Special relativity is weird because it
involves travelling at speeds no one has ever come close to. General relativity is weird because it tries
to understand the universe as a whole, or at least things much more massive than
ourselves; ambitions few of us have ever gotten the itch for.
It’s time now for a different
direction. From now on, we will be
focussing on the realm of the vastly small and, often, extremely
short-lived. To give you an idea of just
how small, how many molecules of water are in a glassful? If the volume is about a cup, or, more usefully,
let’s use the metric system and call it 100 milliliters or 0.1 liter, then that
number is about 33,000,000,000,000,000,000,000,000 or 3.3×1024
molecules. To capture a feel for just
how large a number than is (i.e., how small water molecules are), all of the
waters of this planet Earth would take somewhere between 1019 and 1020
glasses to hold it. That number is at
least ten thousand times smaller than
the number of water molecules in an ordinary drinking glass.
My
opening salvo is, objects that small definitely fall deep into our notions of
weird. You really can’t imagine it all,
you can only do calculations with the numbers.
I’m tempted to try to give you a feeling for it; say, by going down to
the beach and building a sand castle, and all that. But sand grains themselves are so large that
they fall into the range of common-sense physics, and are barely one iota along
our quest for the ultra-small. If sand
grains behaved as atoms and molecules do, this would be a very, very different
world indeed.
* * *
If you were to go down to the beach
anyway, and try to follow the whims and whereabouts and whichways of a sand
grain buffeted about by the sea breezes, you would probably feel like a cat
chasing the spot of a laser pointer about the house (they go nuts chasing it). Nevertheless, come, let me take you – a child
version of you works best because of its innocence – down to the sea shore to
chase a grain of sand. You will no doubt
quickly report to me that it is impossible, you will never be able to do it. In response, I say, “I know what you say sounds right, but it
can’t be. Think about it. At every moment in time the sand grain
has: a) an exact mass; b) an exact
speed; c) an exact direction it’s travelling; d) an exact momentum; e) an exact
kinetic energy; f) an exact temperature; g) an exact albedo (how much of the
sun’s light it reflects); and – I’m sure I could come up with other
measurements, but you get the idea. You
just need to apply yourself more diligently to follow these quantities.”
A true child would not comprehend this
quandary, but just find something else to play with. But the adult you gets the point: while
in practice we simply cannot move quickly enough to follow a sand grain’s various
dances, in theory it could be
done. Could be. Are you certain? What if I were to start shrinking that grain,
ever smaller and smaller, until it was so small it fell beyond the range of the
most powerful light microscopes to see it, down to something of the size of a
water molecule? What would happen then?
We suddenly feel as though we’re in the
middle of Paul McCartney’s song “Lucy in the Sky With Diamonds” or perhaps
“Penny Lane”; a surreal, dreamlike state where all the ordinary rules of reality
have been suspended, and we have to figure out the new ones.
And in truth, that’s not just
metaphor. How reality looks and acts
like at this scale is truly, and astonishingly, very, very little like sand grains
and castles and rail cars. And it is, at
least at first, just a little unsettling.
For it means, for example, we have to throw away things like our
previous, common-sense idea of exact measurements. Oh, you can still make them, but at a
price. That price is complete loss of
all information about something else equally measurable.
As a concrete example of this (always
turns abstract concepts into concrete visions, when possible, remember), try to
measure the position where our sand grain scaled down to water molecule can be
found at any given moment. Yes, you can
still set up a measuring system that gives you a 100% accurate value for that
position. But only for that moment when
you make the measurement. And where has
the grain gotten too after that?
Strangely, oddly, bizarrely, utterly incomprehensibly we have no idea! In order to measure its position exactly, we
have completely scrambled all information about its speed and direction from
that position. It, literally, and I mean
literally in the literal sense here as in quite absolutely literally, could now
be anywhere in the known universe!
You see, we must hand over our
magnifying glasses and microscopes, and radar guns, and all other mechanisms
for measuring the quantities of things, over to the police of this Brave
(perhaps with a few shots of gin) New World, and accept new tools for spying on
the world about us. These tools only
allow us to make probabilistic approximations on things. By accepting limits on how well we can
measure, we can measure other things too, albeit within their limits as well.
If
I sound heavy-handed in repeating again that we are not dealing with metaphor
or analogy, but with nature as it actually is, please be patient. We are so accustomed to our “normal” rules of
reality that different ones do sound like artistic inventions or flights of the
imagination. I merely want to emphasize
over and over that what we are faced with is as real as the computer I’m typing
these words on. But it is only in the
land of the ultra-miniature do our new rules come into play, that is, in a way
that we cannot fail to notice even if we try.
* * *
I just used the word probabilistic, I
suspect not for the first time. In
talking about probability we all, I think, have a good grasp of what this
concept means. We don’t know – sports
analogies always work best – who is going to win the World Series next year but
we can analyze the different teams and the caliber of their players and
compute, even if roughly, probabilities – likelihoods, odds, where to place our
money, and so on. In a similar fashion
we cannot predict the height of a human being chosen at random, but we can use
probability to come up with a good ballpark (pardon the pun) figure. In fact, the particular method of probability
used here is simply called the Gaussian Distribution function, illlustrated below:
A measurement of a large number of human heights, plotted
on a graph, would look much like this.
There would be a high central region where most of the heights found
themselves, and then declining tails of progressively shorter and taller – and
fewer – people. A person chosen at
random has a 68.2% chance of falling within one standard deviation, or 1σ, of
the highest, most central height, called μ or the norm or average; a 95.4% of
coming within 2σ of the norm; and so forth.
And that’s it. All
we can say. We simply cannot speak with
certainty the height of an individual chosen randomly. There is a big difference, however, between
this problem, which is part of ordinary reality, and the “same” problem in the quantum
world. In the world we’re used to we can
still measure the selected individual and reveal himself to be an individual
(e.g., David Strumfels) with a height of six feet two inches. But in the quantum world, no. First, there are no individuals; and if we
try to make measurements which with to create individuals, then other important
information about them is irretrievably lost.
In the quantum world, probability rules with an iron fist.
We can place this in more prosaic contexts however. According to the laws of the quantum world,
we cannot say exactly where a water molecule is in a glass; but we can
calculate a probability distribution, similar to the Gaussian but not eactly
the same. There is, in fact, a famous calculation chemistry
and physics students perform that is quite similar: a lone electron in a (one-dimensional) box. You cannot say precisely where the electron
is, you can only construct an equation showing the probabilities of being in
the box somewhere at any given moment.
Given that I’ve raised the subject of electrons …
* * *
Atoms. We’ve all
heard about them. Unless you truly slept
through all your schooling you heard that:
a) everything is made from them (not true); b) they’re so tiny nobody
can even see them (also not true); and c) they’re composed of a tiny, dense
nucleus made from protons and neutrons, with even tinier electrons spinning
around it (partially true, but critically untrue for where this discussion is
headed).
It is time to continue on our journey. We were at the level of water molecules, and
as as we work our way down past that, past individual atoms even, down finally to
electrons, things change little. Mainly,
those probability curves just keep getting bigger. They get so big, in fact, that by the time
we’ve reached the level of electrons the curves are complete masters of their
behaviors.
Our interest in electrons is simple enough; they are what
make chemistry possible, and control most of what it can do. Not everything – remember the discussion on
organic chemistry and geometry before – but without electrons there is no such
thing as chemistry at all.
Some further evidence of just how strange the quantum
world is worth the effort here. In the
quantum world probability is truly just that; by that I mean that if, say, we
knew all the forces and effects working on a coin toss we could predict, with
absolute certainty, whether a given toss would be heads or tails. And it is always one or the other, nothing …
well, nothing in-between, curious as that may sound. In quantum probabilities, no such luck
however. All we can do is make
observations, in which we will see this or that or something else, an infinity
of possibilities with each one having its given probability.
This might sound purely academic, but study the picture
below:
Figure VI.
The gist of this, Erwin Schrödinger’s famous quantum cat
experiment, is a sealable container (usually a box), within which resides: a) a live cat; b) a vial of deadly poison; c)
a hammer and a lever which can drop the hammer, breaking the vial and killing
the cat; and d) a single radioactive atom with a 50% probability of decaying
within an hour and which, if it does decay will trip the lever, again killing
the cat. Unlike the picture however, the
box is not open but completely sealed so that no information on the cat’s state
(alive or dead) can get out.
You see, the quantum world is even stranger than I’ve
described. I’ve spoken of making
observations and getting results according to probabilities, as though the results were already there
waiting for us to unveil them. It’s
worse than that. The results don’t even exist until we make the observation! Quantum physicists speak of things like “collapsing
the state function” among other arcane words and phrases, but what they mean is
making an observation. The act of
observation reduces the infinitely many probabilities into a single, actual
result, and that result doesn’t exist until the observation is made.
Sounds like magic, I know, or some kind of
tomfoolery. But it is the simple truth. A truth that has been demonstrated with
absolute certainty by thousands of experiments, making measurements to many
naughts beyond the decimal point.
* * *
Let’s return to our concrete example, the quantum
cat. Seal the box, wait an hour –
remember, I said that there is a 50% probability of the atom within decaying
within that time span – then open it again.
What will you see?
This part of the answer is simple. Either the atom has decayed, the lever
tripped, the vial containing the point released, and the cat dead; or the atom
didn’t decay and the cat is very much alive.
The hard part of the answer is, what’s the state of the
cat just before opening the box? Your
first reply I’ll bet is the same as before, though you’re probably suspicious
by this point. You smell a rat following
the cat and you think I’m trying to pull a fast one on you.
If you feel that way, that’s actually good; it means
you’re following the discussion. (See, I
told you you could do it!) Remember I
just said that, at the quantum level, the result of an observation doesn’t
exist until it is made. The cat isn’t
dead or alive prior to opening the box – unless of course, you decide that the
cat is an observer who saw the vial being smashed or not, which, to be honest,
is a valid argument here. So: what does exist?
Recall that I said that the result of the coin toss
experiment is never somethin “in-between?”
I meant that it’s in a head state or tail state even before we lift our
covering hand and make the observation.
Again, no such luck with the quantum cat. The convenient way of treating it, that is
what scientists do, is to say that it is in a “superposition” of both
states. It’s both alive and dead; until
you open the box that is, and collapse the state fuction into one of the other
component states.
I advise not over-thinking about this. Scientists and philosophers and other pundits
to this day argue about what quantum physics implies about reality and our
relation to it. Even the cat experiment
is taken seriously by some, while a joke by others. My personal take about the experiment, by the
way, is that it is not physically possible; you can’t construct a container
which allows no information to leak out about the cat being dead or alive,
meaning that it is always possible to make a measurement about the cat’s state
without opening the box; whether or not the measurement is actually made or not
(what exactly is a measurement or who or what makes it, anyway?) is
irrelevant. But their are problems with
this stance too. So please – don’t lose
any sleep over it.
The pragmatic truth about quantum physics is that
scientists use it because it works. And
there are very real versions of the cat experiment that can and have been done,
experiments which have shown that the superposition of states with the result
not existing until observation is made is correct. As for the philosophical implications, most scientists
are too practical-minded to worry about them.
Too much.
* * *
It is now time to return to electrons, and how they
behave in atoms and molecules. Remember,
this is the fundamental requirement just to have chemistry, so we need a
reasonable feel for this before we can continue. Fortunately, a reasonable feel is all we
need. To get that, all is needed is to take
what we learned about cats in the quantum world and apply them to this new
arena.
Like cats in a box, electrons in atoms and molecules exist
in quantum states; these states describe such (indirectly) observable
quantities as their energies, magnetism, where they’re likely to be found, and
so on. Also, like the cat’s fate, the states are quite distinct from each
other, and the electron can only move from one to the other, never being found
in-between.
If you have been wondering why the world of the
infinitesimally small is called quantum, the answer lies here. In the quantum world things generally exist
in discrete states and not somewhere along a continuum. To give another example, imagine getting in
your car, pulling out along along a highway, and accelerating to 50 mph. It seems to you that the acceleration is
continuous, with no breaks anywhere; there are no discrete sepeed states, and
you don’t make jumps from one state to another.
But if quantum rules applied strongly at “ordinary” levels of reality,
you might find that you could only do the speeds 0, 10, 20, 30, 40, 50 mph
exactly, jumping from 0 – 10 – 20 instantly.
You could not do 36.568 mph, for that “state” doesn’t exist. Of course, it’s possible that the actual
states don’t comprise a continuum anyway, they are just too close together to
measure. In fact, this is the actual
case, and quantum physics applies at the medium, large, and enormous scales of
reality too. It’s just that as the
masses involve increases, the states grow ever and ever close together. Even at the level of sand grains they are far
too close together to be observed.
So: electrons in
atoms and molecules do not orbit around nuclei, like planets around the sun,
but exist in appropriate (usually lowest energy) quantum states. All the images of atoms being like miniature
solar systems are innacurate in this respect.
So why do we use them? Simply,
the actual picture of an atom is too difficult to illustrate at our reality
level, and the solar system works best here.
It is accurate in a number of ways that justify its use. Just bear in mind that, when it comes to
electron behavior, and it’s relationship to chemistry, it really doesn’t work
at all that way.
Before I go on, let me remind you about thermodynamics
and entropy again. This is a very
complex subject, but all we need to know is that electrons, like everything
else, “perfer” to be in their lowest energy states. In a multi-atom molecular, electrons are
attracted to more atomic nuclei (made of protons, and particles of opposite
charge attract one another), and this alone would be large molecules the most
stable states of matter. Careful. The larger an atom/molecule the more
electrons there are too, and these tend to repel each other, thereby raising
the energy. Exactly what types and
number atoms and molecules which from atoms is largely a balancing act between
these two forces.
The simplest, and most abundant atom in the universe is
hydrogen, consisting of a one proton nucleus and one electron distributed about
it. This is not the most stable atom;
it’s abundance is purely the result of how the universe has evolved. Next to hydrogen, helium, with a two-proton
nuclear and two electrons about it, is most common; it is more stable that
hydrogen, but is still far from maximum stability and again it’s abundance is
explain by cosmic evolution.
As for molecules, hydrogen usually exists in diatomic
molcules, in this case H2.
This is because the two electrons are more strongly drawn toward the two
nuclei than they repel each other.
Helium exists as monoatomic He atoms, however, because the four electrons
exert a more profound effect than the two, albeit di-proton nuclei, can
counter.
As we climb the periodic
table of elements (just a diagrammic listing of all elements, to be read
left to right and top to bottom) you see hydrogen and helium in the top row:
Figure VII.
The elements Li through Uuo don’t follow an easily made
out trend when it comes to stability and molecule making, but there are general
trends that can be worked out and understood, usually with a lot of hairy math
and computer time.
As this section is in fact about molecules and not lone
atoms, what we want is a feeling for how the constituent atoms form them, with
the particular shapes they have. A good
place to start is to return to good old water, a simple molecule that can
illustrate many of the principles.
Water is made from two hydrogen and one oxygen
atoms: the famous H2O. But why that particular formula, and why do
water molecules take on a bent shape:
Figure VIII.
No need to go into the details here. Basically, the available basic shapes of the
oxygen atom make up a tetrahedron (remember, I said you need to recall some
basic geometry):
Figure IX.
The ogygen
nucleus sits at the center of the tetrahedron, and the electrons are “at”
(actually around) the vertices (the connecting lines are the chemical bonds). If you imagine sticking the hydrogen atoms
(whose electrons make up spherical states) at two of the vertices, the water
water molecule shape should pop out at you.
Each electron on the hydrogen atoms pair up with a “lone” electron on
ogygens atoms pair up; this is another quantum oddity in which a state can hold
only up to two electrons; any more and they spill over into other states,
usually larger, more energetic ones.
This limitation, by the way, is why atoms and molecules can build up.
A more complex example of a molecule is glucose (blood
sugar):
Figure X.
I am not implying that all bonds have tetrahedral
shapes. Far from it. It is merely one of a myriad shapes which is
fairly common, especially among atoms in the top two rows of the periodic
table. The HC=O group, for example, is
flat and triangular. Another shape is
represented below
Figure XI.
This molecule (which has a negative three charge) is
comrpised of two shapes. The 6 CºN portions are linear (the º is called a triple
bond), while the central FeC6 part is an octahedron, the carbon at
the center:
Figure XII.
As said, there are numerous other shapes as well. And shapes can be bonded to shapes, creating
large, complex molecules, such as proteins and DNA and many other molecules
found in our bodies. Still, other
molecules possess even simpler shapes, such as O=O (O2), H-H (H2), NºN (N2), O=C=O (CO2), and so
on. The possibilities are, for a
practical purposes, infinite, and many millions of molecules are now known
about.
Many of these molecules are called “organic” and it is
time to know why.
