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Weird Physics

Section 3 – How Big Is an Atom? – Wave-Particle Duality

Which came first, the Auto or the Atom?  This sensationalist heading from a grad-school poster I put together made a point that the legitimate debate about whether atoms really existed is very young in our culture.

The original idea that there might be a point where you can’t divide matter any further came from the ancient Greeks, as does the root of the name “Atom”.  However, up until the turn of the twentieth century, many respectable theories about the nature of matter vied for first place.  The main opponent of the Atom concept was that matter was continuous and that the properties including spectral lines had to do with resonances of vibration of this continuous matter.  Rutherford’s scattering experiments showed that the positive charge was concentrated in lumps, giving rise to the famous “plum pudding” model, with protons imbedded in a continuous negatively charged matrix.  It was not until 1908 that Perrin did the experiment that put the question to rest, measuring Brownian Motion predicted by the theoretical equations in one of Einstein’s 1905 “Miracle Year” papers.

1908 was also the year that Henry Ford put the Model T into production.  A tie?  Not so – Ford founded his company in 1903, and the first mass-produced automobile was the Oldsmobile Curved Dash, which began production in 1901.  So, the American Car Culture began before we firmly believed that Atoms really existed.  No wonder we continue to struggle with wrapping our minds around what atoms really look like.

Most people still think about chemicals in terms of the models used in school – balls and sticks.  Forget for the moment that there are 602,000,000,000,000,000,000,000 of them in a single gram of Hydrogen (about a balloon full), making them extremely tiny.  Typical chemical bonds are 1-2 angstroms, or about 170,000,000 per inch.  So that’s the region of space normally taken up by the electron.  How about the proton?  The stick-and-ball model suggests maybe half that but it’s really even tinier – 10 to the minus 15th power meters or 10,000 protons per chemical bond.  So in the model, if the stick is 1 inch, a ball representing the proton would be 1/1000 of a millimeter.  The stick-and-ball model is, to say the least, misleading.

Oh, it’s better than some of the other models, with electrons being little ping-pong balls orbiting a bowling ball (to get relative masses right), but it still misses the nature of the electron.  We cling to the idea that it’s a particle, and talk about the probability distribution of where it might be, creeping towards telling ghost stories around the camp fire. (This appeals to some primal instinct in us, and many theoreticians seem to be in it only for the thrill of the story telling, with no interest in whether their gee-whiz stories correspond with reality, so we get black holes, dark matter, wormholes, and all that…. but that’s another story).

You can’t understand electrons until you let go of the idea that it is a particle.  It’s a wave – also.  Smack!  We’re up against the infamous wave-particle duality.  But it’s not campfire mumbo-jumbo.  It’s just a case of trying to choose which coordinate system to use to describe reality.  The best computer monitor for painting a picture of a tiny bug is an LED screen – one pixel on.  The best choice for painting a picture of a snake is an old-style oscilloscope, like is used for monitoring wiggly heart-beats in the hospital.  But you can make a sharp blip on an oscilloscope and you can paint waves on an LED screen.  Similarly, particles are easy to describe as a “point” in x-y-z coordinates where waves are best describe by talking about the wave’s frequency on a spectrum.  An elegant choice of coordinate system gives easier math.

In the real world, nothing is a perfect wave or a perfect particle, but somewhere in-between.  A proton or neutron is pretty darn close to being a perfect particle (I’m ignoring finer structure on quark scale here), but they actually have some size, so they’re not just points in space, their mass is distributed more like a very narrow bell curve.  Light is the other extreme, which we think of as a wave, but even laser light is not perfect and to some degree it is “choppy” waves we’re talking about.  The perfect wave of a single frequency would have no imperfections and would go on oscillating regularly right past the horizons – forever.  The perfect wave would have to be infinite, go on forever, to be a pure wave.  Never say never, but perfect doesn’t happen in the real world.

So electrons are somewhere in-between.  Think of them as clouds, squishy exploded marshmallows, bursts of sci-fi blaster fire, bundles of static, or packets of waves.  Think of them as what you see when you shake a jump rope, and a bump runs down the line.  This is the animal we’re trying to describe.

Fortunately for us, we can divide these electron wave packets into two sorts.  One is a standing wave, like a vibrating string on a guitar that repeats itself and doesn’t go anywhere.  The other is in motion, like that blast of laser light shooting off into space.  The standing wave is what you get when an electron is bound to the nucleus of an atom that attracts it, the speeding wave packet is what you get when the electron is ionized or free to wander.  One choice is more like a wave, the other choice more like a particle.  But the identity crisis is in what coordinate system to use in describing it, how to paint the picture of the electron, NOT in terms of the electron itself.  It is what it is.

The standing wave form is most relevant to human existence since under ambient conditions (almost anything around you except fire and fluorescent lights) the electrons are firmly bound to atoms and find standing-wave forms of existence.  To describe these forms, we need pretty heavy math, and we’d be doing quantum mechanics.  I’ll keep it simple though.

Long story short, somebody found a coordinate system that does elegantly describe the standing-wave form of electrons – provided there is only one in the universe, and it is kept company only by a single nucleus (like Hydrogen, but the positive charge on the nucleus may be changed with out messing up the math).  Anything more complex you have to use a computer to track it.  These are called the Spherical Harmonics, and they turn out to be a useful tool in explaining the behavior of atoms, how they make ions and molecules, and the structure of the Periodic Table of the Elements.  The lowest energy standing wave is the 1S orbital (bad name due to historical baggage, don’t confuse them with “orbits”) is completely symmetric around the center – think of the letter “o”.  If there is one “node” (zero-crossing wiggle point) to the wave, it can be along the x, y, or z axis so you get the three 1P orbitals, and you should think of a figure “8” on it’s side.  Alternatively, the node can be like a shell and you get the 2S orbital (think of a 3-D donut).  The more nodes, the more wiggly the wave is, and the more energy there is in it.  This is what you get out of solving the Schrodinger equation for the given situation.  It takes a number-crunching computer to estimate it for anything more complex that Hydrogen.   The point is there are an infinite number of these orbitals possible for the electrons to adopt, each with a well-defined energy level and progressively more wiggle as the energy goes up.  If an electron moves from one orbital shape to another, it changes energy, and the excess must be accounted for by exact book-keeping, absorbing or giving off light of a very specific energy, which translates to frequency or color.  This is why elements have very characteristic spectra, and we say that light comes in well-defined packages called “quanta”.  This is where the name “Quantum Mechanics” comes from.

So, how big is the electron?  If you bleed off all the extra energy that you can, making it absolute zero in temperature, then the electron will go to the lowest possible energy level, occupying the 1S orbital.  It’s a wave, so there is no sharp cut-off, but we can describe it like a bell curve, using the math of statistics, and say it is mostly “here” and get a 99% confidence limit statement that 99% of it is in this region here: about 1 angstrom across.  Now let it warm up a little bit to temperatures above zero.  There is a chance that the electron will occupy one of the higher energy levels – in accordance with Thermodynamic’s temperature-dependant Boltzmann distribution.  Higher energy orbitals have more wiggles (nodes) in them and are more spread out.  We can do the math.

In grad school at Stanford, a teaching assistant set the problem of deriving an equation to give mean size of a hydrogen atom as a function of temperature.  So you do a weighted average which becomes adding up an infinite series, multiplying the answer for each possible orbital times the probability that the electron is in that particular orbital.  Do the math, sum the series, and the average number comes out — infinity!  Well, that was not the answer they wanted, so this guy did the thing any engineer would do.  He pointed out that each term became smaller and smaller to the point you could ignore any one term, truncated the series at that point, and got the textbook answer of a little more than an angstrom, expanding a wee bit with temperature.

This did not sit well with me – the math had to have some real significance.  Thinking about it, if there really was a single hydrogen atom in the universe, what would happen?  Unless it really was at absolute zero, the electron really would wander away and never find it’s way back home to the neutron.  It would wander off into deep space and the size of the “atom” (distance from electron wave packet to nucleus) really would be infinite.  This is sort of what you see in outer space – the hydrogen really is ionized, called a plasma like is in your fluorescent light bulbs.  But in reality, you go about a meter and bump into another atom taking up space.  I used this fact as a basis for saying the extreme orbitals were not available to our original atom’s electron, they are occupied by other atoms, used this as a physical justification for where to truncate that infinite series, and ended up with the same textbook answer.  I had to argue with the TA to get credit since I didn’t do it his way, and in a later year I ended up being the TA for that class.

So the answer is – the electrons, like junk in a garage, really do expand to fill up all available space.  In your hand, they are the building blocks of molecules.  Beyond the atmosphere, they ionize and fill all available space.

One important take-home here is that outer space is not really a perfect vacuum.  It is a sea of electron’s wave functions, spotted with an ionized proton every meter or so.  Not so far from that Plum-Pudding model, now that I think of it.  But understanding this is critical to the next chapter, and the understanding of the Hubble Red-Shift.

[© Copyright 2016 by Gerald Keep.  All Rights Reserved.]



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