant, into a large mass $M = E/c^2$. (After a short time $t \sim \hbar/E$, the mass will have to disappear again and give its energy back to the vacuum).  If that mass is large enough, though, it can create a small black hole.  A black hole gobbles up all other photons around it if they have a wavelength smaller than the black hole’s Schwarzschild radius, $R_s \sim G M/c^2$.  (Here, $G$ is Newton’s gravitational constant).  This starts to be a real problem when $R_s$ gets as large as the wavelength of the photon that first created the black hole, because then the black hole can consume the original photon and all other photons with smaller wavelength.

Do you see the problem with that (semi-contorted) logical sequence?  If very short length scales exist, then very high energy photons exist.  But if high energy photons exist, then they should be able to create, for a brief moment, very high masses.  Those high masses will create black holes.  And those black holes will eat up all the high energy photons.

It’s sort of a logical inconsistency.

As mentioned above, this problem first arises when the Schwarzschild radius $R_s$ becomes equal to the wavelength $\ell$ of the photon that created it.  If you work through the chain of algebra above, this will bring you to a length scale of

$\ell_p = \sqrt{\hbar G/c^3} \approx 10^{-35}$ m.

At any length scale smaller than $\ell_p$, we don’t know what’s going on.  At such small length scales either quantum theory should be different from what we know, or photons should be different, or gravity should be different.

Or maybe at such small length scales there is no good notion of continuous space at all.  Such thinking, as I understand it, gives rise to lots of picturesque ideas about “quantum foam”, and is the playground of (as yet mostly non-existent) theories of quantum gravity.

This picture is supposed to illustrate the idea that space might smooth and continuous until you try to look at it at the scale of the Planck length.

## What shall we say about Lenin?

So in the end, was Lenin right about the electron being “inexhaustible”?  For the moment, it looks like the answer is no, in the sense that there isn’t really any serious candidate for the intrinsic size of the electron.  In that sense, we could all have saved some time by just accepting the standard dogma that an electron is a sizeless point in space.

But I personally tend to resist dogmatism in all its forms, even the kind that is almost certainly correct.  Because sometimes those heretical questions lead you through all sorts of interesting ideas, ranging from from $10^{-10}$ meters down to $10^{-35}$ meters.

And if it becomes clear some day that Lenin’s statement really only works on the Planck scale, then we can probably say that his prediction came several centuries before its time.

I generally don’t like to use this blog to discuss my own scientific projects.  (Physics as a whole is much more interesting than my own meager contributions to it.)  But there is a recent project I was involved in that is getting a decent amount of attention from the popular press.  So I thought there might be some value in giving a description of it from the horse’s mouth (or, at least, the mouth of one of the horses).

The basic question behind this project is this:
Is it possible to model a crowd of people as a collection of interacting particles?
If so, what kind of “particles” are they?

Now, this question may seem silly to you.  Obviously, a human being isn’t a “particle” like an electron or a billiard ball.  Human motion (in most cases) arises from the workings of the human brain, and not from random physical forces.  So what would motivate someone to talk about humans as interacting particles?

The answer, at least for me personally, is that the motivation comes from watching videos like this one:

Basically, when you watch the movement of crowds at a large enough scale, the motion starts to look beautiful and familiar.  Perhaps something like the flow of a liquid:

Or maybe a granular fluid:

These apparent similarities are pretty exciting to a person like me (and to many others who are perhaps not a lot like me), in part because they imply the possibility of bringing old knowledge to a new frontier.  We have centuries of knowledge that was built to describe the physics of fluids and many-body systems, and now there is the hope that we can adapt it to say something useful about human crowds.

But all that those videos really demonstrate is that crowds and particle systems are visually similar, and so the question remains: is there really a good analogy between particle systems and human crowds?  If so, what kind of “particles” are we?

When you come down to it, the defining feature of a particle system is the “interaction law”, which is the equation that relates the energy $E$ of interaction between two particles to their relative positions.  For example, for two electrons the interaction law is the Coulomb law $E \propto e^2/r$, while for two neutral atoms it is something like the Lennard-Jones potential, $E \propto C_1/r^{12} - C_2/r^6$.

So, in some sense, the question of “how do we describe human crowds as particle systems?” comes down to the question “what is the interaction law between pedestrians?”.

In fact, scientists have been interested in that question for a few decades now.  And, generally, the way they have approached it is to make some hypothesis about how the interaction law $E(r)$ should look, and then make a big computer simulation of pedestrians following that law and see if the simulation behaves correctly.

This approach has gotten us pretty far — it has helped to save lives in Mecca and given us fantastic CGI crowd animations in movies and video games.  But it has also given rise to a sort of messy situation, scientifically, in which there are many competing models for crowds and their interaction law, with no generally accepted way of adjudicating between them.

What my colleagues and I eventually figured out is that we could take a different approach to this problem.  Instead of guessing what we thought the interaction law should be and then checking how well our guess worked, we decided to look first at real data and see what the data was telling us.  If there is, indeed, a universally correct equation for the interaction law between people, then it should be encoded in the data.

I’ll spare you the technical details of our data-digging, but the basic idea is like this.  In many-body physics, there is a general rule (the Boltzmann law) that relates energy to probability.  This law says, in short, that in a large system, any configuration of particles having a large energy is exponentially rare: its probability is proportional to $\exp(-E\times \text{const.})$.  So, for a crowd of people, looking at the relative abundance of different configurations of people tells you something about how much “energy” is associated with that configuration.  This allows you to infer the correct quantitative form of the interaction energy, by correlating the properties of different configurations with their relative abundance in the dataset.

So what we did is to first amass a large amount of crowd data.  This data was generally in the form of digitized video footage of people walking around crowded areas.  For example, we had data from students milling around on college campuses, shoppers walking around shopping streets, and even a few controlled experiments where people were recorded walking out of a crowded room.

An example of some of the data we looked at, showing different pedestrian “tracks”. The color corresponds to the time-averaged pedestrian density.

When we finished the analysis, there were two results that jumped out from the data very clearly: one obvious, and one surprising.

The first result is that the interaction law between pedestrians is not a function only on their relative distance $r$.

In hindsight, this result is  pretty obvious.  For example, two people walking headfirst into each other will feel a large “force” that compels them to move out of each other’s way.  On the other hand, two people walking side-by-side may feel no such force, even if they are relatively close to each other.

The implication of this result, though, is important.  It means that human pedestrians are very different from other, non-sentient “particles.”  An electron, for example, feels a force that is based on the physical proximity of other electric charges to itself at that particular instant.  Humans, on the other hand, respond to the world not as it is currently, but as they anticipate it to be in the near future.

Given the first, obvious result — that humans are not particles — the second result is much more surprising.  What we found is that there is, in fact, a very consistent interaction law between pedestrians in a crowd.  And it looks like this:

$(\text{interaction energy}) \propto 1/(\text{projected time to collision})^2$.

In other words, as pedestrians navigate around each other, they base their movements not on the physical distance between each other, but on the extrapolated time $\tau$ to an upcoming collision.  What’s more, the form of their interaction energy has the very simple form $E \propto 1/\tau^2$.  That this interaction law could have such a remarkably simple form was quite surprising, and that it holds across a whole range of different environments, densities, and cultures, was even more surprising.

It’s remarkable that something so mathematically simple could describe what is essentially a psychological phenomenon.

That, in a nutshell, was the finding of our paper.  My colleagues went on to show how this simple rule could immediately be used to make fast and accurate simulations of pedestrian crowds.  You can check out some of their simulation videos here, but I’ll also embed one of my favorite ones:

Here, two groups of people are asked to walk perpendicularly past each other.  They manage to resolve their imminent collisions by spontaneously forming diagonal “stripes” that cut through each other.

What’s nice about this result is that it gives us, with some confidence, the first major building block needed for making a real theory of human crowds.  Now that we know the nature of the interaction between two individuals, we can start putting together a kinematic theory of how crowds move in the aggregate.  This has all sorts of practical importance, in terms of understanding and predicting crowd disasters before they happen, but it also opens up a variety of fun problems to the language of condensed matter physics.  Maybe, in addition to describing the bulk “flow” of crowds, we can talk about the emergent features (“quasiparticles“) of crowds, like lane formation and mosh pit vortices.

It will be fun to see how this field develops in the near future.

## Credits:

Of course, most of the credit for this work belongs to my co-authors, Ioannis Karamouzas and Stephen Guy, at the Applied Motion Lab at the University of Minnesota.  They did the hard work of suggesting the problem, doing (most of) the data analysis, and writing the computer simulations.  My role was mostly to insist on making the problem “sound more like physics.”

There’s a great joke in Futurama, the cartoon comedy show, about a horror movie for robots.  In the movie, a planet of robots is terrorized by a giant “non-metallic being” (a monsterified human).  The human is finally defeated by a makeshift spear, which prompts the robot general to say:

“Funny, isn’t it?  The human was impervious to our most powerful magnetic fields, yet in the end he succumbed to a harmless sharpened stick.”

The joke, of course, is that the human body might seem much more fragile than a metallic machine, but to a robot our ability to withstand enormous magnetic fields would be like invincibility.

But this got me thinking: how strong would a magnetic field have to be before it killed a human?

$\hspace{1mm}$

Unlike a computer hard drive, the human body doesn’t really make use of any magnetic states — there is nowhere in the body where important information is stored as a static magnetization.  This means that there is no risk that an external magnetic field could wipe out important information, the way that it would for, say, a credit card or a hard drive.  So, for example, it’s perfectly safe for a human (with no metal in their body) to have an MRI scan, during which the magnetic fields reach several Tesla, which is about $10^5$ times stronger than the normal magnetic fields produced by the Earth.

A computer hard drive stores information in a sequence of magnetically aligned segments.

But even without any magnetic information to erase, a strong enough magnetic field must have some effect.  Generally speaking, magnetic fields create forces that push on moving charges.  And the body has plenty of moving charges inside it: most notably, the electrons that orbit around atomic nuclei.

As I’ll show below, a large enough magnetic field would push strongly enough on these orbiting electrons to completely change the shape of atoms, and this would ruin the chemical bonds that give our body its function and its structure integrity.

$\hspace{1mm}$

## What atoms look like

Before I continue, let me briefly recap the cartoon picture of the structure of the atom, and how to think about it.  An atom is the bound state of at least one electron to a positively charged nucleus.  The electric attraction between the electron and the nucleus pulls the electron inward, while the rules of quantum mechanics prevent the electron from collapsing down completely onto the nucleus.

In this case, the relevant “rule of quantum mechanics”  is the Heisenberg uncertainty principle, which says that if you confine an electron to a volume of size $r$, then the electron’s momentum must become at least as large as $p \sim \hbar/r$.  The corresponding kinetic energy is $p^2/2m \sim \hbar^2/mr^2$, which means that the more tightly you try to confine an electron, the more kinetic energy it gets.  [Here, $\hbar$ is Planck’s constant, and $m$ is the electron mass\$.]  This kinetic energy is often called the “quantum confinement energy.”

In a stable atom, the quantum confinement energy, which favors having a large electron orbit, is balanced against the electric attraction between the electron and the nucleus, which pulls the electron inward and has energy $\sim -e^2/\epsilon_0 r$. [Here $e$ is the electron charge and $\epsilon_0$ is the vacuum permittivity].  In the balanced state, these two energies are nearly equal to each other, which means that $r \sim \hbar^2 \epsilon_0/me^2 \sim 10^{-10}$ meters.

This is the quick and dirty way to figure out the answer to the question: “how big is an atom?”.

The associated velocity of the electron in its orbit is $v \sim p/m \sim \hbar/m r$, which is about $10^6$ m/s (or about a million miles per hour).  The attractive force between the electron and the nucleus is about $F_E \sim -e^2/\epsilon_0 r \sim m^2 e^6/\hbar^4 \epsilon_0^3$, which comes to ~100 nanoNewtons.

$\hspace{1mm}$

## Who pulls harder: the nucleus, or the magnetic field?

Now that I’ve reminded you what an atom looks like, let me remind you what magnetic fields do to free charges.

They pull them into circular orbits, like this:

The force with which a magnetic field pulls on a charge is given by $F_B \sim e v B$, where $B$ is the strength of the field.  For an electron moving at a million miles per hour, as in the inside of an atom, this works out to be about 1 picoNewton per Tesla of magnetic field.

Now we can consider the following question.  Who pulls harder on the electron: the nucleus, or the external magnetic field?

The answer, of course, depends on the strength of the magnetic field.  Looking at the numbers above, one can see that for just about any realistic situation, the force provided by the magnetic field is much much smaller than the force from the nucleus, so that the magnetic field essentially does nothing to perturb the electrons in their atomic orbitals.  However, if the magnetic field were to get strong enough, then the force it produces would be enough to start significantly bending the electron trajectories, and the shape of the electron orbits would get distorted.

Setting $F_B > F_E$ from above gives the estimate that this kind of distortion happens only when $B \gtrsim 10^5$  Tesla.  Given that the strongest static magnetic fields we can create artificially are only about 100 Tesla, it’s probably safe to say that you are unlikely to experience this any time soon.  Just don’t wander too close to any magnetars.

$\hspace{1mm}$

## Distorted atoms

But supposing that you did wander into a magnetic field of 100,000 Tesla, what would happen?

The strong magnetic forces would start to squeeze the electron orbits in all the atoms in your body.  The result would look something like this:

So, for example, an initially spherical hydrogen atom (on the left) would have its orbit squeezed in the directions perpendicular to the magnetic field, and would end up instead looking like the picture on the right.  This squeezing would get more and more pronounced as the field is turned up, so that all the atoms in your body would go from roughly spherical to “cigar-shaped,” and then to “needle-shaped”.

Needless to say, the molecules that make up your body are only able to hold together when they are made from normal shaped atoms, and not needle-shaped atoms.  So once the atomic orbitals got sufficiently distorted, their chemistry would change dramatically and these molecules would start to fall apart.  And your body would presumably be reduced to a dusty, incoherent mess (artist’s conception).

But for those of us who stay away from neutron stars, it is probably safe to assume that death by magnetic field-induced disintegration is pretty unlikely.  So you can continue lording your invincibility over your robot coworkers.

UPDATE:

A number of people have pointed out, correctly, that if you really subjected a body to strong magnetic fields, something would probably go wrong biologically far before the field got so ludicrously large fields as 100,000 Tesla.  For example, the motion of ions through ion channels, which is essential for nerve firing, might be affected.  Sadly, I probably don’t know enough biology to give you a confident speculation about what, exactly, might go wrong.

There is another possible issue, though, that can be understood at the level of cartoon pictures of atoms.  An electron orbiting around a nucleus is, in a primitive sense, like a tiny circular electric current.  As a result, the electron creates its own little magnetic field, with a “north pole” and “south pole” determined by the direction of its orbital motion.   Like so:

Normally, these little electron orbits all point in more or less random directions.  But in the presence of a strong enough external magnetic field, the electron orbit will tend to get aligned so that its “north pole” points in the same direction as the magnetic field.  By my estimate, this would happen at a few hundred Tesla.

In other words, a few hundred Tesla is what it would take to strongly magnetize the human body.  This isn’t deformation of atoms, just alignment of their orbits in a consistent direction.

Once the atomic orbits were all pointed in the same direction, the chemistry of atomic interactions might start to be affected.  For example, some chemical processes might start happening at different rates when the atoms are “side by side” as compared to when they are “front to back.”  I can imagine this subtle alteration of chemical reaction rates having a big effect over a long enough time.

Maybe this is why, as commenter cornholio pointed out below, a fruit fly that grows up in a ~ 10 Tesla field appears to get mutated.

## Footnote

I have been assuming, of course, that we are talking only about static magnetic fields.  Subjecting someone to a magnetic field that changes quickly in time is the same thing as bombarding them with radiation.  And it is not at all difficult to microwave someone to death.

[Update: A number of people have brought up transcranial magnetic stimulation, which has noticeable biological effects at relatively small field strengths.  But this  works only because it applies a time-dependent magnetic field, which can induce electric currents in the brain.]

Here’s a strange math problem that I encountered as an undergraduate:

What is the solution to the following equation?

[Note: The order of exponents here is such that the upper ones are taken first.  For example, you should read $2^{3^2}$ as $2^{(3^2)} = 512$ and not as $(2^3)^2 = 81$.]

As it happens, there’s a handy trick for solving this equation, and that’s to use both sides as an exponent for $x$.  This gives

From the first equation, though, the left hand side is just $2$.  So now we’re left with simply $x^2 = 2$, which means $x = \sqrt{2}$.

Not bad, right?  Apparently the conclusion is that

Where things get weird is when you try to solve an almost identical variant of this problem.  In particular, let’s try to solve:

We can do the same trick as before, using both sides of the equation as an exponent for $x$, and this gives

so that we’re left with $4 = x^4$.  The solution to this equation is, again, $x = \sqrt{2}$.

But now you should be worried, because apparently we have reached the conclusion that

So which is it?  What is the correct value of $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}$?  Is it 2, or is it 4?

## Yes, but which one is the real answer?

Maybe in the world of purely abstract mathematics, it’s not a problem to have two different answers to a single straightforward mathematical operation.  But in the real world this is not a tolerable situation.

The reasoning above raised a straightforward question — what is $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}$? — and provided two conflicting answers: $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 2$ and $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 4$.  Both of these equations are correct, but which one should you really believe?

Suppose that you don’t really believe either of those two equations (which, at this point, you probably shouldn’t), and you want to figure out for yourself what the value of $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}$ is.  How would you do it?

One simple protocol that you could do with a calculator or a spreadsheet is this:

• Make an initial guess for what you think is the correct value of $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}$.
• Take your guess and raise $\sqrt{2}$ to that power.
• Take the answer you get and raise $\sqrt{2}$ to that power.
• Repeat that last step a bunch of times.

Try this process out, and you will almost certainly get one of two answers:

If you initially guessed that $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}$ was any number less than 4, then you will arrive at the conclusion that $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 2$.

If your initial guess was something larger than 4, though, you will instead get to the conclusion that $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = \infty$.

The situation can be illustrated something like this:

Only if your initial guess was exactly 4, and if your calculator gave you the exact correct answer at every step, will you ever see the solution $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 4$.  A single error in the 16th decimal place anywhere along the way will instead lead you to a final answer of either $2$ or $\infty$.

In this sense $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 2$ is a much better answer than $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 4$.  The latter is true only in a hypothetical world of perfect exactness, while the former is true even if your starting conditions are a little uncertain, or if your calculator makes mistakes along the way, or (most importantly) there’s some small additional factor that you haven’t taken into consideration.

## Scientists adjudicate between mathematical realities

For the most part, this has been a silly little exercise.  But it actually does illustrate something that is part of the job of a physicist, or anyone else who uses math as a tool.  Physicists spend a lot of time solving equations that describe (or are supposed to describe) the physical world.  But finding a solution to some equations is not the end of process.  We also have to check whether the solution we came up with is meaningful in the real world, which is full of inexactnesses.  For example, the equation that describe the forces acting on a pencil on my desktop will tell me that the pencil can be non-moving either when lying on its side or when balanced on its point.  But only one of those two situations really deserves to be called a “solution”.

So, as for me, if you ask me whether $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 2$ or $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}} = 4$, I’ll go with 2.

Because, as Napoleon the pig understood, some equations are more equal than others.