Quantum Mechanics

by Jim Al-Khalili

  Heisenberg and the uncertainty principle

  Werner Heisenberg’s contribution to quantum mechanics was profound. In 1925 he formulated a new and strange kind of mathematics to describe atoms. In contrast with Schrödinger’s wave mechanics, his far more abstract approach was called matrix mechanics and suggested that the spread-out nature of quantum particles wasn’t a real wave at all, but an abstract mathematical entity that becomes physically tangible and solid only when we observe it.

  Schrödinger, on the other hand, preferred to think of the quantum world as physically real, even if his wave function could only give us probabilities: that an electron was ‘really’ spread out, even when we were not looking. Heisenberg hated this, arguing that we must give up on ever being able to picture atoms accurately.

  Today we have learnt to live with these two complementary ways of viewing the quantum world: Heisenberg’s abstract mathematical way and Schrödinger’s wavy way. Other quantum pioneers went on to show that these two seemingly incompatible approaches were, in fact, equivalent.

  In 1927 Heisenberg proposed his famous uncertainty principle, which states that we can know either the location of, say, an electron, or its speed, but not both at once. This is not a result of the experimenter giving the electron an unavoidable kick through the very act of measuring its position. Rather it gives us a limit on what we can predict about the quantum world.

  Explaining chemistry

  In the late 1860s Dmitri Mendeleev came up with the periodic table, in which elements of similar physical and chemical properties were grouped into families. But it was only in 1925 that the Austrian prodigy Wolfgang Pauli discovered that these properties were dependent on how an element’s electrons occupied its atoms’ quantum orbits, or shells. Each electron has a set of numbers that define its ‘quantum state’. He pointed out that no two electrons in the same atom can be in the same quantum state.

  This explains why electrons don’t all drop down to the lowest-energy orbit. Instead, they fill successive ‘shells’, the number in each governed by those quantum rules first laid out by Bohr. Once a shell is full, further electrons must occupy the next one out. The electrons in the outermost shell then govern how atoms bond together to make the huge variety of possible chemical compounds as well as explaining their physical properties, such as how well they conduct heat or electricity.

  Broadly speaking, particles of matter like electrons, along with the constituents of the nucleus, protons and neutrons, which obey Pauli’s rule, known as the Exclusion Principle, are called fermions. Conversely, particles of pure energy, such as photons, which do not obey this principle are called bosons. The difference, Pauli explained, is due to the way they ‘spin’ – not in a logical way like a basketball, but rather in a very strange quantum way that, for example, even allows for electrons to spin in both directions at once!

  How the Sun shines

  An intriguing yet important quantum concept is called tunnelling. It is a process that explains some of the most fundamental mechanisms in the universe.

  This is when a quantum particle, such as an electron, proton or alpha particle, can hop from one side of an energy barrier to the other in a way that should be impossible. Consider trying to roll a ball up over a hill. You would need to give it enough energy to get it to the top. But in the quantum world, there would be a small probability that the ball could disappear from one side of the hill and reappear instantaneously on the other.

  To visualize this, we must think of the electron not as a tiny localized particle, but as a fuzzy cloud that can seep through the barrier such that, at any given time, there is a non-zero probability that it will ‘materialize’ on the other side.

  Tunnelling plays a vital role in sustaining our Sun, hence all life on Earth. The process that allows the Sun to shine is called thermonuclear fusion, whereby two protons fuse together in the first stage of turning hydrogen into helium, releasing large amounts of energy. You might expect that the positive charge of the protons means they repel each other and cannot stick, just like two north poles of magnets. But thanks to their wave nature and quantum tunnelling, they can sometimes leak through their repulsive force field and get close enough together to fuse.

  Our Sun is only hot enough to fuse hydrogen to form helium and a few other light elements. More massive stars can make heavier elements like lead and gold when they explode in a supernova.

  Dirac and antimatter

  In a recent poll, the Englishman Paul Dirac was voted the fifth greatest physicist of all time, after Newton, Einstein, Maxwell and Galileo, which gives you some idea of the high regard in which this unsung hero is held in science.

  Dirac was a shy man who was often more concerned with the elegance of his mathematics than the results of experiments. He once said:

  I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.

  He showed that the two different ways of describing the quantum world, due to Heisenberg and Schrödinger, were equivalent, and went on to modify quantum mechanics to take into account particles moving at close to the speed of light. He had to invent a new equation that now bears his name.

  The Dirac equation famously predicted the existence of antimatter. Today, we know that for every matter particle there can exist its antimatter counterpart, but if a particle and its antiparticle come into contact they will annihilate in a burst of pure energy. However, the notion of an antimatter bomb is still very much science fiction, so don’t worry. The process of matter-antimatter annihilation can also happen in reverse, whereby a quantum of energy, such as a photon of light, can be converted into a pair of particles, such as an electron and its antimatter partner, a positron.

  Proof of antimatter: the spiral lines show tracks of an electron/positron pair being created and bent in opposite directions by a magnetic field because they have opposite electric charges.

  Clash of the titans

  The nature of reality described by the new quantum mechanics espoused by Niels Bohr and the Copenhagen school of thought was so strange and counterintuitive that many physicists, including Einstein himself, were unhappy with it. Things came to a head at the now famous Solvay Conference in Brussels in 1927.

  For several days over the course of the week there, Einstein would present Bohr with arguments in the form of thought experiments in which he claimed to show that quantum mechanics was not the whole story and that the only way to avoid all the weirdness was to insist that it could not be complete in its current form. Each day, Bohr would go away and mull over the problem, only to return the following morning and demolish Einstein’s argument. Eventually, Bohr even used Einstein’s greatest contribution to science, his general theory of relativity, against him by showing that it was consistent with the predictions of Heisenberg’s uncertainty principle.

  History books tend to record that the Solvay Conference essentially marked the completion of the mathematical foundations of quantum mechanics. They also claim that Einstein’s insistence that ‘God does not play dice’, referring to the unpredictability and fuzziness of the quantum world, was shown to be a forlorn hope – that we must accept the strangeness of reality at the tiniest scales.

  Nevertheless, the arguments over the meaning of it all still rage on to this day.

  Cats in boxes

  In 1935, Erwin Schrödinger proposed one of the most famous thought experiments in science to highlight its weirdness.

  He asked what would happen if a cat were shut in a box with a radioactive substance and a container of poison that would be released when the radioactive material emitted a particle.

  Since quantum mechanics tells us we cannot know the moment of decay of a radioactive atom, when the box is closed we can only ascribe probabilities to the process, and thus to whether the cat is alive. Thus, we must describe the particle as having been both emitted and not emitted at the s
ame time. We say it exists in a quantum ‘superposition’.

  Since the state of the cat now rests on a quantum event, it too must be both dead and alive simultaneously. We only force it to ‘choose’ when we open the box to look.

  Is this really any different from the situation of not knowing what birthday presents you have until you unwrap them? Is it indeed just a metaphysical debate?

  One sensible way of resolving the issue is to assume that such quantum superpositions exist only at the atomic scale and ‘leak away’ in a process called ‘decoherence’ that occurs when a tiny isolated quantum system is forced into contact with its surroundings. So the superposition no longer survives once we get to large objects like cats, made up of trillions of atoms, which can never be in two states at once.

  Digging deeper

  By the mid-1930s just a handful of elementary particles were known to exist. They included the proton and neutron that make up the atomic nucleus. But powerful accelerators (the most well-known example of which today is the Large Hadron Collider at CERN, near Geneva), were soon smashing these particles together at ever higher energies, creating new ones in the process.

  Soon, so many new particles were discovered that a new classification scheme was required. To restore order, Murray Gell-Mann and George Zweig proposed that protons and neutrons were, in fact, not elementary at all, but were composed of tinier constituents called ‘quarks’. Their hypothesis was confirmed in a series of experiments between 1967 and 1973 at the Stanford Linear Accelerator in California.

  Today, we know there to be six ‘flavours’ of quark: ‘up’, ‘down’, ‘strange’, ‘charm’, ‘top’ and ‘bottom’. Protons and neutrons are made of just up and down quarks. Along with quarks, another group of particles, called leptons, includes the electron and its two heavier relatives, the muon and the tau, and three types of neutrino.

  The Standard Model of particle physics is like a periodic table for elementary particles. As well as the matter particles, it includes the Higgs boson and force-carrying particles like the photon and gluon.

  The Large Hadron Collider in CERN is hunting for an entirely new family of particles called supersymmetric particles that may or may not exist, but which would help solve a number of mysteries in physics.

  Spooky action

  Possibly the strangest feature of the quantum world is the notion of entanglement – so strange that even Einstein refused to believe it, calling it ‘spooky action’. This is the process whereby two separated particles remain ‘connected’ such that anything happening to one of them will instantaneously affect its partner. This is referred to as a non-local connection, which is not possible in classical mechanics involving everyday objects because communication faster than the speed of light is forbidden.

  But non-locality and entanglement are commonplace in the quantum world. Mathematically, this is just the extension of the idea that particles can sometimes behave like spread-out waves. If two particles come into close contact with each other they can become correlated and behave as a single entity, even if they are then moved to opposite sides of the universe. Even more incredibly, if one of them is put into a quantum superposition of being in two states at once, then the second particle will also be forced into a superposition. So, by measuring one we destroy the superposition of its remote partner, instantaneously and regardless of the distance between them.

  A word of warning, though: please do not think you can appeal to quantum entanglement to explain non-scientific ideas such as telepathy. Like other quantum phenomena, it is constrained to the subatomic realm. We will see later, however, that entanglement can be used in some ingenious applications.

  Quantum field theory

  In the late 1940s three physicists, including the great Richard Feynman, came up with a powerful theory called quantum electrodynamics, or QED for short. It was a generalization of quantum mechanics and provided a new way of describing the way matter interacts with light – in fact, how all matter is held together through the electromagnetic force.

  QED is an example of what is called a quantum field theory, an idea that goes back to the late 1920s with the work of Paul Dirac, who wrote a pioneering paper that combined quantum mechanics with Maxwell’s theory of electromagnetism.

  However, throughout the 1930s and 40s, quantum field theory was plagued by a troublesome mathematical problem: calculations always led to infinite answers when they should have been finite. The basic premise is that empty space is never truly empty, but is, in fact, a froth of particles and anti-particles continuously appearing and disappearing. However, this means that even something as straightforward as the electrical force between two electrons has to be written as an infinite series of ever-more complex processes taking place in the space between them.

  The issue of the infinities was solved by QED, which is today regarded as the most accurate theory in all of physics. Feynman’s approach is particularly appealing, and typical of his skills as one of the greatest scientific communicators of all time, because it makes use of powerful pictorial representations called Feynman diagrams.

  But it works! Applying quantum mechanics

  Quantum mechanics lies at the heart of so much of modern physics and chemistry, and plays a central role in our everyday lives, often in ways you might not realize. The rules of quantum mechanics explain how electrons arrange themselves in atoms and how atoms fit together to make molecules, and hence govern the nature of all the materials we see around us. For example, without quantum tunnelling we would not have understood how electricity is conducted in semiconductors, so we would not have created the silicon chip, and so would not have the computer or the internet. And the way electrons spit out photons of light led to the invention of the laser, which is used in all sorts of medical and industrial applications, not to mention leisure and entertainment – your DVD player reads the data using a laser.

  We have quantum mechanics to thank not only for the technology behind the ubiquitous microchip, but also for another device found in electronic circuits that exploits quantum mechanics more directly: the tunnelling diode, which is used as a very fast switch in microprocessors. Quantum tunnelling also gave us nuclear power, electron microscopes and MRI scanners, which make use of quantum spin. Even the smoke detectors in your home rely on quantum tunnelling of subatomic particles.

  And just consider that smartphone of yours that you’ve been using to google something on those rare occasions when I’ve lost you. It is packed with components that only do what they do because of our understanding of quantum mechanics.

  Quantum 2.0

  Devices that fall into the generic term of ‘quantum technologies’ are considered part of the second quantum revolution (or Quantum 2.0) – to distinguish them from the first quantum revolution of lasers, microchips and MRI scanners. Quantum technologies encompass a class of devices that can manipulate exotic states of matter through quantum superposition, tunnelling or entanglement.

  Advances in areas such as quantum information theory, quantum electronics, quantum optics and nanotechnology are helping to develop such devices as highly accurate sensors, atomic clocks, quantum processors and secure communication instruments that use quantum cryptography. It’s all very exciting.

  Quantum cryptography, for example, relies on something called quantum key distribution to guarantee secure communication, because the key to encrypt and decrypt messages and information relies on pairs of quantum-entangled particles. To obtain the key, any eavesdropper or hacker would have to do the equivalent of intercepting and measuring one of the entangled particles, but in doing so would inevitably destroy their delicate quantum state, thus raising the alarm.

  For now, we can make do with ‘non-quantum’ public key cryptosystems. The public key encryption, which is still virtually impossible to crack, means we can safely submit our credit card details online. However, if and when we invent quantum computers then public key cryptosystems would be compromised and we would then need to move to quan
tum encryption.

  How to build a quantum computer

  Quantum computers make direct use of quantum mechanics to perform operations on data, and differ from the ‘classical’ binary computers that we use today.

  A quantum computer is based on the idea of the ‘quantum bit’ or qubit. In a classical computer, the basic component is the bit, which is either on or off (0 or 1). However, a qubit can exist in both states at once: a quantum superposition of 0 and 1 at the same time. By entangling many qubits together, we can achieve the ultimate in parallel processing, carrying out all computations simultaneously. Quantum computers would be able to perform certain tasks many times more quickly than even the most powerful classical computer. They are therefore expected to have many important applications.

  There are currently several approaches to building a practical quantum computer. All rely on the idea of manipulating entangled superpositions of atoms, but all ultimately suffer with the same problem: how to prevent these delicate superpositions from leaking away and decohering, just like Schrödinger’s cat, before the quantum task is complete.

  Significant advances have been made recently, but it is debatable whether anyone has built a true quantum computer yet. However, in 2013 a consortium including NASA and Google began looking into how quantum computers might be applied in areas like artificial intelligence. It is anticipated that many applications such as this, which sound like science fiction today, will soon completely transform our world.