Brad DeLong's Semi-Daily Journal (2004)

My eight year old is learning algebra and geometry. Im continually stunned by the things she takes for granted.

Wonderful!

You could go with the anthropomorphic explanation. Any universe where electrons weren't fermions atoms couldn't exist, so neither could we.

Here's a derivation of the solutions for the hydrogen atom:

http://scienceworld.wolfram.com/physics/HydrogenAtom.html

The fact that electrons obey Fermi-Dirac statistics follows from the Dirac equation for the relativistic electron:

http://scienceworld.wolfram.com/physics/DiracEquation.html

The connection between spin and 'statistics' (aka The Spin-Statistics Theorem) is one of those frustrating truths that are fundamental but don't appear to have a non-technical explanation-- at least I've never heard one. The key to the connection seems to be causality-- when you apply relativistic causality to fields with spin, the 'correct' statistics just sort of fall out. Not very satisfying, I know.

"But where does the Pauli Exclusion Principle come from? What reason is there for particles to obey Fermi-Dirac statistics?"

In "ordinary" non-relativistic quantum mechanics
you just have to accept that electrons and other particles with half-integral spin obey Fermi- Dirac statistics as an empirical fact.

In relativistic quantum mechanics (quantum field theory) you can prove something called the spin-statistics theorem which says that
half-integral spin particles (like electrons and protons) must obey Fermi-Dirac statistics and inetegral spin-particles (like photons and gluons) must obey Bose-Einsetin statistics.

The basic reason is "locality". Phsyical theories
should (we think) be local (an influence can't be felt instantaneously across a distance - some time must elapse for the influence to propagate).
If the spin-statistics theorem didn't hold, locality would be violated.

It is nonetheless quite interesting that the stability of matter in the ordinary non-relativsitic world depends on a principal like locality.

I'm not sure how you can answer a question like that. What I can tell you is that particles in quantum mechanics are classified by representations of the Poincare group. Unpacking that, it has to do with how the particle (more properly the wavefunction of the particle) behaves when you do things like rotations and Lorentz boosts. For massive particles, it turns out that this classification comes down to the spin which is either a natural number or a natural number divided by two. Finally, in quantum field theory, one can try to write down a theory of particles of a given spin. The famous spin-statistics theorem tells you that half-integer spin particles can only be quantized with Fermi-Dirac statistics and that integral spin particles need Bose-Einstein statistics.

The magic number 8 has to do with representation theory, too. You can separate out the Schrodinger equation into a radial and an angular part. The radial part gives a number 'n'. The angular part, in a sense, lives on the sphere. Given a function on the sphere, we can decompose it under how it behaves under representations. These are the spherical harmonics. Anyways, these can be labeled by a number 'l' and it turns out that n>l. The dimension of the representation labeled by 'l' is 2l+1. Thus, the total number of slots in the level 'n' is

1 + 3 + ... + 2n - 1 = n^2

Two electrons can fit in each slot, so the magic numbers are therefore of the form 2n^2. There's actually a hidden larger symmetry in hydrogen which makes this even easier or more abstruse depending on what direction you're going at it from. But it's all just group theory at the bottom.

JOhn Baez's explanation

http://math.ucr.edu/home/baez/spin.stat.html

While we're with Dr. Baez, I'll pass along his page on the secret extra symmetry of the hydrogen atom:

http://math.ucr.edu/home/baez/gravitational.html

Well, some others have tried, so let me have a go as well. After all, it is one of the few topics on this blog I am actually qualified to say anything about.

"Why the magic number 8"? The numerology of this is not too complicated. Why the rules are the way they are is another question. But anyway, here are the rules that lead to the magic numbers.

Each shell is labelled by a number N.

The n'th shell can support angular momentum values (actually L^2) of 0, 1, 2, up to N-1,

For each angular momentum, the orbitals are classified by the direction of their angular momentum (the axis around which the electron is, loosely speaking, orbiting). This number is written ml (actually the l is a subscript). This is the projection of the angular momentum along a reference axis. This can take values of L, L-1, ... 0 ... -L. (L=pointing along the axis one way, 0 = orthogonal to the axis, -L = pointing along the axis in the opposite direction).

Each orbital can hold two electrons.

So, do the arithmetic:

For N=1, there is just L=0, ml=0 -- therefore two electrons.

For N=2, there is L=0 (two electrons) plus L=1. L=1 has ml=+1, 0, -1. That's a total of six electrons in L=1 orbitals (p orbitals) plus 2 in the L=0 orbital (s orbital) for a total of 8.

As for N=3, the number is 18. The L=2 orbitals (d orbitals) can hold 10 electrons. These d orbitals are partially filled in the transition metals and give rise to all kinds of wonderful things, such as our ability to breathe.

As for "why Fermi statistics"? Well, as Jeffrey Miller says, you need relativistic quantum mechanics to do that one. But we can say that wavefunctions represent a "probability amplitude" and that the square of the wavefunction is what is directly observable. So the wavefunction itself has a single degree of freedom, just as the square root of 1 can be either + or -1. One of these corresponds to Fermi Dirac statistics, and is obeyed by anything with a spin of a half. The other is Bose Einstein statistics and is obeyed by anything with spin 0 or 1. So an unlimited number of photons (spin 1) can fit in a single energy level.

Well, probably anyone who has read this far knows the answer as well as I do, so I'll stop.

I don't know a great answer for where the Pauli Exclusion Principle comes from. It's a result of the nature of the wavefunction of Fermions, which are "spin 1/2" particles, which is that they interfere destructively - if you try to calculate the probability of having two fermions in the same state, that probability works out to zero. This is obviously not any kind of real explanation (WHY are they anti-symmetric?) - it comes from quantum field theory, which I do not understand at all, and, every time someone has tried to explain it to me, I have nodded sagely, remarked "yes, I see" at what I assumed were the appropriate points in the discussion, and not understood one word of what was said. I now regret this. This page is a not-very technical description of why this is, which may be helpful. I just muttered "yes, I see" while reading it through force of habit.

http://newton.umsl.edu/~philf/candles.html

Why the electron shells hold the number of electrons they do is easy. First, accept that electrons can only exist in certain "modes" in the atom, which are analogous to the notes on a guitar string. You can imagine a string held essentially motionless at each end, and imagine the ways you can make it oscillate in a standing pattern: as a big hump, as two humps with a motionless "node" in the middle, as three humps with two nodes, and so on. You can index these modes: call the one hump mode "mode one", the two hump mode "mode two", and so on, with each higher numbered node having higher energy. These numbers of the modes - 1,2,3 ... - are calle dthe "quantum numbers" in our analogy, and completely describe the way the rope is oscillating, assuming it's a stable oscillation, and assuming you know the boundary conditions of the rope (its ends are held down, it has a certain mass per unit length, etc.)

An atom is more complicated than a string, and so requires 4 quantum numbers, which must be integers: n, the principle quantum number, which gives the "shell" the electron is in; l, which gives the total angular momentum of the electron, which must be less than n (but positive, of course); m, the amount of angular momentum the electron has along a certain axis (the choice of which is essentially arbitrary), which may range from -l to +l; and s, the spin, which may be either +1/2 or -1/2. The lowest shell of the electron is n=1, which can only have l=0, m=0, and s=+/-1/2. Any subshell with l=0 is called an 's shell', and, as you can see, there are only two possible arrangements of the quantum numbers, so at most 2 electrons can be here.

Next in energy is n=2, l=0, m=0, s=+/-1/2 - the 2s subshell. Shell 2 can also have l=1 - the "p shell" - which, in turn, can have m=1, m=0, and m=-1 a unique quantum number sets, with each one allowed to have one spin +1/2 and one spin -1/2 electron in it - a total of six. So shell 2 can hold 8 electrons. And so on.

http://jeries.rihani.com/index9.html

Here are pictures of the modes of an electron in an atom, which look identical to the transverse modes of a laser in a laser cavity, although, really, the atomic modes are arranged in 3-D.

http://www.rwc.uc.edu/koehler/biophys/6b.html

The fact that particles with integer spin obey Bose-Einstein statistics (multiple particles can occupy the same quantum state--no Pauli exclusion) while particles with half-integer spin obey Fermi-Dirac statistics (only 1 particle per quantum state--Pauli exclusion) is generally called the spin-statistics theorem and was first seriously established by Wolfgang Pauli.

Despite the occassional claim to the contrary, there appears to be no simple proof of the spin-statistics theorem, i.e., none that exists rigorously outside of relativistic quantum theory. Indeed, you can open up some pretty heavy-duty graduate textbooks like Stephen Weinberg's _The Quantum Theory of Fields_ and see that the actual proof of spin-statistics theorem is in one of those infamous starred sections that "lie somewhat outside of the book's main line of development" (It's specifically Section 5.7 of Volume I in case you're curious).

For it appears that the spin-statistics connection is intimately tied to Poincare invariance (i.e., the group of transformations that leave equations of motion in special relativity unchanged). Specifically, the spin-statistics relation is tied to how the irreducible representations of the Poincare group are associated with spin. The reason for believing that Poincare invariance is necessary is that if you just assume merely Galilean invariance (i.e., the groups of transformations that leave equations of motion unchanged in Newtonian Mechanics), then you can construct for any given spin value--be it integer or half-integer--consistent theories with either Bose-Einstein or Fermi-Dirac statistics.

I get these facts from the grand old man himself of rigorous quantum field theory, Arthur Wightman, in his American Journal of Physics review [Am. J. Phys. 67, 742 (1999)] of Ian Duck and ECG Sudarshan's book _Pauli and the Spin-Statistics Theorem_ (World Scientific, 1997... Berkeley library
PHYS QC793.3.S6.D83 1997 Not checked out as of 21:41 EST Sunday 2/15) which is 500+ pages of most everything that was ever written on the subject, including some dead ends.

Sidenote: You might have heard that the clear distinction between bosons and fermions, the spin-statistics relation we know and love, only holds in 3 or more spatial dimensions. In 2 dimensions, exotic "fractional" statistics and "anyons" are possible and these play a role in funky phenomena like the "quantum Hall effect". There is a relatively simple proof this fact that the # of dimensions is crucial for the spin-statistics connection if one adopts the path integral view of quantum mechanics and one posits indistinguishability of particles. (Basically, the trajectories of 2 indistinguishable particles that can't occupy exactly the same point have new topological possibilities if the particles can move 3 or more dimensions rather than 2.) The simple proof of this can be found on pages 17-22 of

Richard MacKenzie "Path Integral Methods and Applications" http://arxiv.org/abs/quant-ph/0004090

which is a swell intro to path integrals in general.

Granted all this is a lot for a 10 year old, but at some point every 21st Century human being should be introduced into the awe-inspiring facts that:

1) Mathematics describes the natural world absurdly well (in this case, the fundamental particles that we infer from experiments seem classified awfully well by the irreducible representations of certain algebraic groups).

2) The path integral picture shows that a perfectly workable calculational scheme to predict what dynamics actually occur is to sum over ALL CONCEIVABLE dynamics (including things that are never seen to happen like virtual particles popping in and out of existence that have energy-momentum relations different than those predicted by special relativity and particles going alternately forward and backward in time) with complex weights exp(-iS) of equal amplitude but variable phases depending on the classical action S = (momentum times distance integral) for hypothetical trajectory.

This is marvelous stuff. I wish I were ten years old again.

I think it was Heisenberg who once remarked to his colleagues: “if we can’t explain quantum mechanics to a ten-year old boy, then we really don’t understand it ourselves” (or something very similar). When my daughter was young she was always asking me to explain things (including physics and mathematics) in terms she could understand. Often I just couldn’t, and she would sometimes think I didn’t want to, which made me feel bad. I’m always telling the guys at work that if we really understand something then we could explain it to children and managers.

I go to comment, thinking this is a rather recent post, and there's a half-dozen explanations already.

It appears an exceptional number of physicists / grad students visit this site daily...

Many of these physicists have gotten jobs as software developers, they are well educated, they devoted themselves to what everyone said would give them a long enjoyable career. And that and their mortgages gives them a substantial barrier to switching industries. And many of them are unemployed right now due to outsourcing.

But otherwise liberal fetishistic economists with tenure stick to their story: it's better that American's make their ten year old daughters grind diamonds right now, so that in the future a Cambodian girl will not have to.

But the economists don't have the math to prove this. Their dismal science is not nearly as rigorous, observable, testable, falsifiable as physics.

Luckily though, the economists have tenure. And faith.

Jeffrey Miller's explanation pretty much nailed it. I don't think it's fair to say this is something we don't really understand. We do understand it perfectly within quantum field theory - which isn't much of a qualifier since quantum field theory describes reality (a reality that needs some expansion to include gravity, but that's another topic).

I'm curious, are you the Jeff Miller that was a grad student with John Cardy?

Speaking of Periodic Tables, you may want to check out a "real" Periodic _Table_ Hours of surfing fun.

http://www.theodoregray.com/PeriodicTable/

Actually, there is a fairly "simple" explanation. The answer is that if you try to "add" the wave functions and spinnors of two spin-1/2 particles together then they cancel out. Bose-Einstein statistics determine the behavior of particles with integer spin, and Fermi-Dirac statistics determine the behavior of particles with n-tuples of spin-1/2.

The spherical harmonics essentially follow simple rules of geometric complexity increase. The simplest one is a sphere. That gives you your "s" level. Two electrons - spin-1/2 - particles can fit in this. Why just two? There are two possible spin-1/2 states that add up to integer spin.

Anyway from there it becomes more complex:
http://odin.math.nau.edu/~jws/dpgraph/Yellm.html

But the point is that it's all in the spherical harmonics and spinnors. These are mathematically put in directly into the wave functions. When you "exchange" a particle, a fermion will be anti-symmetric or flip its orientation mathematically canceling out the other particle. So while the electrons can switch places, if they try to be in one place at the same time their wave functions cancel out and the probability of finding them is zero. However since the electrons are indistinguishable particles, you can't tell if they've switched. You might think this is intuitive, but bosons such as light photons don't obey this rule. Under mathematical exchange they are symmetric, so they can all congregate at a given energy level.

So you have this electron, it settles after a while into the lowest energy state. If you pop in another electron, if you "exchange" the particles their wave functions become anti-symmetric and cancel each other out. So the only place for the new particle to go is an unfilled orbital position.

Here's a decent intro:
http://library.thinkquest.org/C0110925/html/theatom/manyelectronatoms/manyelectronatoms6.html

Brad, beware. The proper explanation of angular momentum is hard, very hard. You have explained the s orbital correctly, but your explanation of the three p orbitals ("because we live in a three-dimensional world") is incorrect, and dangerously so: now you are stuck with trying to explain why there are five d orbitals and seven f orbitals. ("The d orbitals were designed by aliens who live in five dimensions.")

Rather than torture the Ten-Year-Old (and you!) with the full-blown explanation... dare I point out that there's nothing wrong with a few arbitrary rules? If Max Planck had insisted on a sensible, consistent explanation of energy quantization before he used it and published it, he would have died in obscurity. I would encourage the Ten-Year-Old to think like a chemist: the numbers are, indeed, magic, and have been given to us by Wizards like Max Planck, Albert Einstein and Jeff Miller. Someday you too may become a Wizard... but meanwhile, if you learn the proper accounting (the magic numbers are 1, 3, 5 and 7, plus the fact that electrons come in spin-up and spin-down flavors) you can explain ionic bonds and covalent bonds, which in turn helps to explain... etc, etc.

The ocean of truth is all well and good, but it's important not to let the presence of that large and daunting ocean detract from your appreciation of the smooth pebbles and the pretty shells.

My chemistry was all of the stinks and bangs type. We did the physical stuff, but it was simpler those days, who now speaks of "Crum Brown's Rule". We were also taught that the helium group could not react with anything, then some idiot managed to make xenon fluoride, ruining another theory.

Where does the Pauli exclusion principle come from? Intuitively, it is just another way of stating the familiar idea of impenetrability: two things cannot occupy the same space at the same time. Clearest discussion of issue of orbitals, their numbers, shapes, etc. is in Linus Pauling's "General Chemistry" (Dover) written for beginners.

BTW, this whole discussion strikes me as an example of physics envy, one of the besetting sins of contemporary (mathematically oriented) economics. What we need is a generation of economists who DON'T learn calculus but concentrate on an intuitive understanding of the basic economic relationships. That way they wouldn't be so surprised when they discover, for example, that free trade between rich countries and poor countries leads to falling wages (or its equivalent, growing unemployment) in the rich countries.

I think the best way to explain the octet rule to a child involves Andrew Northrup's approach above. The harmonic series of standing waves on a stretched string represent that possible ways that energy can exist on the string given its boundary conditions. There must be nodes (no displacement) at the fixed ends.

So the waves on a string show energy bounded in one dimension.

On a drumhead there is also a series of possible modes of vibration (Google "standing waves drum" for a picture). These are standing waves in two dimensions.

s, p, d, or f electron orbitals represent the possible ways for 'electron standing waves' to exist around an atom. Just like the equations for the waves forms of the harmonic series on a guitar string are determined as a series of possibilities you describe using 'quantum number' n=1,2,3...., with electon waves you also need quantum numbers to describe the series of possibilities. With three dimensions, there are three quantum numbers to describe the solutions to the Shroedinger wave equations (n, l, m), and there is a fourth number that is a different kind of quantum number (spin).

When an atom is in the form of an octet, 2 electrons have filled the single orbital (wave form) of the s subshell (within a particular shell) and 6 electrons have filled the three orbitals of the p subshell.

How can you explain why the octet is special to a ten year old? There is the 'real' physicist's explanation involving advanced mathematics, and there is the 'intuitive' explanation that is helpful for a ten year old. (Physicists on the board, please correct or modify if possible):

Draw out horizongal lines for the 2s and the three 2p orbitals. Now ask her to imagine if that there were seven (a pair with opposite spins in the 2s and then five in the 3 2p orbitals). Which of the p orbitals goes unfilled? Which spin is the single, up or down? She will see that there is a whole range of possible configurations for seven. This is unsettled and disharmonious. The electrons are in a situation akin to notes lacking harmony. The interference of the electron waves causes the system to have a higher internal energy.

But now look at an octet. There is only one way. The electrons waves are able to exist together much more harmoniously, so the system settles down into a lower overall energy state that is hard to undo.

That's how I would try to explain it.

JOhn Baez's explanation

I initially misread this as "Joan Baez's explanation" and was really looking forward to it....

That Isaac Newton quotation is one of the reasons why I think it is crime to put youngens through four years of engineering education without giving them at least a taste of Harvard FAS style arts and science education.

For the drum head alluded to above: you live in the SF Bay Area, so you can take your daughter to the Exploratorium in SF. There, you can see many examples of 2-d standing waves, including the kettle drum:

http://www.exploratorium.edu/xref/exhibits/kettle_drum.html

When she gets a bit older (High School), your daughter can become an Explainer there. Best job I ever had in high school.

Interesting, very interesting. I liked wetzel's guitar string best. But beware! In physics there is a tendency to mix up the 'real world' and the models.

The truth is that we don't live in a three dimensional world, 3-D geometry just does a pretty good job for us at scales smaller than astronomic ones. Remember that math usually does a bad job at describing the 'natural world', but the few phenomenom it does describe absurdly well it really does describe absurdly well.

And I do think wetzel has many good reasons to quote the words trying to separate the physicist's models from the 10'yr old's. The difference is probably much smaller than you think!

Brad - "because we live in a three-dimensional world"
Bill - "Mathematics describes the natural world absurdly well"
wetzel - "There is the 'real' physicist's explanation involving advanced mathematics, and there is the 'intuitive' explanation that is helpful for a ten year old."

Mats is, I believe, onto the right track. Might I recommend, an historical/chemical approach to the issue, rather than starting with the most abstract model (in the sense of alienated from common experience).

In this way, the "rules of the game" come not from an mathematics, but as a set of observations, which were found (later, simultaneously, and even in a few cases predicted) to agree with relatavistic quantum physics. One thus can say that if the child or adult wants to understand the underlying model, he or she will have to learn a fair amount of mathematics. Or, as most, one can simply accept that the predictions of the models agree with the classification, and reduce them to a set of rules for allowed quantum numbers.

http://www.meta-synthesis.com/webbook/30_timeline/timeline.html
has a nice development of atomic and molecular structure. In discussing this with a child, even a precocious child (were we not all so), I would emphasize the role that chemistry and ideas of chemical bonding played in the development of the periodic table, and how Bohr's theory, and the concept of atomic numbe were used by G.N. Lewis to formulate the octet rule and the idea of covalent bonding. That lead, in turn to a more powerful periodic table. Pauling's formulation of electronegativity provided a stronger base yet for the periodic table and a better explanation for ionic bonding. I emphasize these points, because bothe Pauling and Lewis did their work at Berkeley and the chemistry department is likely to have some of their stuff on display (or it might be at the exploratorium). Taking the child to see such an exhibit might be a good thing in the 1066 sense.

Apropos of nothing, John Baez is Joan Baez's cousin. He's often confused with his physicist uncle Albert Baez, who was in some educational films.

I remember trying to get some sort of intuitive handle on the spin-statistics theorem in the quantum field theory course I took in graduate school, but it never quite came; the cute topological "proofs" always seem to me as if they're missing something or other. The three-dimensionality of space is definitely key, though.

Speaking of which, Mike Booth said that the existence of three p orbitals has nothing to do with the three-dimensionality of space. In one sense he's right, but in another he's wrong: there is a connection there. The different orbitals correspond to the different representations of the classical rotation group, SO(3), which are also the different kinds of tensorial objects that can exist in space. The simplest is a number without direction: that is the s orbital, one component. Then there's the vector, which has three components, one for each dimension of space: that's p. The higher orbitals correspond to various types of tensors on three-dimensional space, with the bits whose rotational behavior is already accounted for subtracted out. Might be a bit too hairy an explanation for a ten-year-old, but the connection does exist.

Also, if you're interested, there are some online java (or something) widgets called "physlets" which are very good illustrations of these and other fancy science things:

http://www.thepoorman.net/archives/001542.html

And you can make your own. Also, I believe it's true that the functions which describe the various orbitals are called "Spherical harmonics", which I further believe are built-in to Mathematica, which can graph them. If you've got a copy, that's some more hands-on stuff.

"BTW, this whole discussion strikes me as an example of physics envy"

Why? Aren't economists alllowed to have an interest in physics?

"What we need is a generation of economists who DON'T learn calculus but concentrate on an intuitive understanding of the basic economic relationships"

I.e., we need feelgood "economists" who can't rigorously demonstrate the chains of reasoning that lead them to their conclusions. Gotcha.

"That way they wouldn't be so surprised when they discover, for example, that free trade between rich countries and poor countries leads to falling wages (or its equivalent, growing unemployment) in the rich countries."

Yeah, and the moon is made of green cheese. This is the sort of absurdity "intuitive understanding" leads one to. To paraphrase Pauli, your statement isn't right - it isn't even wrong!

--- "I.e., we need feelgood "economists" who can't rigorously demonstrate the chains of reasoning that lead them to their conclusions. Gotcha." ---

Wow -- I wondered what I had wandered into. What a delightful interlude between the political arguments!

And then you guys had to return us to reality.

Ducktape: Remember Hagar the Horrible's description of life. You are born, you work, you die. Lucky Eddy, depressed by this asks: Isn't there anything else, to which Hagar replies: Yeah, if you're lucky, you can have a few beers.

Another good one:

Hagar's wife says: Kids nowadays are getting smarter and smarter!

To which Hagar responds: Bullshit! From where are all those stupids parents coming then?

"The connection between spin and 'statistics' (aka The Spin-Statistics Theorem) is one of those frustrating truths that are fundamental but don't appear to have a non-technical explanation-- at least I've never heard one"

This is a BAD analogy, but it is at least comprehensible for a child.

Imagine an oval shaped toilet seat, floating or falling freely thru the air. It can spin around an axis thru the hole - A. It can spin around an axis drawn longwise thru the longest part of the oval - B. It can spin around a similar axis drawn thru the shortest length of the oval - C.

Now let's imagine that "interaction" with the toilet seat consists of hitting it with a tennis ball.

In case 1, as it happens, we see the oval edge on, spinning on Axis A, with no spin on B or C. The probability of hitting the narrow edge with the ball is fairly small. (1)

In case 2, the thing is spinning on A and axis B, but not C. Now the flat of the seat is periodically at a maximum, facing on to the pitcher. So the probability of hitting the seat is greater.

In case 3, the thing spins on all three axis. Now the face is available to bounce the ball off even more often, a higher probability.

(1) In this first trivial case, the rate of spin has no bearing on the probability of interaction. But as we add other dimensions, it begins to matter a lot. The faster the thing spins -- in relation to how long the ball is in transit thru the volume occupied by the spinning toilet seat --the more likely the ball will hit something.
A sufficiently quickly spinning toilet seat is
almost indistinguishable from a solid oblate
spheroid -- a big football shaped thingee. You are no more likely to pitch a tennis ball thru such a spinning toilet seat than you are to get that rising helium ballon safely thru the turning blades of a ceiling fan.

All that said, working out the PARTICULAR likelihood of hitting the seat involves a bunch of statistical tools and math. If you're really that interested, do your homework.

(Do not mistake the math that describes the map for the math that describes the territory. However, the ability to WORK such math may be applicable.)

Now I understand why I was taught to lift the seat!

No, you haven't taught your daughter anything about electrons. You've just told her about 8 little blobs going round 1 big blob. The crucial qualities of electrons that enable them to have such an impact on chemistry are 1) charge 2) obey Schrodinger equation 3) have spin 1/2 and obey Pauli exclusion. Without any of these properties, the little blobs orbiting the big blob are nothing more than a fancy way of counting up to 8.

Again, without some concept of why they exist, 'orbitals' are nothing more than fancy ways of counting.

So what you're actually teaching is the idea that the numbers 2,8,8,18,... have some sort of mystical significance. In other words, the Mendeleevian situation, that there is a periodicity, but we don't know why.

Except that the ten-year-old believes in Mendeleevian periodicity because her daddy said so, whereas Mendeleev believed in it because he knew the properties of the elements.

The toilet seat is indeed a bad analogy. Statistics doesn't mean likelihood here, it means whether every solution can be occupied by at most one particle (Fermi), or by indefinitely many (Bose).

Concerning the exclusion principle - bosons *can* be in the same place at the same time. Bose-Einstein condensation means that lots of particles are in *exactly* the same state.