by Tim James
When we think of waves, we can calculate how much energy they are carrying from their frequency (how many waves hit you per second) and their wavelength (how far apart each wave peak is).
Since we can also calculate the energy of a moving particle given its mass and velocity, de Broglie posed the question: why not set these two energies equal to each other? If we know the properties of something in particle terms we can calculate its energy and then switch our brains to think of it as a wave, the energy of which we have just calculated. Energy serves as a translator between wave physics and particle physics.
Initially this suggestion was met with scepticism. Were we seriously going to say every particle had a wavelength and every wave had a mass? Fortunately for de Broglie, Albert Einstein liked his idea a lot and began endorsing it in lectures (which never hurts).
The de Broglie approach says you can take any particle you like and calculate its ‘associated wavelength’. Once you have that, you can build a double slit of the appropriate size and fire your particle at it, getting an interference pattern on the other side. While we might not be able to visualise how something can be a particle and a wave at the same time, we can certainly do calculations on it and get reliable data.
And it works too. In 1944 Ernest Wollan used de Broglie’s theory to diffract neutrons, thousands of times heavier than electrons, through a crystal of table salt.1 Protons can also be diffracted the same way, although surprisingly there are no records of who did this experiment first. With hindsight, I probably should not have claimed it was me in that university application letter.
A BIT EXTREME
To learn that protons, neutrons and electrons all behave like waves is both profound and peculiar. Every object in the world is made from those particles so everything we think of as matter, including our own bodies, is wavelike as well. Your body has a wavelength and if we somehow fired you at an appropriate double slit you could be diffracted.
If you’re curious, an average human being launched out of a cannon at 30 metres a second would have a de Broglie wavelength of about 0.0000000000000000000000000003 metres. If we could somehow find a way of getting every atom in the human body to line up with a double slit that size we could genuinely get diffraction occurring. I reckon you could use your volunteer from the earlier experiment where you blasted their hand off with a cannonball.
The current world record for diffracting objects bigger than a single particle is held by Sandra Eibenberger who, in 2013, managed to wave-interfere a whole molecule of C284H190F320S12N4. That’s 810 atoms going through both slits at the same time, superpositioning with themselves on the other side.2 Not quite a whole person, but we have to start somewhere.
ENTER HEISENBERG
Before he was a law-breaking New Mexico chemistry teacher played by Bryan Cranston, Werner Heisenberg was one of the finest mathematicians in the world. Unlike Planck, Einstein and de Broglie, who focused on experimental results, Heisenberg was more interested in taking well-established theories and twisting them to breaking point without worrying what it meant for the lab workers.
He was notoriously ignorant of real-world physics and during his doctoral viva was asked how a simple battery worked and had no idea3 (which I find reassuring since evidently even Heisenberg faced humiliating interviewers).
Despite this practical incompetence, his mathematical brilliance was unparalleled and in 1920 he was hired to work for Arnold Sommerfeld, one of the physicists who had helped Bohr devise his atom theory.
Sommerfeld gave him a puzzle to work on which involved the mathematics of how light splits itself, which Heisenberg cracked in under a fortnight but his solution was so complicated that Sommerfeld rejected it, assuming he could not have hit the answer so quickly. That was until a few months later when the far more established physicist Alfred Landé published the exact same idea, thus getting the credit.4
Soon after that experience, Heisenberg transferred to work with Niels Bohr at the Copenhagen Institute in Denmark, fast becoming the world citadel for quantum research. Perhaps he felt bummed about the way Sommerfeld could not accept his genius, or maybe he just fancied working for a Nobel Prize winner (Sommerfeld was nominated eighty-four times but never won). Either way, Heisenberg relocated and became Bohr’s top student and one of his closest friends.
These were Heisenberg’s golden years as he found himself surrounded by the sharpest minds in Europe, devising much of the methods and equations we still use in quantum physics today. Heisenberg and Bohr would go hiking together in the mountains, go out on the town in search of women, and spend every other waking moment discussing the peculiar behaviour of particles. It was a time when Heisenberg was truly respected and happy.
Sadly, the latter half of his life was more controversial. As Nazism spread across Europe, many scientists fled the onslaught and relocated to America. Heisenberg stayed put, however, and was recruited to help the Nazis build an atom bomb.
According to some historians, Heisenberg tried to sabotage these efforts from within because in post-war interviews he described accurately how such a bomb could be built, even though his project never succeeded. Perhaps he figured it all out but kept his mouth shut to stifle the Nazi effort.5
However, in 2002 a series of letters between Heisenberg and Bohr was uncovered, and they paint a murkier picture. It appears Heisenberg was perfectly comfortable working on the bomb and it only failed because he did not have a good team working for him (the good scientists were Stateside) and he was himself clueless in the lab.6 Presumably, all the equipment was battery operated.
Nobody knows for sure what Heisenberg’s ethical stance was during this time. He did get himself in hot water for promoting the work of Albert Einstein, a Jewish physicist, and only avoided serious trouble because his mum got him out of it. Mrs Heisenberg was good friends with Mrs Himmler, the mother of Heinrich Himmler, head of the SS, and when Heisenberg was in trouble she phoned her friend and effectively said, ‘Tell your son to leave mine alone!’7
When it comes to Heisenberg’s politics, I suspect he simply was not thinking very hard about the ethical implications of what he was doing and just wanted to solve physics problems people gave him.
Judging him as good or evil in retrospect is difficult because we do not have much to go on, and let us not forget that the Allied side was hardly composed of saints. Robert Oppenheimer, Heisenberg’s opposite number in America, was a man who once tried to poison his doctoral supervisor in true fairy-tale fashion by covering an apple in toxic chemicals – what we’d probably call attempted murder.8
But if we leave aside the political overtones of his later life, Heisenberg’s contribution to quantum theory is still invaluable.
ELIMINATE ALL DISTRACTIONS
One summer, while suffering from a severe bout of hay fever, Heisenberg decided to spend a holiday on the island of Heligoland, which has no pollen-producing plants. During this break he hit upon a new approach to the mathematics of quantum theory, which included the origin of the thing for which he is best known today: the Heisenberg uncertainty principle.9
Particles have clearly defined properties we can measure. Things such as location, velocity, mass and so on. If we know everything about the initial state of a particle we can, in theory, predict everything it is going to do in the next moment. And the next. And the next.
This philosophy of determinism began with Isaac Newton and is what made physics so important. Whereas mystics of old claimed you needed to slaughter virgins and drink dove’s blood to predict the future, Newton showed you could do it with a few equations and 100 per cent accuracy.
But de Broglie and co. discovered that all particles have wave properties too. Asking ‘where’ a particle is and ‘how fast’ it is going are separate questions in Newton’s physics but because waves are, by definition, in motion constantly and their location is spread across a region, concepts such as speed and location are no longer independent of each other.
If you know something
about the location of a wave this also contains information about its velocity and vice versa – the two properties are linked. So Heisenberg applied a similar idea to particles. Their motion and location could not really be treated as separate because they were only partly particulate.
If we have a particle and we have not yet taken any measurements on it we can specify what momentum and location it probably has with a graph such as this:
All we know is that the particle’s physical characteristics will be somewhere inside that hump. In normal, classical terms, when we take a measurement on a particle we are squashing that hump into a dot somewhere on the grid, which pinpoints both of the values and tells us where the particle is and what momentum it has.
In this diagram we read along the horizontal axis to find the particle’s location and then read vertically to discover its momentum. Simple enough.
But Heisenberg knew waves were different. What happens when you try to pinpoint a wave is you get a spike such as this:
We know exactly where the particle is because we’ve narrowed its location to a single value on the horizontal axis, but if you read vertically you find a whole variety of momenta occurring at the same time. Because the wave’s position and momentum are not separated like they would be according to everyday classical physics, if you know the location of your particle you lose all certainty about its momentum.
Or if we imagine squeezing things in the other direction we might be able to measure a precise value for a particle’s momentum, but we end up with the particle existing in lots of locations at once:
Momentum and location are linked properties in quantum theory, so we can either know where a particle is or what its momentum is, but never both at the same time.
ARE WE QUITE CERTAIN OF THAT?
This was the first ‘uncertainty relationship’ and the most oft-quoted one: you can never know a particle’s location and momentum at the same time and if you know one of them you lose any knowledge of the other. As quantum theory evolved we discovered other pairs of properties that are linked together (we shall meet some later) but Heisenberg’s original still shook the foundations of physics.
Putting it bluntly, quantum theory says it is impossible to know everything about an object because there will always be something we cannot measure. If you know enough about one property, you automatically lose information about something else.
Newton’s view of a universe in which we know the future by knowing the present is thus slaughtered by a maths nerd with hay fever. You cannot know everything about the present, so correctly predicting the future is never going to happen. Ever.
Sometimes the Heisenberg uncertainty principle gets misrepresented by saying our equipment is not good enough to know all of a particle’s properties but this is selling it short. It does not matter how well we build a detector and it does not matter how precisely we make a measurement. We can never pin down a particle’s properties in full because it is not exclusively a particle. It is a wave too.
Asking an electron about its particle properties is like asking, ‘Which letter is War and Peace?’ or ‘What colour is a rainbow?’ It is trying to narrow something that cannot be narrowed.
As Heisenberg himself put it: ‘We cannot know, as a matter of principle, the present in all its details.’10 Quantum theory forces us to give up Newton’s dream of predicting the future with accuracy.
In the everyday world it is easy to know where an object is and how fast it is going. That forms the basis of every sport, in fact. A game of quantum basketball would be totally unfeasible (although hilarious) because if we threw a ball with a known momentum we could no longer be sure where it was. It would blur into a ballish-cloud mid-air and, although we could watch how fast this cloud moved, we could not specify exactly where we would need to stand to catch it. The only reason we do not notice the uncertainty principle in our everyday lives is because we are very big compared to the size of the ‘uncertainty clouds’. It does not cause an issue when we are doing everyday tasks but if we want to take measurements of a single particle we hit a wall of ignorance.
It also means a particle can never be brought to rest. If an electron were to stop moving around a nucleus it would have a clearly defined position and a single momentum (zero). It would be behaving as a particle only and its wave character would vanish. That does not happen and thus particles have to be in perpetual motion.
In Heisenberg’s own words:
It was almost three o’clock in the morning before the final result of my computations lay before me… At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise.11
CHAPTER FOUR Taming the Beast
OH, DOCTOR SCHRÖDINGER!
Bohr’s theory of the atom with its electron shells and quantum leaps was still not sitting well in everyone’s stomach. It did not explain why only certain energy shells were allowed and the data it predicted did not always match experiment.
He had mashed his atom together from different theories like a child playing with dolls, forcing them to kiss, and he knew it would not be a permanent fix. There had to be a neater, more elegant theory nobody was thinking of. The person who cracked it was Erwin Schrödinger, a man whose personal life was the talk of the town even more than his theories.
Schrödinger was a socially eccentric, spiritually liberal, bow-tie-wearing genius who wrote prolifically on many topics including science, the arts and philosophy over a career spanning fifty years. He was an outcast in polite society and caused a scandal for living in a three-way relationship with his wife and mistress, Anny and Hilde, as well as fathering children from at least two other women (contrary to popular myth, science nerds can have very active personal lives and do not have trouble finding people with whom to be active).1
As interesting as he was though, the hanky-panky of Schrödinger’s personal life was not the reason he got a Nobel Prize (it would have been quite the acceptance speech); it was for making a superior recipe for explaining the atom.
HUMBUG
Schrödinger hated Christmas. He was notoriously against religiously tinted events and so, in December 1925, decided to isolate himself from the festivities at a remote villa in Switzerland. His wife stayed home, but he was accompanied in his hibernation by ‘an old girlfriend from Vienna’ the identity of whom has been lost.2 Although he stopped recording in his diary during this time – so nobody really knows what he got up to – we do know he brought a physics problem in his luggage to work on over the holidays.
Schrödinger initially had little interest in quantum theory. He was a talented physicist but his area of expertise was in the behaviour of waves, not particles. Electrons, photons and protons frankly bored him. That was, of course, until people realised we needed to think about these particles as waves some of the time.
Schrödinger knew there were dozens of equations to predict the behaviour of electrons as if they were particles, but nobody had invented a method to describe them as waves. He later wrote of this time (paraphrased from German), ‘this extreme idea may be wrong… on the other hand, the opposite point of view, which neglects waves, has led to such difficulties that it seems desirable to lay an exaggerated stress on the opposite approach’.3
Everyone else was using particle physics to describe atoms, but if he could come up with a wave equation to do the same thing, maybe it would give new insights. Essentially Schrödinger was trying to be awkwardly different. Given his success with the opposite sex and the Nobel Prize sitting casually on his mantelpiece, however, maybe he was onto something.
By his own ad
mission, Schrödinger did not know enough mathematics to come up with a new law,4 but he obviously surprised himself over that mysterious winter because as the new year dawned, he emerged with exactly what everyone had been searching for: an equation that accurately predicted the energies of electrons going around the nucleus.
Schrödinger put aside the question of how wave–particle duality worked and just focused on the wave side of things. He imagined stretching each electron across the surface of an atom like a piece of butter spread over toast, and this electron membrane could wrap around the nucleus and vibrate at certain frequencies.
Given a few input values such as mass or pull from the nucleus, Schrödinger’s equation accurately predicts, using something he called a ‘wavefunction’, the shape an atom’s electrons ought to vibrate in three dimensionally.
A wavefunction is an equation you solve to generate a list of properties an electron has at a particular point in space or time: the height of the wave, its wavelength, the speed at which it is rippling and so on.
Schrödinger’s equation does a calculation on this wavefunction (it’s an equation within an equation) and predicts how the electron’s properties and behaviour evolve with time. Not only that, it finally explained why only certain energy shell values were allowed.
WHERE WE’RE GOING WE DON’T NEED PARTICLES
Because every electron in an atom is trapped by the nucleus, there are restrictions on its behaviour. For example, a wave can only exist in whole numbers of ripples, shown in the two images below. The left image is showing a single wave and the right is showing a double wave.
