Fundamental

by Tim James

  You cannot have three-quarters of a wave because it would not fit on the line. Only specific shapes are permitted.

  These permitted waves are called ‘harmonics’ and that name, which sounds vaguely musical, is no coincidence. The shapes in the diagrams above correspond to the notes you can play on a stringed instrument, with each shape of the wave producing a different sound in the air.

  When you pluck a guitar or banjo string, it vibrates with a very specific energy. A different note is a different harmonic (permitted wave) and Schrödinger’s equation shows that electrons bound to a nucleus are equally musical.

  We have to go into higher dimensions, of course, because atoms are not straight lines, but if you can calculate the harmonics of a three-dimensional vibration you should get the shapes electron waves can take. It’s rather hard to imagine how you might actually calculate this (it was beyond Schrödinger’s talents) but fortunately other mathematicians soon calculated the answers.

  The first electron harmonic is spherical (shown on the left below). The second harmonic looks like two balloons squashed against each other, one at the front and one at the back (shown on the right):

  These spherical and dumb-bell shapes are telling us where we are going to find an electron wave. Rather than going around the nucleus like little pellets, we have to think of electrons as vibrating surfaces taking on bulbous shapes with the nucleus at the centre. And as we go up in energy, more complicated shapes are adopted (too fiddly for me to draw… I just about exhausted my artistic limitations with those sphere sketches but, if you’re curious, Google the term ‘s p d f shape’).

  These regions around a nucleus where the electrons vibrate are no longer thought of as orbits, so we call them ‘orbitals’ instead. Schrödingles sounds better of course.

  What this does is finally explain where energy quantisation comes from. Only certain electron harmonics will fit around a nucleus so only certain energy values are possible for a given atom. In Bohr’s shell theory, energy levels were something you had to stick in out of nowhere. In Schrödinger’s more advanced wavefunction theory, you get the energy levels as a prediction.

  Better still, Schrödinger’s equation tells us these orbitals only bond together at certain angles, a prediction confirmed by the whole of chemistry.

  The Schrödinger equation also gets rid of those pesky quantum leaps too. What happens when an electron moves from an inner orbital to an outer one is like a vibrating skipping rope changing wavelength as your hand changes speed. The transition looks ugly but it is a smooth process rather than an instant one. The photon emitted or absorbed when an electron changes from one orbital to another is the result of an electron wave vibrating to a new shape.

  THE ONLY CATCH

  Schrödinger’s equation performs calculations on a wavefunction and predicted how it would change. The wavefunction itself then provides a full description of whatever we want to know about our electron and it was without doubt a triumph of mathematical beauty. Provided you did not ask what any of it meant.

  All you needed to do was plug in relevant numbers, crank the handle and symbols on the page spat out reliable data, telling you what happens to an electron in a given circumstance. But what… exactly… is an electron wave?

  More worrying still, the Schrödinger equation did not give the correct answers unless the sums included something called an ‘imaginary number’ in the sum. I know we are taking a non-mathematical approach to quantum physics in this book, but imaginary numbers are important to the story and we cannot easily bypass them. So buckle up, baby, we’re gonna get math-tastic!

  USE YOUR IMAGINATION

  Here’s the deal. If you take a negative number such as minus two and double its size, you get minus four as an answer. We can write that sum as: −2 × 2 = −4.

  If we multiply minus two by minus two itself, however, we get the opposite. A minus number multiplied by another minus number turns out to be a positive because we’re minusing a minus, i.e. turning a minus number inside out. Two negatives multiplied together yield a positive answer, which we write as: −2 × −2 = 4.

  Everyone knows the square root of four is two (if not… spoiler alert!) but this is not the whole answer. The square root of four is actually two and minus two, since they are both multiplied by themselves to get four.

  That means the square root of minus four is not minus two because minus two does not square itself to give minus four. So where are the square roots of minus numbers? There does not seem to be any number you can multiply by itself to give a negative number as its answer.

  In order to solve this paradox, the Greek mathematician Heron of Alexandria invented a new type of number that lives at right angles to the number scale we are used to picturing. These are numbers defined as the square roots of negatives and René Descartes referred to them as ‘imaginary’ because they seem pretty unrealistic.5

  We represent them with a letter i, defined as the square root of −1. The number i2 is the root of −4, i3 is the square root of −9, i4 is the square root of −16 and so on.

  Imaginary numbers look like cheating to some people, but then again, mathematicians often invent concepts that do not appear to make sense until science invents a use for them. After all, there was a time when negative numbers would have sounded silly. Can you hold negative five objects?

  Negatives are not ‘real’ in the sense we can count them out in our hand, but they are definitely useful to have. The charges on electrons and protons cancel to zero, so positive and negative numbers are a good system to use.

  Similarly, electron wavefunctions only work if you include imaginary numbers as part of the equation. If Schrödinger’s method works (and it does) then an electron’s properties are vibrating not just in three dimensions around the nucleus, but in an imaginary dimension as well. What the hell, nature?

  BORN FREE

  The first man to try and give meaning to what an electron wavefunction truly represented was the German physicist Max Born. Born was intrigued by the randomness of quantum physics, which was a direct consequence of Heisenberg’s uncertainty principle.

  When we take a measurement on a particle, we end up discovering its properties such as location, momentum, etc. (within Heisenberg limits), but what is really peculiar is that because these properties are sort of fuzzy prior to measurement, repeating the measurement over and over can give different results each time.

  If you repeat a classical (normal) experiment over and over you get the same result. Roll a ball down a ramp and you can comfortably predict where it will arrive. As far as someone such as Isaac Newton was concerned, there is no such thing as true randomness or chance. Just predictable laws of physics.

  Even the toss of a coin is not random to a classical physicist. The impulse applied from your thumb, the angle of the coin as it arcs through the air and its interaction with the ground all predict how it will finally land.

  If you have a powerful enough computer and feed it all the data, you can predict the outcome of a coin flip with perfect accuracy. The only reason we treat coin flips as random is because we cannot do such intensive calculations on the spur of the moment. But quantum physics is different. Quantum physics appears to have genuine randomness in its outcomes.

  You may have heard the proverb that ‘insanity is repeating the same mistake and expecting different results’, often misattributed to Einstein (it actually comes from a 1981 pamphlet printed by Narcotics Anonymous).6 It makes sense though. How crazy would you have to be to repeat the same thing over and over expecting a different thing to happen? As crazy as a quantum physicist it turns out.

  Take the double-slit experiment. At the start you fire a bunch of electrons/protons/whatever towards a two-slit junction and end up with a zebra pattern on your detector screen. But you cannot tell, ahead of time, which stripe a particle is going to land in when it arrives at the screen – all you can do is make a guess based on probability.

  There could be a 40 per cent chan
ce of your particle arriving in the central band, then a 20 per cent chance of it arriving in the bands either side, a 10 per cent chance of landing in the next two, and so on (remember those numbers for a moment).

  Effectively, when you roll an electron down a track its destination can change on each repeat. Heisenberg’s uncertainty principle forces us to abandon the idea of predictable futures and accept that things happen based on probability, at the whim of some elegant although deranged quantum goddess. A particle’s location is not precise until we actually measure it (it is uncertain) and when we do the measurement we can only predict a probable location, not an absolute one.

  Born decided, therefore, to calculate the Schrödinger answers for an electron wave as it goes through a double slit, and discovered that the wave ‘amplitude’ (how high the wave is) corresponded to familiar numbers.

  The peak in the centre of the detector screen (where the wave amplitude is highest) winds up with a value of 6.32. The next two bands either side have an intensity of 4.47, then the next are 3.16. These numbers do not seem to follow a pattern but they do: they are the square roots of the probability percentages we saw a moment ago – 40 per cent, 20 per cent, 10 per cent and so on.

  If we do the Schrödinger calculation on an electron wave, then multiply the answers by themselves (square them), they match the probable locations for where particles are located in an experiment.

  It would appear that Schrödinger’s wavefunction is calculating the square root of an electron’s probable location. So yeah. Gee. Thanks, Born. That makes it perfectly clear. What was I confused about?

  WHAT DOES THAT EVEN MEAN?

  Born’s interpretation of the Schrödinger equation seems to imply that probability – a human invention for saying which horse might win a race – is an inherent law of the universe that particles have to obey.

  Particles are definitely particles but their locations are determined by waves of probability itself, constantly in flux. In this view of quantum theory we have to abandon the idea of anything having a definite location and say that locations are determined at random, based on laws of probability.

  Areas where the Schrödinger wave is peaking are where a particle is likely to be detected, and where the Schrödinger wave is dipping are where a particle is less likely to be. Electrons, protons and photons are not really waves themselves, but their probable locations are.

  We can never predict exactly where a particle is at any given time, but by using the Schrödinger equation (and assuming particle locations vibrate in an imaginary direction as well as real ones) we can calculate a probable outcome to our measurement.

  Particles can therefore be pushed through the same experiment over and over but finish in a different place because their fate is never set in stone. Every cause has many potential effects and the quantum goddess is choosing which ones will happen entirely at random. Sometimes an electron will be on one side of an atom, but as its location vibrates through space it can find itself existing on the other side.

  If this is all getting a bit heavy and confusing then here’s a bit of fun trivia to lighten the mood: Max Born’s granddaughter was Olivia Newton-John, the singer and actress who starred opposite John Travolta in the 1978 movie Grease.

  Sadly Grease is light on references to quantum physics, unless we decide ‘chills’ is a metaphor for ‘wavefunction’ and when our chills are multiplying, this means multiplying the wavefunction by itself (squaring it) to get the probability of the electron’s final resting place.

  If so, it would be accurate to say that once we solve the Schrödinger equation we are faced with a philosophical crisis in which we have to accept the outcome of an experiment is truly random and we are thus ‘losing control’.

  TUNNEL TO VICTORY

  Born’s interpretation of the wavefunction is bold, but there is a way to test it. We can throw a particle at a wall and see if it sticks.

  If you imagine a classical (ordinary) object such as a tennis ball hurtling towards a barrier, there’s no question about what is going to happen – it will collide with the wall, stick there for a moment and then bounce back.

  Born’s interpretation for quantum particles suggests otherwise. A particle’s location can be described as a constantly vibrating wave of probability.

  If we throw our electron at a wall we have to consider it as a wave. Each peak on the wave means ‘particle is likely to exist here’ and each dip means ‘particle unlikely to be here’. So if our wave approaches a wall, some of its peaks might be on the other side of the wall, like so:

  When the particle reaches the wall, as in the diagram above, most of its location stays on one side (the peaks of the wave are mostly on the left) but a tiny fraction crops up on the other side every once in a while. This is not a common occurrence because the wavefunction’s value is pretty low on that side (it is only a tiny bump), but occasionally an electron should appear on the far side of a wall by chance.

  This effect, known as quantum tunnelling, has been observed and documented many times, exactly in accordance with Born’s prediction. There is even a device in electronics called a Josephson junction, which consists of two conducting materials sandwiched either side of an insulator. An electron would normally be blocked by such a barrier but thanks to the tunnelling effect it is possible to transport electrons through the barrier and control the flow of current by altering its thickness.

  It takes a hundred billionths of a second for tunnelling to occur7 and it is similar in essence to the double-slit experiment. A particle has a choice to bounce off the wall or tunnel through, just like it has a choice about which slit to go through. The only difference is that in the double-slit experiment the probability for each slit is 50 per cent likely but in the tunnelling scenario above, the probability of moving through the barrier is very low so we do not see it happen very often.

  Tunnelling also explains where radioactivity comes from. An atomic nucleus can sometimes spit out a couple of protons and neutrons at random (called alpha radiation). This would not be possible classically since the protons and neutrons would never make it past the other protons and neutrons encasing them in the nucleus.

  But now we can describe all particle locations with a wavefunction and a small part of it can dangle outside the atom. Every once in a while, at random, we should see particles tunnelling their way to the edge of an atom and magically appearing outside, which is precisely what we observe in radioactive emission.

  CHOOSE YOUR WORDS

  Prior to 1926 there was a bunch of loosely connected strands to quantum theory and Schrödinger was the guy who braided it all together. He showed that wave–particle duality was linked to the energy levels of electron shells… which explained the shapes of atomic orbitals and all of chemistry… allowing us to make predictions about particles via probability.

  In some circles it is loosely agreed that the early generation of quantum physicists such as Planck, Einstein, de Broglie and Bohr were working on ‘quantum theory’ whereas the more sophisticated stuff of Schrödinger, Born and Heisenberg was ‘quantum mechanics’.

  Most people are not picky about this distinction and quantum mechanics usually serves as an umbrella term for both pre- and post-1926 physics. But, for the purists, quantum theory started with Planck and quantum mechanics began with Schrödinger.

  CHAPTER FIVE Things Get Even Weirder… Again

  EVERYTHING WE KNOW IS WRONG

  You may have noticed by now that every theory in quantum mechanics soon turned out to be incorrect. For people not familiar with science this can seem disconcerting, as if scientists are in a constant state of uncertainty (can I interest anyone in a Heisenberg pun?) but this is the normal state of affairs.

  Scientists work best by taking an idea and pushing it to the limit to see when it no longer works because nothing is above question and no fact is sacrosanct. It is always better to have confidence in an idea than certainty because then it is easier to admit you might be wrong. The Schröd
inger equation, for all its brilliance, is no different.

  When he published it, Schrödinger openly ignored the electron’s charge because it stays constant and does not need to be adjusted for. This does mean, however, that his equation dissolves if you bring your electron near a magnet.

  Magnetism and electric charge have a strong influence on each other. A moving magnet can coax electrons in a wire to flow, and an electric current going in a circle will blossom a magnetic field around itself. An equation that describes the electron but ignores magnetic effects is therefore incomplete.

  YOU SPIN ME RIGHT ROUND

  We will explore the link between electric charge and magnetism in Chapter Eleven, but what it comes down to for now is that electrons have magnetic fields around them like tiny bar magnets, one end being north and the other being south. I like to think of them as having little harpoons skewered through their middle indicating which way their magnetic field points:

  A magnetic field can be generated by spinning an electric current around in a loop of wire, so the assumption was made that the magnetic property of an individual electron must arise in a similar way. Electrons must be spinning constantly like a gyroscope to generate magnetic poles.

  Unfortunately, when we calculate how big the electron’s radius would have to be to generate a magnetic field of the right size, we get an answer bigger than a whole atom, so trying to explain an electron’s magnetic field as the result of it spinning on an axis is clearly wrong.