Wondering About Our Place

Wondering About Our Place

To be, or not to be, — that is the question: —
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles,
And by opposing end them? — To die, to sleep, —
No more; and by a sleep to say we end
The heart-ache, and the thousand natural shocks
That flesh is heir to, — 'tis a consummation
Devoutly to be wish'd. To die, to sleep; —
To sleep, perchance to dream: — ay, there's the rub;
For in that sleep of death what dreams may come,
When we have shuffled off this mortal coil,
Must give us pause: there's the respect
That makes calamity of so long life;
For who would bear the whips and scorns of time,
The oppressor's wrong, the proud man's contumely,
The pangs of despis'd love, the law's delay,
The insolence of office, and the spurns
That patient merit of the unworthy takes,
When he himself might his quietus make
With a bare bodkin? who would these fardels bear,
To grunt and sweat under a weary life,
But that the dread of something after death, —
The undiscover'd country, from whose bourn
No traveller returns, — puzzles the will,
And makes us rather bear those ills we have
Than fly to others that we know naught of?
Thus conscience does make cowards of us all;
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought;
And enterprises of great pith and moment,
With this regard, their currents turn awry,
And lose the name of action.

William Shakespeare, Hamlet Act 3, scene 1, 19–28, circa.1600

Bolero by Ravel. An der schönen blauen Donau by Strauss. Rhapsody in Blue by Gershwin. Yesterday I listened to these three pieces of music, among the most beautiful and thrilling that I know of. Each has its own peculiar emotional impact, quite different from each other and yet all calling to me in ways that I am quite sure I could never put words to. I would give anything to know exactly what they have done to my brain and nervous system, which neurons they fire in which sequence, which neurotransmitters – serotonin? dopamine? – they released or absorbed in exactly the right structures and cells of my limbic system and cerebral cortex. There any many other wondrous pieces, from Beethoven to Mozart, to Benny Goodman, the Beatles, and Bob Dylan, and more which provoke the same questions.

There is more. Today I spent several hours driving along River Road in Bucks County, Pennsylvania. The road curvingly parallels the Delaware river in many places, in others the old Delaware Canal. It is carved out of the ancient rock which lines the river, and after several days of rainfall there are numerous small and medium rivulets and waterfalls cascading from the rocks, onto the road surface, and then across it to join the river and its way to the sea. Even without these added splendors, there are the carved, ancient rocks themselves, the trees and other wild flora of May, and the occasional animal, although I did not see any deer, or wild turkey, or any of the other wild animals that inhabit the woodlands on this particular day.

I know – I know as a scientist and as a rational human being – that what I have experienced these last two days would not be possible without millions of years of Darwinian evolution sculpting senses and a nervous system and brain to allow me to experience them. If I were but a rock, I would know none of them. Even if I were a cockroach, perhaps even a fairly evolved organism such a mouse … but because I am human – a sentient being – I experience all of it; all of what gives my life so much of its meaning.

And yet I am missing something.

It is a conundrum that has been known for centuries. One that philosophers have spun and spiraled in their minds to resolve, one that scientists in the relevant fields have grappled with to this day. Some think they have solved it. Yet I beg to differ. Some very straightforward thought experiments show how perplexing it is, how much it defies simple solutions. Theists and other religious pundits think that they solved it long ago, but I believe they are just as deluded. It is the problem of the soul.

What’s this? A scientist speaking of the soul?

Soul is perhaps a bad term. It conjures up the supernatural and the religious, and that, above all, is precisely what we are trying to avoid here too, as in all the previous chapters of this book. Better words are sentience and consciousness. Sentience is somewhat the better of these two because consciousness can refer to the mind and its workings, and what we want to grab hold of is that, however our bodies and minds work, there is an indisputable “we” inside, somewhere, that experiences those workings. This we has a more or less continuous existence, minus deep sleep and any periods of anesthesia or coma we might have had, going back to as far as … well, as far as we have memories of being.

We must concede an undeniable connection to mind and body, for, as I have been emphasizing, without these things there is nothing to experience, and sentience, the experiencer, must have something to experience if it is to exist. At the same time, however, as strong as this connection is, its strength does not reach to identity. Or at least I believe I have good reason for thinking it does not. Naturally, this only deepens the mystery; how can mind / body and sentience be at the same time the same thing and yet two separate things? The answer is that it cannot, yet we struggle mightily to resolve this seeming contradiction.

Don’t think there really are contradictory aspects to it? A few thought experiments should illustrate them nicely. Here’s one: imagine we have a machine, a lá science fiction, into which you step into one booth and out pops in a different booth, by some magical technology we shall in all probability never have, an atom-by-atom exact duplicate of yourself. This, of course, is the basic idea behind matter / energy beaming devices in Star Trek, and though I heartily doubt it will ever be accomplished, it seems at least possible in theory.

Well, what would you expect? Would you still be you? I expect all of you would agree that you would be. But how about this other “person” (I put this in quotes for a specific reason), stepping forth from the other booth? Would you be him / her as well? The answer to this question would seem to have to be an unqualified no, if only for the reason that there are no neural or any other connections between the two brains, which we are quite certain is absolutely necessary for you to experience being two bodies / brains at the same time. On the other hand, if you aren’t both you, then clearly you are the original you and the duplicate, although it would have all your memories, thoughts, and feelings, and be utterly convinced it was the real you, is just as clearly someone else. All this assumes, of course, that they are anyone at all and not a non-sentient simulacrum of you – which can only be true if making at atom-by-atom-duplicate of you is still missing something, something that we have no conception of as of yet. Either way, it isn’t the real you, however identical from a known science point of view it is.

Let me illustrate the problem a different way. I often read by those working in the fields of neurology, psychology, philosophy, and all the ways these fields can be conjoined (neuropsychology, cognitive science, etc.), that sentience is a consequence of brain action, an emergent phenomenon or epiphenomenon, one deriving from brain structure from the macroscopic to the microscopic, from the whole down to neurons and axons and dendrites and neurotransmitters and synapses and, well, and the laws of physics and chemistry as we know them. But there is something wrong with this picture, something, I think, that is actually quite obvious. It is that the Me (hereafter capitalized) that experiences being me does so now in a brain that is different from the brain it experienced being me yesterday, and even more different from the brain it experienced being me a year ago, and ten years ago, twenty, forty, fifty years … all the way back as far as I can remember being sentient.

All I know is this: Richard Dawkins’ statement in his preface to his most inspirational book The Blind Watchmaker, that “Our existence once presented the greatest of mysteries, but it is a mystery no longer because it has been solved,” is both true and false. It is true in the sense that Darwinian evolution, combined with the laws of physics and chemistry in this universe, neatly explains why at this moment some six point seven billion of us humans are running around on the surface of this planet, trying to survive and more, toward what consequences we are both uncertain and afraid of. But it is false in the sense of explaining why we billions experience ourselves doing so – assuming all of us do. Yes, yes, our highly complex and massive brains are part of the solution to this part of the mystery, but – well, is it enough?

* * *

This book being largely composed of scientific ideas and arguments, I wish like anything that I could present some for this most defiant of all mysteries. Alas, I find that after half a century’s worth of reading, exploring, thinking, and probing I cannot. Which leaves me in the position of wishing it would go away, so that it might not torment me, but it refuses to do that either. It is not, mind you, that I am afraid of dying and there being nothing left of either me or Me at all, perplexing and somewhat despairing I find that prospect to be; no, it is a true intellectual riddle, one that has defied all attempts not merely to solve it but even to adequately frame it. At least the reason for this can be stated in a straightforward way. The scientific method is an objective approach to reality, combining observation with hypothesis formation and testing, using both reductionism and holism when appropriate, in the never ending quest to determine just what is out there, all around us, to the ends of the universe. And it is a noble and even, dare I use the word, holy endeavor. But how and in what ways can this method be applied to the subjective reality of experience? How can it explain Me, or You, or any of Us? The answer I keep coming up with is that it cannot, cannot explain Me, You, or any of Us, solely because these are not objective phenomenon “out there” for us to explore and dissect. We can and should dissect and explore brains, and how they work, yes. But in the end, no matter how much we discover doing so I fear we will still not have solved the problem.

The conundrum is very real, and very serious, because we know of no method but science that can reliably reveal truths about reality to us. Mysticism and religion have no chance, in fact don’t even pretend to have a chance however many pseudo-arguments their proponents hurl at us. Yet science and reason can’t will or doubletalk the issue away, either, however.

* * *

Still, I have invited you to read a chapter about this subject, and merely repeating how dumbfounded I am about it is going to wear thin very quickly. So I must make some attempt(s), some approach(es), that have a plausible chance of leading us somewhere toward understanding.

And yet, I must proceed carefully. For example, certain writers, notably Roger Penrose (The Emperor’s New Mind) have suggested that sentience emerges from some of the properties of quantum mechanics. He has apparently even identified structures in the brain, known as neural microtubules, which he claims account for consciousness / sentience in a quantum mechanical brain; part of his argument, as I understand it, is that the human mind is able to solve problems in a non-algorithmic way. While I do not claim to fully understand his arguments, other writers, notably Daniel Dennett and Stephen Pinker, have challenged Penrose, saying that in fact all the things the human mind can do can be reduced to algorithms, albeit highly complex ones, without any consideration of the physical hardware (brains, computers, etc.) that these algorithms are executed in.

Personally, I find both approaches inadequate. We really don’t have any good reason to think that a sufficiently complex computer, one that can fully emulate all the properties of a human brain, will actually be sentient. On the other hand, the mysteriousness of much of quantum mechanics shouldn’t seduce us into thinking it has anything to do with the mysteriousness of our own awareness. That is an argument that sounds powerful on first hearing, but is really quite feeble. Lots of things in this universe are still mysteries, at least to some extent, but that is no reason to assume that they are interrelated simply because they are mysterious.

Of course, this doesn’t prove that quantum processes don’t have anything to do with sentience either, so I don’t want to grind my heel into any such speculations. It’s just that there are so many other mysteries as well. For example, why do so many of the natural constants of nature happen to have the value they have – the “fine-tuning” problem that vexes so many scientists? Why are there four fundamental forces, and why do they have the relationships they have? Why is the speed of light in a vacuum what it is? Why does Planck’s constant have the value it has? And so on. Some people, even scientists, note that all these, and other, constants, have values that are absolutely necessary for intelligent beings like us to exist, so perhaps there is some kind of higher intelligence or will that has ordained them so. Other scientists shake their heads at this kind of semi-mysticism and insist that, as we understand the cosmos and the laws of physics better, we will see how they had no choice to be what they are. Or perhaps there are many, many universes – perhaps an infinite of universes – so some simply had to turn out to have the right conditions; and of course we must be living inside one of those universes, or we would not be here to ask the questions and debate the answers.

* * *

My own personal feeling – and personal feeling is exactly what it is – suggests something else to me. A hundred years ago, at the beginning of the twentieth century, there were certain phenomena that stubbornly defied explanation by the then known existing laws of nature. The structure of the atom, as I have already mentioned, is probably the most famous. The conflict between Maxwell’s laws of electrodynamics and Newton’s laws of motion were another. As was the spectrum of blackbody radiation. The heat capacity of multiatomic gasses, and the photoelectric effect were a third and a fourth.

The solutions to these vexing problems involved, not merely new theories based on the existing laws of physics, but new paradigms, new ways of thinking, which opened up a new universe of laws and theories and hypotheses. These new paradigms were so challenging that many scientists have had a hard time accepting them even to this day, while those who do still sometimes puzzle and scratch their heads at what they really mean. Quantum mechanics. Special and General Relativity. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD). The expanding universe and the notion of a beginning to everything, the Big Bang (though this is being challenged today in some quarters), and perhaps an end to all things, including time. The idea that space and time, matter and energy, are related in ways that you cannot treat them as separate phenomena. The use of mathematical group theory to explain the plethora of mass-bearing and force-bearing particles in nature, and the relationships between those particles. The idea of inflation in the very early universe, and how it might have led to many universes forming. And now of strings and supersymmetry.

Standing here, at the opening of the twenty-first century, I can envision a similar revolution in paradigms arising to answer the questions I address in this chapter. But as I said in chapter seven, looking at it now, it is science fiction. Perhaps even fantasy. For example, here’s one possibility: perhaps we will create a “super” brain, one composed of electronics and neuronics, that we can all interface with or even become part of. This brain might eventually spread throughout the solar system and then beyond, perhaps to ultimately fill the entire universe. Perhaps this is when humanity learns its meaning and destiny, and all questions are answered. Even those billions who have lived and died may be reincarnated into this star-spanning mind, and not just humans but every other sentient race that has lived and died, here and elsewhere in the universe.

Following this line of prognostication, maybe sentience is something like another property of the universe, one which requires certain conditions, such as those that occur in our brains, to manifest itself. But if it is that, a property, then what kind of property is it? It isn’t a force, or a kind of particle. Something interwoven into the fabric of spacetime itself? But how? And in what way?

* * *

Sometimes I wonder if the Buddhist concept of Maya and Enlightenment can help us here. Maya is the illusion we all experience, that of being separate beings, apart from each other and the rest of the universe, struggling to find our way through life, and ultimately dying in this illusion. The experience of Enlightenment is supposed to be one in which all Maya drops away and you are fully aware of being one with everyone and everything – an experience regarded as impossible to capture in words or any other physical medium. Yes, I wonder if Buddhism is on to something here. It would have to defy explanation by language or any other form of normal communication. One would have to either experience it, or have no idea what it is. That does sound like it has a sporting chance of being right, or at least it does to me.

But if so, then this does imply that there are laws and properties of reality that we do not, and perhaps can never, understand intellectually, because they are not susceptible to scientific analysis? That they work beneath, or above, the radar of our intellects, however hard we try?

If all this is true, however, then what should we do? What can we do?

What we must do, I maintain yet again, is not give in to despair simply because we don’t know the solution to the puzzle, and may never know the solution to it. Also, remember that many mysteries have resisted solution for centuries, only to finally be solved by an application of new paradigms and ways of looking at things. Above all, we must not give up, even if things appear hopeless. A hundred years from now, we may find ourselves shaking our collective heads at our current confusion. I am tempted, however, to call this question – the question of sentience – the ultimate question, to which all others are sublimated. I really do believe that if and when we solve it, there will be a collective sigh of satisfaction greater than the solution to any question that has proceeded it.

* * *

Somehow or other, whether by luck or design or an intermingling of the two, we find ourselves where and when we are. We inhabit a planet orbiting a yellow dwarf star at the edge of a rather typical spiral galaxy. The star is but one among billions in the galaxy it has found itself in, and the galaxy may be one of trillions in a universe many billions of years old and perhaps far, far older. In all that, our individual lives occupy only a few decades of time, a century if we are fortunate. There seems to be nothing particularly special about this where and when we exist, except that is one of the few places we could be in the universe, perhaps the only even, and perhaps one of the few universes we could be in. Maybe the only one. Moreover, we do not know what will happen, not merely to ourselves as individuals, but to us as a species over the next few centuries.

We have spent thousands of years beating our heads against an invincible wall, wondering what the answer to all this is, and for all our pounding still pretty much have no idea. Of course, the answer may well be that “this is all there is”, that once our bodies cease to function that is the end of both us and Us, and no beliefs, religions, philosophies, or wishful thinking can change that. Sad though that is in one respect, even if it is true I believe we should be grateful, grateful for the opportunity to have existed at all and had the opportunity to marvel at this universe we have manifested in. It is even really not so sad either, when you think about it; after all, in the billions or trillions or infinity of years before we existed we suffered not one iota for not being, so certainly after we are gone we will not suffer at all then either. It is only sad, to me at least, in that We will cease to exist with so many wonderful questions unanswered. That, I have to admit, is a bitter pill to force down.

But let us assume that this is not the case. Let us imagine that sentience, while inactive without a brain to model the universe about it, nonetheless still exists in some potential form. I use the word potential with a very specific meaning. We speak of potential energy, as when an object is raised to a certain height, or an elastic material stretched, or as a chemical potential that can lead to an energetic reaction. The energy does not exist in any active form, yet it is still there, waiting to be manifested. Quite possibly, sentience without a brain with which to experience some kind of reality, can be held in an analogous potential form. What would that mean? One possibility is the repeated incarnations of the “soul” as claimed by many Eastern religions, although I am not certain I can believe in that.

I have difficulties with this, because in Eastern religions, the soul can reincarnate as almost anything: another person, an animal, a plant, or even a rock. Yet rocks and plants, and probably even most animals, do not possess the capacity for sentience, as they lack a sufficiently complex brain and nervous system. There are other practical problems as well. Even if we reincarnate as human beings, since the number of human beings on this planet has been exponentially increasing over thousands of years, where are all the new souls to come from to inhabit all these new bodies? There is a disparity here that is hard to reconcile.

There is another tack I would like to try. I am an aficionado of the television series House, which, if you aren’t (fie on you!), is about the brilliant but renegade and rather misanthropic Dr. Gregory House and the characters and cases which spin around him in a mythical teaching hospital between Princeton and Plainsboro, NJ. One of the episodes involves Dr. House temporarily reviving a patient who has been in a coma for ten years, for the purpose of extracting family background in order to save the coma patient’s son’s life (it ends with the coma patient committing suicide in order to donate his heart to his dying son – now you know why I say fie on you if you don’t watch it). Before I begin, I have to say I find the premises of this episode highly dubious at the least: someone who has been in a coma for ten years will have undergone so much muscle atrophy and coordination loss that I doubt he could walk, let alone drive a car to Atlantic City and basically act like someone who has just woken from a short nap. But that is beside the point I want to make.

No, my question is: is the sentience that results from the coma awakening, and spends his last day in a quest for the perfect hoagie then ends by sacrificing his life for his son’s, the same sentience that ended ten years earlier? An even better question might be, does this question even make any sense? The re-awakened father would of course insist that he his in every way conceivable the same person, but how much does that utterly sincere insistence count for? And what possible tests and / or measurements could we make to settle the issue?

I have to confess to something. This is not a mere academic issue to me. I was once in a coma, from which I fortunately awoke after several days. But does that make any difference? Like that father in House, I absolutely insist that I am the same Me that fell into that coma, but how can I, or anyone, really know? And again I ask, does the question even make sense?

Maybe it is an absurd question. Or, not so much absurd as worded incorrectly. Perhaps what seems to happen to Us in those moments, or days, or years, when we still exist but We do not is that time ceases to exist for Us. Just like, according to Einstein’s Special and General Theories of Relativity, time ceases to exist under certain conditions – if we were to ride on a beam of light or (if I understand what I have read correctly) fall into an infinitely deep gravity well – time comes to a complete stop for Us whenever the conditions needed to manifest Us ceases to exist. The question then is, do those conditions exist only within our own brains, for if so, then our current lives are the only ones We can ever manifest in?

* * *

I suspect that I have frustrated and dissatisfied you, dear reader, for I keep promising answers to this deepest of questions, but invariably find myself only circling about and finding myself at my own beginnings, my own head-shaking ignorance and failure of my own imagination and curiosity to solve this most impenetrable of puzzles

Will I give up then? No, first of all because I see no way of letting go of my curiosity and wonder and imagination, without letting go of what it means to be a living, sentient mind in a universe we still have so much to explore within. If there are places and times I have no concept of how to reach, then I am simply going to accept them for the time being, and hope that at some point in the future my eyes will start to open about them. Nor will I relinquish the scientific approach to thinking about reality, for it has served us so well, and has provided answers to what appeared to be impenetrable mysteries, and so I cannot give up hope on it, certainly not at this time and place in humanity’s evolution. Perhaps, of course, these things will lead to my death with so many important questions unanswered, and, yes, as I have admitted, that disturbs me. But, as I said, to stop now and lay down all of the weapons and tools of the mind and surrender to ignorance; that is something I cannot even conceive of doing. I would certainly die of despair if I even so much as tried.

So we have come around and around, and it the end must still admit that this greatest of mysteries has not yielded to science, at least not yet. And yet, that is all right. Mysteries are the lifeblood of science, and indeed of all our wonderings and imaginative escapades. Maybe, like the character in the Monty Python sketch I mentioned early in this book, we even need them, need these challenges to our curiosity, as though they are part of what gives our lives meaning. I know that they have given my life at least a healthy part of its meaning.

* * *

"There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened."

Douglas Adams, The Hitchhiker's Guide to the Galaxy (1979)

"Now my own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose."

JBS Haldane, Possible Worlds and Other Papers (1927), p. 286

As I said at the beginning, a large part of this book is about what it means to be human, with curiosity, wonder, and imagination being fundamental parts of the answer. I also stressed the special importance of imagination, supplemented by technology, along with the warning that if we really wish to understand the universe we live in, we must not limit ourselves to our sensory experiences and our intuitions about them. We saw how important that became once we started deviating from the norms of our existence, whether in space or time. When we are dwelling in the world of the ultra-small or large, slow or fast, the laws of physics deviate from common sense in ways we would never have predicted. Phenomena such as the uncertainty principle and the depths of geologic time, time dilation and the bending of spacetime become increasingly important as we move further and further away from the norms of our everyday existence. We found that if we allowed those deviations to take us logically wherever they went then, however strange our discoveries, they could be integrated into the whole of understanding.

We also came to understand that the paths we took were our personal ones, each unique to us even if, ultimately, we all found ourselves in the same place in the end, that end being still finding ourselves facing the same ages old mysteries of our own existence. This is one of the crucial paradoxes of the human condition, I believe; that we all experience our lives as infinitely separated individuals, while underneath we are all tied together by the same laws, the same processes, the same foundations. It is as though each of us perceives ourselves as alone in a tiny boat on the open ocean, winds whipping and waves constantly washing water into the boat, forcing us to bale with all our strength and persistence just to stay afloat, while in fact, ironically, we are all collectively in one huge boat, with each of us making our tiny contribution to keeping the boat afloat and headed for – what land we are uncertain, but whatever it is we shall all arrive there together, in the end.

In the end, maybe this is our place in the scheme(s) of things. I am not the first person to speculate that we may be nothing more than reality’s attempt to comprehend itself. If so however, then we are faced with another mystery, that of how reality can have intentions or goals at all instead of being nothing more than the blind working out of physical laws. A mystery which only becomes deeper if we assume that intelligence, in some form, is itself part of that reality.

I stated at the outset of this book that I do not intend to give in to nihilism or despair, and I will take the time to reaffirm this promise again. Somehow we reasoning, questioning, imagining animals have found ourselves in this universe, and that alone should provoke our minds to keep trying to discover how and why. Indeed it is my view that we are probably still closer to the beginning of our quest than the end. I will also take the time to state my personal gratitude that we are in the middle of it.

We are born as, and grow up into, creatures of curiosity, wonderment, imagination, and rational thought. I do not care what nation or culture you were raised into, what you were taught, or what experiences you have had. Merely by being human, you still have all these traits within you, each one waiting to boil up to the surface at any time. I know that I have been astonishingly fortunate in this respect, in one sense more than most in this world, but at the same time I can’t believe that I have been any more gifted in these things than anyone else. I have just had the good fortune to have these things nurtured and encouraged.

I remember being a child with all these things within me, and nothing gives me more pleasure than today, at fifty-three years of age, to discover that same child just as strong. Though I have spent a half-century’s worth of growing, experiencing, maturing; though I have married, raised children, and known “The heart-ache, and the thousand natural shocks that flesh is heir to” including pain I thought I would never recover from or survive; though I have stared into space and wondered what the point of those pains were … that part of me has never been diminished or defeated in any way.

And so there is nothing more for me to do except present myself as an inspiration, and as a hope. If you have any doubts, then go somewhere where the lights and pollution of the city cannot find you. Wait until the sun goes down, and then lie on the grass, staring skywards at the stars. Stare, and remember that for each one you see, there are trillions beyond your sight, beyond the sight of the most powerful telescopes for that matter. Gaze at the fierce beacons pouring their fires down upon you, and wonder. Though this universe we live in is far vaster than our imaginations can even begin to encompass, I believe you will know what I mean. Though we are but the most mortal of beings, barely eking a century’s worth of experience of the billions of years those beacons have shown, each of us has still our own meaning, our own purpose, whether we know it or not. I believe this will dispel all those doubts.

The Implied Hypothesis in Science

In the standard scientific paradigm, observations lead to hypothesis (educated guesses to explain the observations), which lead to further observations and experiments designed to prove or disprove the hypothesis.  If the hypothesis continues to pass the tests against it and be supported by the tests for it at some point we call it a theory -- which in scientific parlance, means a fact.  Oh, of course it must not contradict any other theories, otherwise we will have to dig deeper and discover which is true; or it might turn out, as with General Relativity and Quantum Mechanics, they contradict but are true within their own respective realms.  If this happens we suspect an even deeper theory, connecting the two and making sense in all realms.  This is where we are in physics right now, though we are making progress.

I want to talk about a different subject:  Darwinian evolution.  Since Darwin's publication of The Origin of Species in 1859, it was swiftly accepted by most scientists as true, even though there were still a number of details yet to be worked out:  for example, Darwin's ideas on inheritance were in conflict with his theory, mainly due to ignorance of Gregor Mendel's publication on particle gene inheritance two years earlier in an obscure article Darwin (or hardly anyone) read at the time.  It was only until 1900 that Mendel's work was rediscovered and that chestnut laid to rest.

However, there was a more serious problem with Darwinism at the time, and that concerned the sun.  You see, at the time, nobody knew where the sun's prodigious energies came from; there was two prevailing hypotheses, neither of them adequate.  One is that the energy came from chemical burning, like a huge sphere in space of carbon or some flammable material.  Never mind we knew by then there was no oxygen in space and that this element was essential for combustion.  The other hypothesis was that the gravitational contraction of the sun provided the energy.  This stretched the sun's energy out some millions of years, but was still not enough to satisfy evolution, which required hundreds of millions to billions of  years

If Darwin was right then an automatic, or implied hypothesis came from it:  there had to be a source of energy for the sun and stars which could power them for billions of years.  Yet is was not until 1905 that Albert Einstein suggested one with his famous equation:  energy(E) = mass(M) times the speed of light(C) squared.  some twenty years later the late Hans Bethe described the hydrogen fusion reactions (which create helium) in the core of the sun, and which would keep the sun alive for at least ten billion years (We are now about halfway through its lifetime).  Gravity had already explained how stars are formed from interstellar dust and gas, yielding the final piece to the puzzle in place.

The Idiot’s Guide to Making Atoms

Avagadro’s Number and Moles

Writing this chapter has reminded me of the opening of a story by a well-known science fiction author (whose name, needless to say, I can’t recall): “This is a warning, the only one you’ll get so don’t take it lightly.” Alice in Wonderland or “We’re not in Kansas anymore” also pop into mind. What I mean by this is that I could find no way of writing it without requiring the reader to put his thinking (and imagining) cap on. So: be prepared.

A few things about science in general before I plunge headlong into the subject I’m going to cover. I have already mentioned the way science is a step-by-step, often even torturous, process of discovering facts, running experiments, making observations, thinking about them, and so on; a slow but steady accumulation of knowledge and theory which gradually reveals to us the way nature works, as well as why. But there is more to science than this. This more has to do with the concept, or hope I might say, of trying to understand things like the universe as a whole, or things as tiny as atoms, or geological time, or events that happen over exceedingly short times scales, like billionths of a second. I say hope because in dealing with such things, we are extremely removed from reality as we deal with it every day, in the normal course of our lives.

The problem is that, when dealing with such extremes, we find that most of our normal ideas and expectations – our intuitive, “common sense”, feeling grasp of reality – all too frequently starts to break down. There is of course good reason why this should be, and is, so. Our intuitions and common sense reasoning have been sculpted by our evolution – I will resist the temptation to say designed, although that often feels to be the case, for, ironically, the same reasons – to grasp and deal with ordinary events over ordinary scales of time and space. Our minds are not well endowed with the ability to intuitively understand nature’s extremes, which is why these extremes so often seem counter-intuitive and even absurd to us.

Take, as one of the best examples I know of this, biological evolution, a lá Darwin. As the English biologist and author Richard Dawkins has noted several times in his books, one of the reasons so many people have a hard time accepting Darwinian evolution is the extremely long time scale over which it occurs, time scales in the millions of years and more. None of us can intuitively grasp a million years; we can’t even grasp, for that matter, a thousand years, which is one-thousandth of a million. As a result, the claim that something like a mouse can evolve into something like an elephant feels “obviously” false. But that feeling is precisely what we should ignore in evaluating the possibility of such events, because we cannot have any such feeling for the exceedingly long time span it would take. Rather, we have to evaluate the likelihood using evidence and hard logic; commonsense can seriously mislead us.

The same is true for nature on the scale of the extremely small. When we start poking around in this territory, around with things like atoms and sub-atomic particles, we find ourselves in a world which bears little resemblance to the one we are used to. I am going to try various ways of giving you a sense of how the ultra-tiny works, but I know in advance that no matter what I do I am still going to be presenting concepts and ideas that seem, if anything, more outlandish than Darwinian evolution; ideas and concepts that might, no, probably will, leave your head spinning. If it is any comfort, they often leave my mind spinning as well. And again, the only reason to accept them is that they pass the scientific tests of requiring evidence and passing the muster of logic and reason; but they will often seem preposterous, nevertheless.

First, however, let’s try to grab hold of just how tiny the world we are about to enter is. Remember Avogadro’s number, the number of a mole of anything, from the last chapter? The reason we need such an enormous number when dealing with atoms is that they are so mind-overwhelmingly small. When I say mind-overwhelmingly, I really mean it. A good illustration of just how small that I enjoy is to compare the number of atoms in a glass of water to the number of glasses of water in all the oceans on our planet. As incredible as it sounds, the ratio of the former to the latter is around 10,000 to 1. This means that if you fill a glass with water, walk down to the seashore, pour the water into the ocean and wait long enough for it to disperse evenly throughout all the oceans (if anyone has managed to calculate how long this would take, please let me know), then dip your now empty glass into the sea and re-fill it, you will have scooped up some ten thousand of the original atoms that it contained. Another good way of stressing the smallness of atoms is to note that every time you breathe in you are inhaling some of the atoms that some historical figure – say Benjamin Franklin or Muhammad – breathed in his lifetime. Or maybe just in one of their breaths; I can’t remember which – that’s how hard to grasp just how small they are.

One reason all this matters is that nature in general does not demonstrate the property that physicists and mathematicians call “scale invariance.” Scale invariance simply means that, if you take an object or a system of objects, you can increase its size up to as large as you want, or decrease it down, and its various properties and behaviors will not change. Some interesting systems that do possess scale invariance are found among the mathematical entities called fractals: no matter how much you enlarge or shrink these fractals, their patterns repeat themselves over and over ad infinitum without change. A good example of this is the Koch snowflake:

which is just a set of repeating triangles, to as much depth as you want. There are a number of physical systems that have scale invariance as well, but, as I just said, in general this is not true. For example, going back to the mouse and the elephant, you could not scale the former up to the size of the latter and let it out to frolic in the African savannah with the other animals; our supermouse’s proportionately tiny legs, for one thing, would not be strong enough to lift it from the ground. Making flies human sized, or vice-versa, run into similar kinds of problems (a fly can walk on walls and ceilings because it is so small that electrostatic forces dominate its behavior far more than gravity).

Scale Invariance – Why it Matters

One natural phenomenon that we know lacks scale invariance, we met in the last chapter is matter itself. We know now that you cannot take a piece of matter, a nugget of gold for example, and keep cutting it into smaller and smaller pieces, and so on until the end of time. Eventually we reach the scale of individual gold atoms, and then even smaller, into the electrons, protons, and neutrons that comprise the atoms, all of which are much different things than the nugget we started out with. I hardly need to say that all elements, and all their varied combinations, up to stars and galaxies and larger, including even the entire universe, suffer the same fate. I should add, for the sake of completeness, that we cannot go in the opposite direction either; as we move toward increasingly more massive objects, their behavior is more and more dominated by the field equations of Einstein’s general relativity, which alters the space and time around and inside them to a more and more significant degree.

Why do I take the time to mention all this? Because we are en route to explaining how atoms, electrons and all, are built up and how they behave, and we need to understand that what goes on in nature at these scales is very different than what we are accustomed to, and that if we cannot adopt our thinking to these different behaviors we are going to find it very tough, actually impossible, sledding, indeed.

In my previous book, Wondering About, I out of necessity gave a very rough picture of the world of atoms and electrons, and how that picture helped explained the various chemical and biological behaviors that a number of atoms (mostly carbon) displayed. I say “of necessity” because I didn’t, in that book, want to mire the reader in a morass of details and physics and equations which weren’t needed to explain the things I was trying to explain in a chapter or two. But here, in a book largely dedicated to chemistry, I think the sledding is worth it, even necessary, even if we do still have to make some dashes around trees and skirt the edges of ponds and creeks, and so forth.

Actually, it seems to me that there are two approaches to this field, the field of quantum mechanics, the world we are about to enter, and how it applies to chemistry. One is to simply present the details, as if out of a cook book: so we are presented our various dishes of, first, classical mechanics, then the LaGrangian equation of motion and Hamiltonium operators and so forth, followed by Schrödinger’s various equations and Heisenberg’s matrix approach, with eigenvectors and eigenvalues, and all sorts of stuff that one can bury one’s head into and never come up for air. Incidentally, if you do want to summon your courage and take the plunge, a very good book to start with is Melvin Hanna’s Quantum Mechanics in Chemistry, of which I possess the third edition, and go perusing through from time to time when I am in the mood for such fodder.

The problem with this approach is that, although it cuts straight to the chase, it leaves out the historical development of quantum mechanics, which, I believe, is needed if we are to understand why and how physicists came to present us with such a peculiar view of reality. They had very good reasons for doing so, and yet the development of modern quantum mechanical theory is something that took several decades to mature and is still in some respects an unfinished body of work. Again, this is largely because some it its premises and findings are at odds with what we would intuitively expect about the world (another is that the math can be very difficult). These are premises and findings such as the quantitization of energy and other properties to discrete values in very small systems such as atoms. Then there is Heisenberg’s famous though still largely misunderstood uncertainly principle (and how the latter leads to the former).

Talking About Light and its Nature

A good way of launching this discussion is to begin with light, or, more precisely, electromagnetic radiation. What do I mean by these polysyllabic words? Sticking with the historical approach, the phenomena of electricity and magnetism had been intensely studied in the 1800s by people like Faraday and Gauss and Ørsted, among others. The culmination of all this brilliant theoretical and experimental work was summarized by the Scottish physicist James Clerk Maxwell, who in 1865 published a set of eight equations describing the relationships between the two phenomena and all that had been discovered about them. These equations were then further condensed down into four and placed in one of their modern forms in 1884 by Oliver Heaviside. One version of these equations is (if you are a fan of partial differential equations):

Don’t worry if you don’t understand this symbolism (most of it I don’t). The important part here is that the equations predict the existence of electromagnetic waves propagating through free space at the speed of light; waves rather like water waves on the open ocean albeit different in important respects. Maxwell at once realized that light must be just such a wave, but, more importantly, that there must be a theoretically infinite number of such waves, each with different wavelengths ranging from the very longest, what we now call radio waves, to the shortest, or gamma rays. An example of such a wave is illustrated below:

To assist you in understanding this wave, look at just one component of it, the oscillating electric field, or the part that is going up and down. For those not familiar with the idea of an electric (or magnetic) field, simply take a bar magnet, set it on a piece of paper, and sprinkle iron filings around it. You will discover, to your pleasure I’m certain, that the filings quickly align themselves according to the following pattern:

The pattern literally traces out the, in this case, magnetic field of the bar magnet, but we could have used an electrically charged source to produce a somewhat different pattern. The point is, the field makes the iron filings move into their respective positions; furthermore, if we were to move the magnet back and forth or side to side the filings would continuously move with it to assume their desired places. This happens because the outermost electrons in the filings (which, in addition to carrying an electric charge, also behave as very tiny magnets) are basically free to orient themselves anyway they want, so they respond to the bar’s field with gusto, in the same way a compass needle responds to Earth’s magnetic field. If we were using an electric dipole it would be the electric properties of the filings’ electrons performing the trick, but the two phenomena are highly interrelated.

Go back to the previous figure, of the electromagnetic wave. The wave is a combination of oscillating electric and magnetic fields, at right angles (90°) to each other, propagating through space. Now, imagine this wave passing through a wire made of copper or any other metal. Hopefully you can perceive by now that, if the wave is within a certain frequency range, it will cause the electrons in the wire’s atoms to start spinning around and gyrating in order to accommodate the changing electric and magnetic fields, just as you saw with the iron filings and the bar magnet. Not only would they do that, but the resulting electron motions could be picked up by the right kinds of electronic gizmos, transistors and capacitators and resistors and the like – here, we have just explained the basic working principle of radio transmission and receiving, assuming the wire is the antenna. Not bad for a few paragraphs of reading.

This sounds all very nice and neat, yet it is but our first foot into the door of what leads to modern quantum theory. The reason for this is that this pat, pretty perception of light as a wave just didn’t jibe with some other phenomena scientists were trying to explain at the end of the nineteenth century / beginning of the twentieth century. The main such phenomena along these lines which quantum thinking solved were the puzzles of the so-called “blackbody” radiation spectrum and the photo-electric effect.

Blackbody Radiation and the Photo-electric Effect

If you take an object, say, the tungsten filament of the familiar incandescent light bulb, and start pumping energy into it, not only will its temperature rise but at some point it will begin to emit visible light: first a dull red, then brighter red, then orange, then yellow – the filament eventually glows with a brilliant white light, meaning all of the colors of the visible spectrum are present in more or less equal amounts, illuminating the room in which we switched the light on. Even before it starts to visibly glow, the filament emits infrared radiation, which consist of longer wavelengths than visible red, and is outside our range of vision. It does so in progressively greater and greater amounts and shorter and shorter wavelengths, until the red light region and above is finally reached. At not much higher temperatures the filament melts, or at least breaks at one of its ends (which is why it is made from tungsten, the metal with the highest melting point), breaking the electric current and causing us to replace the bulb.

The filament is a blackbody in the sense that, to a first approximation, it completely absorbs all radiation poured onto it, and so its electromagnetic spectrum depends only on its temperature and not any on properties of its physical or chemical composition. Other such objects which are blackbodies include the sun and stars, and even our own bodies – if you could see into right region of the infrared range of radiation, we would all be glowing. A set of five blackbody electromagnetic spectra are illustrated below:

Examine these spectra, the colored curves, carefully. They all start out at zero on the left which is the shortest end of the temperature, or wavelength (λ, a Greek letter which is pronounced lambda) scale; the height of the curves then quickly rises to a maximum λ at a certain temperature, followed by a gradual decline at progressively lower temperatures until they are basically back at zero again. What is pertinent to the discussion here is that, if we were living around 1900, all these spectra would be experimental; it was not possible then, using the physical laws and equations known at the end of the 1800s, to explain or predict them theoretically. Instead, from the laws of physics as known then, the predicted spectra would simply keep increasing as λ grew shorter / temperature grew higher, resulting it what was called “the ultraviolet catastrophe.”

Another, seemingly altogether different, phenomenon that could not be explained using classical physics principles was the so-called photoelectric effect. The general idea is simple enough: if you shine enough light of the right wavelength or shorter onto certain metals – the alkali metals, including sodium and potassium, show this effect the strongest – electrons will be ejected from the metal, which can then be easily detected:

This illustration not only shows the effect but also the problem 19’th century physicists had explaining it. There are three different light rays shown striking the potassium plate: red at a wavelength of 700 nanometers or nm (an nm is a billionth of a meter), green at 550 nm, and purple at 400 nm. Note that the red light fails to eject any electrons at all, while the green and purple rays eject only one electron, with the purple electron escaping with a higher velocity, meaning higher energy, than the green.

The reason this is so difficult to explain with the physics of the 1800’s is that physics then defined the energy of all waves using both the wave’s amplitude, which is the distance from crest or highest point to trough or lowest point, in combination with the wavelength (the shorter the wavelength the more waves can strike within a given time). This is something you can easily appreciate by walking into the ocean until the water is up to your chest; both the higher the waves are and the faster they hit you, the harder it is to stay on your feet.

Why don’t the electrons in the potassium plate above react in the same way? If light behaved as a classical wave it should not only be the wavelength but the intensity or brightness (assuming this is the equivalent of amplitude) that determines how many electrons are ejected and with what velocity. But this is not what we see: e.g., no matter how much red light, of what intensity, we shine on the plate no electrons are emitted at all, while for green and purple light only the shortening of the wavelength in and of itself increases the energy of the ejected electrons, once again, regardless of intensity. In fact, increasing the intensity only increases the number of escaping electrons, assuming any escape at all, not their velocity. All in all, a very strange situation, which, as I said, had physicists scratching their heads all over at the end of the 1800s.

The answers to these puzzles, and several others, comes back to the point I made earlier about nature not being scale invariant. These conundrums were simply insolvable until scientists began to think of things like atoms and electrons and light waves as being quite unlike anything they were used to on the larger scale of human beings and the world as we perceive it. Using such an approach, the two men who cracked the blackbody spectrum problem and the photoelectric effect, Max Planck and Albert Einstein, did so by discarding the concept of light being a classical wave and instead, as Newton had insisted two hundred years earlier, thought of it as a particle, a particle which came to be called a photon. But they also did not allude to the photon as a classical particle either but as a particle with a wavelength; furthermore, that the energy E of this particle was described, or quantized, by the equation

in which c was the speed of light, λ the photon’s wavelength, and h was Planck’s constant, the latter of which is equal to 6.626 × 10^-34 joules  seconds – please note the extremely small value of this number. In contrast to our earlier, classical description of waves, the amplitude is to be found nowhere in the equation; only the wavelength, or frequency, of the photon determines its energy.

If you are starting to feel a little dizzy at this point in the story, don’t worry; you are in good company. A particle with a wavelength? Or, conversely, a wave that acts like a particle even if only under certain circumstances? A wavicle? Trying to wrap your mind around such a concept is like awakening from a strange dream in which bizarre things, only vaguely remembered, happened. And the only justification of this dream world is that it made sense of what was being seen in the laboratories of those who studied these phenomena. Max Planck, for example, was able, using this definition, to develop an equation which correctly predicted the shapes of blackbody spectra at all possible temperature ranges. And Einstein elegantly showed how it solved the mystery of the photoelectric effect: it took a minimum energy to eject an electron from a metal atom, an energy dictated by the wavelength of the incoming photon; the velocity or kinetic energy of the emitted electron came solely from the residual energy of the photon after the ejection. The number of electrons freed this way was simply equal to the number of the photons that showered down on the metal, or the light’s intensity. It all fit perfectly. The world of the quantum had made its first secure foot prints in the field of physics.

There was much, much more to come.

The Quantum and the Atom

Another phenomena that scientists couldn’t explain until the concept of the quantum came along around 1900-1905 was the atom itself. Part of the reason for this is that, as I have said, atoms were not widely accepted as real, physical entities until electrons and radioactivity were discovered by people like the Curies and J. J. Thompson, Rutherford performed his experiments with alpha particles, and Einstein did his work on Brownian motion and the photo-electric effect (the results of which he published in 1905, the same year he published his papers on special relativity and the E = mc² equivalence of mass and energy in the same year, all at the tender age of twenty-six!). Another part is that, even if accepted, physics through the end of the 1800s simply could not explain how atoms could be stable entities.

The problem with the atomic structure became apparent in 1911, when Rutherford published his “solar system” model, in which a tiny, positively charged nucleus (again, neutrons were not discovered until 1932 so at the time physicists only knew about the atomic masses of elements) was surrounded by orbiting electrons, in much the same way as the planets orbit the sun. The snag with this rather intuitive model involved – here we go both with not trusting intuition and nature not being scale invariant again – something physicists had known for some time about charged particles.

When a charged particle changes direction, it will emit electromagnetic radiation and thereby lose energy. Orbiting electrons are electrons which are constantly changing direction and so, theoretically, should lose their energy and fall into the nucleus in a tiny fraction of a second (the same is true with planets orbiting a sun, but it takes many trillions of years for it to happen). It appeared that the Rutherford model, although still commonly evoked today, suffered from a lethal flaw.

And yet this model was compelling enough that there ought to be some means of rescuing it from its fate. That means was published two years later, in 1913, by Niels Bohr, possibly behind Einstein the most influential physicist of the twentieth century. Bohr’s insight was to take Planck’s and Einstein’s idea of the quantitization of light and apply it to the electrons’ orbits. It was a magnificent synthesis of scientific thinking; I cannot resist inserting here Jacob Bronowski’s description of Bohr’s idea, from his book The Ascent of Man:

Now in a sense, of course, Bohr’s task was easy. He had the Rutherford atom in one hand, he had the quantum in the other. What was there so wonderful about a young man of twenty-seven in 1913 putting the two together and making the modern image of the atom? Nothing but the wonderful, visible thought-process: nothing but the effort of synthesis. And the idea of seeking support for it in the one place where it could be found: the fingerprint of the atom, namely the spectrum in which its behavior becomes visible to us, looking at it from outside.

Reading this reminds me of another feature of atoms I have yet to mention. Just as blackbodies emit a spectrum of radiation, one based purely on their temperature, so did the different atoms have their own spectra. But the latter had the twist that, instead of being continuous, they consisted of a series a sharp lines and were not temperature dependent but were invoked usually by electric discharges into a mass of the atoms. The best known of these spectra, and the one shown below, is that of atomic hydrogen (atomic because hydrogen usually exists as diatomic molecules, H₂, but the electric discharge also dissociates the molecules into discrete atoms):

This is the visible part of the hydrogen atom spectrum, or so-called Balmer series, in which there are four distinct lines: from right to left, the red one at 656 nanometers (nm), the blue-green at 486 nm, the blue-violet at 434 nm, and the violet at 410 nm.

Bohr’s dual challenge was explain both why the atom, in this case hydrogen, the simplest of atoms, didn’t wind down like a spinning top as classical physics predicted, and why its spectrum consisted of these sharp lines instead of being continuous as the energy is lost. As said, he accomplished both tasks by invoking quantum ideas. His reasoning was more or less as this: the planets in their paths around the sun can potentially occupy any orbit, in the same continuous fashion we have learned to expect from the world at large. As we now might begin to suspect however, this is not true for the electrons “orbiting” (I put this in quotes because we shall see that this is not actually the case) the nucleus. Indeed, this is the key concept which solves the puzzle of atomic structure, and which allowed scientists and other people to finally breathe freely while they accepted the reality of atoms.

Bohr kept the basic solar system model, but modified it by saying that there was not a continuous series of orbits the electrons could occupy but instead a set of discrete ones, in-between which there was a kind of no man’s land where electrons could never enter. Without going into details you can see how, at one stroke, this solved the riddle of the line spectra of atoms: each spectral line represented the transition of an electron from a higher orbit (more energy) to a lower one (less energy). For example, the 656 nm red line in the Balmer spectrum of hydrogen is caused by an electron dropping from orbit level three to orbit level two:

Here again we see the magical formula hυ, the energy of the emitted photon, in this case being equal to E, the difference in energy between the two orbits. Incidentally, if the electron falls further inward, from orbit level two to orbit level one – this is what is known as the Lyman series, in this case accompanied by a photon emission of 122 nm, well into the ultraviolet and invisible to our visual systems. Likewise, falls to level three from above, the so-called Paschen series, occur in the equally invisible infrared spectrum. There is also a level four, five, six … potentially out to infinity. It was the discovery of these and other series which confirmed Bohr’s model and in part earned him the Nobel Prize in physics in 1932.

This is fundamentally the way science works. Inexplicable features of reality are solved, step by step, sweat drop by tear drop , and blood drop by drop, by the application of known physical laws; or, when needed, new laws and new ideas are summoned forth to explain them. Corks are popped, the bubbly flows, and awards are apportioned among the minds that made the breakthroughs. But then, as always, when the party is over and the guests start working off their hangovers, we realize that although, yes, progress has been made, there is still more territory to cover. Ironically, sometimes the new territory is a direct consequence of the conquests themselves.

Bohr’s triumph over atomic structure is perhaps the best known entré in this genre of the story of scientific progress. There were two problems, one empirical and one theoretical, which arose from it in particular, problems which sobered up the scientific community. The empirical problem was that Bohr’s atomic model, while it perfectly explained the behavior of atomic hydrogen, could not be successfully applied to any other atom or molecule, not even seemingly simple helium or molecular hydrogen (H₂), the former of which is just after hydrogen in the periodic table. The theoretical problem was that the quantitization of orbits was purely done on an ad hoc basis, without any meaningful physical insight as to why it should be true.

And so the great minds returned to their offices and chalkboards, determined to answer these new questions.

Key Ideas in the Development of Quantum Mechanics

The key idea which came out of trying to solve these problems was that, if that which had been thought of as a wave, light, could also possess particle properties, then perhaps the reverse was also true: that which had been thought of as having a particle nature, such as the electron, could also have the characteristics of waves. Louis de Broglie, in his 1924 model of the hydrogen atom, introduced this, what was to become called the wave-particle duality concept, explaining the discrete orbits concept of Bohr by recasting them as distances from the nuclei where standing electron waves could exist only in whole numbers, as the mathematical theory behind waves demanded:

De Broglie’s model was supported in the latter 1920’s by experiments which showed that electrons did indeed show wave features, at least under the right conditions. Yet, though a critical step forward in the formulation of the quantum mechanical description of atoms, de Broglie still fell short. For one thing, like Bohr, he could only predict the properties of the simplest atom, hydrogen. Second, and more importantly, he still gave no fundamental insight as to how or why particles could behave as waves and/or vice-versa. Although I have said that reality on such small scales should not be expected to behave in the same matter as the scales we are used to, there still has to be some kind of underlying theory, an intellectual glue, that allows us to make at least some sense of what is really going on. And scientists in the early 1920’s still did not possess that glue.

That glue was first provided by people like Werner Heisenberg and Max Born, who, only a few years after de Broglie’s publication, created a revelation, or perhaps I should say revolution, of one of scientific – no, philosophic – history’s most astonishing ideas. In 1925 Heisenberg, working with Born, introduced the technique of matrix mechanics, one of the modern ways of formulating quantum mechanical systems. Crucial to the technique was the concept that at the smallest levels of nature, such as with electrons in an atom, neither the positions nor motions of particles could be defined exactly. Rather, these properties were “smeared out” in a way that left the particles with a defined uncertainty. This led, within two years, to Heisenberg’s famous Uncertainty Principle, which declared that certain pairs of properties of a particle in any system could not be simultaneously known with perfect precision, but only within a region of uncertainty. One formulation of this principle is, as I have used before:

x × s ≤  h / (2π × m)

which states that the product of the uncertainty of a particle’s position (x) and its speed (s) is always less than or equal to Planck’s (h) constant divided by 2π times the object’s mass (m). Now, there is something I must say upfront. It is critical to understand that this uncertainty is not due to deficiencies in our measuring instruments, but is built directly into nature, at a fundamental level. When I say fundamental I mean just that. One could say that, if God or Mother Nature really exists, even He Himself (or Herself, or Itself) does not and cannot know these properties with zero uncertainty. They simply do not have a certainty to reveal to any observer, not even to a supernatural one, should such an observer exist.

Yes, this is what I am saying. Yes, nature is this strange.

The Uncertainty Principle and Schrödinger’s Breakthrough

Another, more precise way of putting this idea is that you can specify the exact position of an object at a certain time, but then you can say nothing about its speed (or direction of motion); or the reverse, that speed / direction can be perfectly specified but then the position is a complete unknown. A critical point here is that the reason we do not notice this bizarre behavior in our ordinary lives – and so, never suspected it until the 20th century – is that the product of these two uncertainties is inversely proportional to the object’s mass (that is, proportional to 1/m) as well as directly proportional to the tiny size of Planck’s constant h. The result of this is that large objects, such as grains of sand, are simply much too massive to make this infinitetesimally small uncertainty product measurable by any known or even imaginable technique.

Whew. I know. And just what does all this talk about uncertainty have to do with waves? Mainly it is that trigonometric wave functions, like sine and cosine, are closely related to probability functions, such as the well-known Gaussian, or bell-shaped, curve. Let’s start with the latter. This function starts off near (but never at) zero at very large negative x, rises to a maximum y = f(x) value at a certain point, say x = 0, and then, as though reflected through a mirror, trails off again at large positive x. A simple example should help make it clear. Take a large group of people. It could be the entire planet’s human’s population, though in practice that would make this exercise difficult. Record the heights of all these people, rounding the numbers off to a convenient unit, say, centimeters or cm. Now make sub-groups of these people, each sub-group consisting of all individuals of a certain height in cm. If you make a plot of the number of people within each sub-group, or the y value, versus the height of that sub-group, the x value, you will get a graph looking rather (but not exactly) like this:

Here, the y or f(x) value is called dnorm(x). Value x = 0 represents the average height of the population, and each x point (which have been connected together in a continuous line) the greater or lesser height on either side of x = 0. You see the bell shape of this curve, hence its common name.

What about those trigonometric functions? As another example, a sine function, which is the typical shape of a wave, looks like this:

The resemblances, I assume, are obvious; this function looks a lot like a bunch of bell shaped curves (both upright and upside-down), all strung together. In fact the relationship is so significant that a probability curve such as the Gaussian can be modeled using a series of sine (and cosine) curves in what mathematicians call a Fourier transformation. So obvious that Erwin Schrödinger, following up de Broglie’s work, in 1926 produced what is now known as the Schrödinger wave equation, or equations rather, which described the various properties of physical systems via one or more differential equations (if you know any calculus, these are equations with relate a function to one or more of its derivatives; if you don’t, don’t worry about it), whose solutions were a series of complex wave functions (a complex function or number is one that includes the imaginary number i, or square root of negative one), given the formal symbolic designation ψ. In addition to his work with Heisenberg, Max Born almost immediately followed Schrödinger‘s discovery with the description of the so-called complex square of ψ, or ψ*  ψ , being the probability distribution of the object, in this case, the electron in the atom.

It is possible to set up Schrödinger’s equation for any physical system, including any atom. Alas, for all atoms except hydrogen, the equation is unsolvable due to a stone wall in mathematical physics known as the three-body problem; any system with more than two interacting components, say the two electrons plus nucleus of helium, simply cannot be solved by any closed algorithm. Fortunately, for hydrogen, where there is only a single proton and a single electron, the proper form of the equation can be devised and then solved, albeit with some horrendous looking mathematics, to yield a set of ψ, or wave functions. The complex squares of these functions as described above, or solutions I should say as there are an infinite number of them, describe the probability distributions and other properties of the hydrogen atom’s electron.

The nut had at last been (almost) cracked.

Solving Other Atoms

So all of this brilliance and sweat and blood, from Planck to Born, came down to the bottom line of, find the set of wave functions, or ψs, that solve the Schrödinger equation for hydrogen and you have solved the riddle of how electrons behave in atoms.

Scientists, thanks to Robert Mullikan in 1932, even went so far as to propose a name for the squared functions, or probability distribution functions, a term I dislike because it still invokes the image of electrons orbiting the nucleus: the atomic orbital.

Despite what I just said, actually, we haven’t completely solved the riddle. As I said, the Schrödinger equation cannot be directly solved for any other atom besides hydrogen. But nature can be kind sometimes as well as capricious, and thus allows us to find side door entrances into her secret realms. In the case of orbitals, it turns out that their basic pattern holds for almost all the atoms, with a little tweaking here, and some further (often computer intensive) calculations there. For our purposes here, it is the basic pattern that matters in cooking up atoms.

Orbitals. Despite the name, again, the electrons do not circle the nucleus (although most of them do have what is called angular momentum, which is the physicists’ fancy term for moving in a curved path). I’ve thought and thought about this, and decided that the only way to begin describing them is to present the general solution (a wave function, remember) to the Schrödinger equation for the hydrogen atom in all its brain-overloading detail:

Don’t panic: we are not going to muddle through all the symbols and mathematics involved here. What I want you to do is focus on three especially interesting symbols in the equation: n, ℓ, and m. Each appears in the ψ function in one or more places (search carefully), and their numeric values determine the exact form of the ψ we are referring to. Excuse me, I mean the exact form of the ψ*  ψ, or squared wave function, or orbital, that is.

The importance of n, ℓ, and m lies in the fact that they are not free to take on any values, and that the values they can have are interrelated. Collectively, they are called quantum numbers, and since n is dubbed the principle quantum number, we will start with it. It is also the easiest to understand: its potential values are all the positive integers (whole numbers), from one on up. Historically, it roughly corresponds to the orbit numbers in Bohr’s 1913 orbiting model of the hydrogen atom. Note that one is its lowest possible value; it cannot be zero, meaning that the electron cannot collapse into the nucleus. Also sprach Zarathustra!

The next entry in the quantum number menagerie is ℓ, the angular momentum quantum number. As with n it is also restricted to integer values, but with the additional caveat that for every n it can only have values from zero to n-one. So, for example, if n is one, then ℓ can only equal one value, that of zero, while if n is two, then ℓ can be either zero or one, and so on. Another way of thinking about ℓ is that it describes the kind of orbital we are dealing with: a value of zero refers to what is called an s orbital, while a value of one means a so-called p orbital.

What about m, the magnetic moment quantum number? This can range in value from – ℓ to ℓ, and represents the number of orbitals of a given type, as designated by ℓ. Again, for an n of one, ℓ has just the one value of zero; furthermore, for ℓ equals zero m can only be zero (so there is only one s orbital), while for ℓ equals one m can be one of three integers: minus one, zero, and one. Seems complicated? Play around with this system for a while and you will get the hang of it. See? College chemistry isn’t so bad after all.

* * *

Let’s summarize before moving on. I have mentioned two kinds of orbitals, or electron probability distribution functions, so far: s and p. When ℓ equals zero we are dealing only with an s orbital, while for ℓ equals one the orbital is type p. Furthermore, when ℓ equals one m can be either minus one, zero, or one, meaning that at each level (as determined by n) there are always three p orbitals, and only one s orbital.

What about when n equals two? Following our scheme, for this value of n there are three orbital types, as ℓ can go from zero to one to two. The orbital designation when ℓ equals two is d; and as m can now vary from minus two to plus two (-2, -1, 0, 1, 2), there are five of these d type orbitals. I could press onward to ever increasing ns and their orbital types (f, g, etc.), but once again nature is cooperative, and for all known elements we rarely get past f orbitals, at least at the ground energy level (even though n reaches seven in the most massive atoms, as we shall see).

Explaining the Periodic Table. Atomic Orbitals.

It hopefully is now beginning to make some sense. Each horizontal row, or period, in the table represents a specific primary quantum number, or n, starting with one at hydrogen (H) and going up to seven at francium (Fr) in the seventh row. As we move from left to right across a period, we are filling the elements in said period, by which I mean their various orbitals, with electrons. For n equals one there is only ℓ equals zero, thus m equaling zero, meaning we only have an s orbital to fill in hydrogen and helium – each orbital can only hold a maximum of two electrons, for reasons we will get to. For the period just below hydrogen and helium, where n equals two, ℓ can equal either zero or one, meaning we have one s orbital and three p orbitals to fill, the latter with m values equaling minus one, zero, and one; this gives us a total of one + three = four orbitals at this level, each orbital containing a maximum of two electrons to give us the eight elements in this row / period, Li through Ne.

The title of this book, The Third Row, refers to the period beginning with sodium (Na) and ending with argon (Ar). The first two columns, or groups, known as the alkali metals and the alkaline earths, represent s orbitals being filled, while the last six groups – of which the last two are called the halogens and the noble gasses – involve p orbital filling. The central, sunken region, or transition metals, are d orbitals being occupied, while the two offset rows at the bottoms are f orbitals being filled. We will get to the reasons why the d and f orbital periods are sunken / disconnected later. The first question you should ask is, how many electrons does it take to fill an orbital? From what I’ve said and you’ve just seen, the answer is two, but we can do a little better than that and explain why. It turns out I have been holding out on you.

Well, no, I haven’t really. We are following the historical development of quantum mechanics, and now is the time to include some important concepts I have been ignoring so far. It turns out that there is a fourth quantum number, known as s or spin (do not confuse this with s orbitals!), which comes about when quantum mechanics is reformulated using Einststein’s special relativity. This turns out to be necessary because the electrons in an atom move at a significant fraction of the speed of light (they can’t move as fast or faster than light speed, as relativity also says) and so relativistic effects cannot be ignored. This new quantum number s is of course also quantized, and so can have only one of two values: +½ħ or -½ħ, where ħ or h-bar as it is called, is equal to h/2π. This work was done by a number of individuals, some of whom we have already met, but the main new name to enter here is Wolfgang Pauli. Using the resulting relativistic quantum field theory devised from special relativity and quantum mechanics, Pauli was able to show in 1925 that no two electrons in an atom can have the same quantum numbers n, ℓ, m, and s. More broadly, he showed that fundamental particles known then (and now) could be divided into two camps: fermions, which obey the Pauli Exclusion Principle, and bosons, which do not. Later it came to be realized that fermions are particles which constitute the main mass of matter, such as electrons, protons, and neutrons, while bosons are force carrying particles, or the glue if you like, holding the whole menagerie of particles together. Photons are a good example of bosons: they carry the electromagnetic force. And there are others, as we learned from the last chapter but did not elaborate much on, such as nuclear forces.

If an orbital can hold only two electrons (same n, ℓ, m, but with different s’s) then we can see how the atoms can be built up, step by step, filling in the lower level orbitals and then expanding out to increasingly higher, or more energetic, level ones. The entire periodic table finally snaps into focus, and we find, to our astonishment, that we can grasp its rationale, or at least some of it. And yet, yes, I still haven’t covered one of the most important topics of all. What do these orbitals look like and how do they behave? And why should we care?

The Shapes and Behaviors of the Atomic Orbitals

What I am about to show you can be misleading, or at least confusing. I will talk about the shapes and sizes of orbitals, and even show pictures of them. The misleading part is in thinking that orbitals, in and of themselves, are actual, concrete things, filling space like any other material object. This image, or illusion I should say, though easily fallen for, is something we have to resist if we are truly to understand orbitals, their meanings, and the functions they serve.

A good place to start is with the s orbital where n equal to one, i.e., the 1s orbital, first because it is the simplest one in shape and also because there is an s orbital for every value of n – that is, every row in the periodic table. Let me begin by reminding you of the Gaussian distribution function, shown several pages earlier:

I do this because this function is essentially the shape of the 1s orbital. The only difference is that this is a two dimensional figure while of course orbitals are three dimensional entities. We should redraw the 1s orbital more appropriately as something like this:

Can you see how this smeared out sphere is the 3D equivalent to the Gaussian curve? It is densest / highest at the center, and then exponentially drops off from there; you get the picture if you will follow the density of the dots, ever closing in but never quite reaching zero as you go out from the center (the dots do not represent electrons themselves, but the probability of them being found at a certain place). Incidentally, the same picture fundamentally applies to all s orbitals, not just those where n equals one; for successively higher values of n (2, 3, 4, etc.) these still have the highest density in the center but the exponential decay decreases progressively more slowly, and circular nodes of zero density start to form rings in the distribution.

What about p orbitals? Remembering that there are three of them, the best description of them is as dumbbell shapes, one dumbbell for each 3D axis (x, y, z), having a node of zero at the center of the atomic nucleus:

Displayed here is a p_x orbital, more specifically the 2p_x as we only start having these orbitals when the principle quantum number is two or higher. There are two other such orbitals, the 2p_y and the 2p_z, which have the same shapes but are oriented along their respective axes. And again, as n goes up, these orbitals become larger and more spread out, and develop nodes.

Technically what comes next are the d orbitals, but again a reminder before we proceed. These orbitals are not material entities in any definition of the word; they are more akin to the various states describing the electrons in an atom. If we were to propose a (admittedly strained) analogy with our own world of space in time, saying that an electron occupies a given orbital is rather like saying a car is going so many miles per hour down a highway. In this analogy, talking about an empty orbital is akin to talking about the state of going so many mph, although no vehicle may actually be in that state. I insert this caution precisely because we will at times speak of orbitals as though they really are physical, even solid, manifestations, for example when we combine them to make new orbitals; but this is just a convenient way of talking about them – don’t lose sight of what they really are, just the complex squares of wave functions found by solving the Schrödinger equation for the hydrogen atom.

With this warning in mind, on to d orbitals. These exist only for n equals three or higher, so they don’t exist at all until we reach the third row of the periodic table, which as I have said, runs from sodium (Na) to argon (Ar). Here is also where we find ourselves faced with the puzzle of why the transition, or d-block, elements don’t begin here but only with the fourth period (n = 4). There are five such orbitals for every period that has them, but unfortunately they cannot easily be described by a few words so the only thing to do is show them in their full splendor:

Here, because it makes them easier to visualize, for instead of the fuzzy pictures made from dots I used for s and p orbitals I am using solid shapes (bearing in mind yet again my warning that orbitals themselves are not solid, material things) which enclose approximately ninety percent of each orbital’s probability distribution. This is as good a place as any to solve the mystery of the sunken d-block elements in the table, not to mention the offset f-block (lanthanides and actinides) as well.

In the beginning was the solution to the Schrödinger equation for the hydrogen atom:

in which the energies of the orbitals was dependent solely on n, the principle quantum number. Recall, however, that some tweaks and calculations are needed as we moved upward through the elements, because they have multiple electrons and so we can’t solve the equation for them directly. One of those changes is that, as soon as we start filling the orbitals with electrons, the kinds of orbitals at each level, s, p, or d, begin to diverge in energy, with the higher ℓ orbitals increasing over their lower siblings. Even by the time we get to n equals two in the table the p orbitals have higher energies than the s, and for n equals three the d orbitals are higher still.

This is what accounts for the sunken transition elements. By the time n equals three the 3d orbitals now lie at higher energies than the 4s orbitals which we naively expected should lie below them. Thus, we must wait for the 4s orbitals to be filled (which they are in the elements potassium or K, and calcium or Ca, through argon, Ar) before filling the 3d orbitals in the first row of transition metals, which goes from scandium (Sc) to zinc (Zn); only then can we move on to the 4p elements, from potassium (K) to krypton (Kr). A similar, even greater disparity in energy accounts for the f-block elements, the lanthanides and actinides, which is why they are set off below the main body of the table. It is a good thing nature doesn’t go as far as g orbitals, or our pretty table would become horrifically complicated!

Let us recapitulate before moving on. We began this chapter by noting that, in general, reality is not scale invariant, meaning that the appearances and behavior of objects, and even the underlying physical laws for them, appear to change as we move either to the world of the immensely large or infinitesimally small. For the latter, we discovered that nature at this level obeys the laws of quantum mechanics, a system of physics that was mainly developed between 1900 and the 1930s. Electrons are so tiny that they fall well within the range of this new system of physics; for example, they can not move in simple orbits about the atomic nucleus as planets do around the sun, but rather, their behavior is determined by wave functions derived by solving the Schrödinger equation for the hydrogen atoms (and then adding some extra tweaks and calculations). All of this has been a considerable trek to understanding the whys and hows of the periodic table of elements. And so, take a breather, and we shall see where this will take us.