angular momentum Momentum is the ‘oomph’ of a travelling body – mass times velocity. Angular momentum is the equivalent for a rotating body, combining momentum with the body’s distance from the centre of rotation.
classical physics The physical theories that held sway before 1900 and the twin twentieth century upheavals of relativity and quantum theory. Newton’s laws of motion is a typical example of a classical theory, which would be superseded by special relativity, but still is a good approximation for most moving bodies.
complementary variables Heisenberg’s Uncertainty Principle links pairs of a quantum particle’s properties. The best known of these complementary paired variables are position with momentum, and energy with time. The more accurately one variable is known, the less accurately the other.
matrix A collection of numbers in a regular array. Often rectangular, matrices can have any number of dimensions. Matrices are used to work on multiple equations simultaneously.
non-relativistic equation An equation that does not take relativity into account. Newton’s second law (force = mass x acceleration) is a non-relativistic equation. For speeds well below the speed of light this is effectively correct, but as velocity increases, relativistic effects become important as, for instance, the mass of an object increases with velocity.
particle accelerators The main tool of particle physics. Accelerators push charged particles close to the speed of light then slam them into other particles or solid objects. The result is a spray of new particles generated by the collision. To date, the biggest such machine is the Large Hadron Collider (LHC) at CERN. Spanning the Swiss/French border, the LHC is in a 17-mile (27-kilometre) tunnel that accelerates streams of protons in opposite directions before colliding them.
quantum states The set of values of a quantum particle’s properties. A state can be ‘pure’, where it has a specific value – for example, the spin when measured will be ‘up’ or ‘down’ – or mixed, in which case the spin might be 40 per cent probability of ‘up’ and 60 per cent probability of ‘down’.
quantum electrodynamics Usually shortened to QED, quantum electrodynamics is our theory of how light and matter (usually an electron) interact. It is a relativistic quantum field theory, because it takes into account special relativity, is quantized and represents a field by imagining each particle accompanied by a rapidly spinning clock. Its hand indicates the particle’s phase and the probability of taking a particular path.
spacetime Relativity treats time as a fourth dimension. In relativity there is no absolute position or absolute time because the way things move influences their position in time, so it is necessary to consider spacetime as a whole, rather than to think of space and time independently.
superposition When a quantum particle has a state with, say, two possible values it will not have an actual value but rather a superposition – a collection of probabilities of being in the states, until it is measured, when it collapses to an actual value. A tossed coin has two states but no superposition. Before we look, the coin already has one of these values. But a quantum particle has no value, literally just probabilities, while in superposition.
virtual particles QED requires virtual particles, which are never seen but take part in quantum processes. The electromagnetic force, for instance, causes an electron to change path because the electron absorbs a virtual photon. Also in empty space, the uncertainty principle means that it is possible for energy levels to fluctuate, briefly bringing a pair of virtual particles – matter and antimatter – into existence before disappearing again to energy.
wavefunction In quantum physics, the wavefunction is a mathematical formula that describes the behaviour of a quantum state of the particle, which evolves over time according to the Schrödinger wave equation. The wave in question, which spreads out over time, does not describe the particle itself, but rather the probability of a quantum state having a particular value – so, for instance, it can describe the probability of finding a particle in different locations. The probability is given by the square of the wavefunction.
Quantum spin is the cause of magnetism in the everyday world. It is an intrinsic property of subatomic particles, a key parameter needed to describe a particle fully and one of four attributes required to define the quantum state of an electron in an atom. Quantum spin, just like everything else in quantum mechanics, comes in fixed packets and particular particles can have only certain amounts of spin, which is represented by their quantum spin number. All sub-atomic particles have a quantum spin number although some can have a spin number of zero. Quantum spin is related to angular momentum, the physical property attributed to rotating objects, in that it affects the measurement of angular momentum in atoms. The effects of quantum spin were first detected in relation to electrons in atoms. Electrons whizzing around the nucleus of atoms impart angular momentum to those atoms through their orbital motion. Quantum spin was discovered by German physicists Otto Stern and Walther Gerlach in 1922 during an experiment that suggested electrons in atoms also have some intrinsic angular momentum in addition to that generated by their orbital motion. This is akin to an electron spinning on its own axis while it is orbiting the atomic nucleus.
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Quantum spin is the reason why magnets work and it allows scientists to distinguish particles from one another.
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The property we call (quantum) spin was given the name because it has some similarities to classical angular momentum, but we have no reason to think that this is to do with particles spinning around (which is hard to envisage for a point particle like an electron), especially as spin’s half integer values are only ever found to be ‘up’ or ‘down’ in any direction in which they are measured.
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3-SECOND BIOGRAPHIES
1900–58
Pioneering Austrian physicist who developed much of the theory on quantum spin
GEORGE UHLENBECK & SAMUEL GOUDSMIT
1900–88 & 1902–78
Dutch physicists who co-wrote the first paper on electron spin
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Leon Clifford
The direction of quantum spin of particles determines their magnetic orientation.
Matrix mechanics is a way of describing the behaviour of quantum systems using a particular mathematical technique known as matrix algebra. The approach was developed in 1925 by German physicists Werner Heisenberg, Max Born and Pascual Jordan who were trying to understand the rules governing atomic spectra. It contrasts with the approach taken by Erwin Schrödinger who used a different mathematical technique involving differential equations to describe quantum systems. The significance of matrices lies in the fact that the order in which things are done affects the outcome. In everyday mathematics, for example, the multiplication of two numbers gives the same result no matter which way round the numbers are ordered: 2 x 3 is the same as 3 x 2. This is not the case with matrices. Instead, if the position of a particle is represented by one matrix and if the momentum of that same particle is represented by another matrix, the multiplication of these two matrices will give different results depending on the order in which it is done. The result of multiplying the position matrix by the momentum matrix is not the same as multiplying the momentum matrix by the position matrix. The difference between the two results gives rise to Heisenberg’s Uncertainty Principle.
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The principle behind matrix mechanics helped Heisenberg and his colleagues to put the uncertainty into quantum mechanics.
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Heisenberg’s matrices offered physicists a new and unfamiliar way of describing quantum behaviour that had no equivalent in the world we observe. But it seemed to be at odds with the more traditional approach, preferred by Schrödinger among others, of using differential equations. In one sense, this was another manifestation of wave-particle duality; differential equations have smoother wave-like features while matrices appear more like discrete objects. Dirac combined the two into a single formalism in 1930.
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HEISENBERG’S UNCERTAINTY PRINCIPLE
3-SECOND BIOGRAPHIES
MAX BORN & PASCUAL JORDAN
1882–1970 & 1902–80
German physicists who developed the mathematics of matrix mechanics
WERNER HEISENBERG
1901–76
German quantum pioneer who used maths to reveal the uncertain and bizarre nature of the quantum world
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Leon Clifford
Heisenberg’s matrix mechanics used the mathematics of matrices to predict quantum behaviour.
Louis de Broglie’s proposal in 1924 that particles such as electrons could behave as waves stimulated Erwin Schrödinger to formulate a mathematical theory of quantum mechanics in which the position and behaviour of particles are described in terms of a wavefunction ψ (‘psi’). This is a kind of wave, but not in the usual sense of a sound wave or an ocean wave. Rather, the wavefunction is a probability wave: its value (or, more properly, the value of the square of the wave function ψ2) at any point in space indicates the probability of finding the particle there. Waves are described mathematically by so-called differential equations, which specify how the size of the oscillations changes over time. But Schrödinger’s equation is unlike an ordinary wave equation, being more akin to the kind of equation used to describe spreading or diffusion processes. In principle, it enables scientists to calculate the wavefunction of any quantum system, and thus the probabilities of its locations, provided its mass and energy are known. In practice, solving the equation exactly is often too hard, and only approximate solutions can be found. Nonetheless, Schrödinger’s equation is the starting point for all attempts to work out how electrons are distributed in atoms, molecules and materials.
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Schrödinger’s equation provides a way to calculate how quantum particles behave as ‘probability waves’: describing where they are likely to be found at any moment.
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Schrödinger’s ‘wave mechanics’ isn’t the only way to write down quantum theory mathematically. While Schrödinger was conjuring up his equation in the 1920s, Werner Heisenberg was also developing a way to express quantum states as tables of numbers called matrices. This ‘matrix mechanics’ is still sometimes used, but Schrödinger’s waves are generally preferred, partly because they offer a more intuitive picture of the ‘appearance’ of quantum states.
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3-SECOND BIOGRAPHY
ERWIN SCHRÖDINGER
1887–1961
Austrian physicist whose work embraced quantum theory, cosmology and genetics
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Philip Ball
Probabilities for locating hydrogen’s electron in different orbits can be predicted by its wavefunction.
Born and raised in Vienna, Erwin Schrödinger was a brilliant young student with an instinctual flair for physics. By the time he secured his first permanent professorship, in Zurich in the 1920s, he had become a firm believer in the wave nature of matter. This culminated in the theory of wave mechanics – a complete and self-consistent formulation of quantum theory that is generally considered to be his greatest achievement.
Schrödinger’s wave equation is as central to quantum physics as are Newton’s laws of motion to classical physics. The mathematical validity and predictive power of Schrödinger’s equation were immediately recognized by his fellow physicists, but to his dismay virtually none of them shared his unmitigated enthusiasm for waves. To him, the prevailing Copenhagen Interpretation, championed by Niels Bohr, with its talk of ‘wave-particle duality’ and ‘collapse of the wavefunction’ was pseudoscientific nonsense. Encouraged by Einstein, one of the few people at the time to share his view, Schrödinger set about devising a thought-experiment that would highlight the absurdity of the Copenhagen Interpretation. The result was the paradox of Schrödinger’s Cat – probably the best-known image in all of quantum theory.
Before the outbreak of the Second World War, Schrödinger escaped the turmoil of continental Europe for neutral Ireland. His maternal grandmother had been British, and he spoke English almost as fluently as he spoke German. At the personal invitation of Éamon de Valera, then Irish Prime Minister, Schrödinger became Director of Theoretical Physics at the newly formed Institute for Advanced Studies in Dublin – a post he held for the next 17 years. He later described his time in Dublin as the happiest years of his life. Arguably Schrödinger’s most important work during this period was What Is Life? – a revolutionary little book that demonstrated how quantum theory and other concepts from fundamental physics could be applied to living organisms. When the secrets of DNA were unlocked a few years later by Francis Crick and James Watson, both men acknowledged their debt to Schrödinger’s book.
Compared with his scientific peers, Schrödinger had an unusual lifestyle. He wrote poetry and had an abiding interest in philosophy and Eastern mysticism. More controversially, he had a string of young mistresses. His three acknowledged children were all born during his 40-year marriage to his wife Anni, but none were hers. Anni seems to have been resigned to her husband’s serial infidelity, however, and she remained his wife until his death in Vienna in 1961 at the age of 73.
Andrew May
Born in Vienna, Austria
1910
Awarded a doctorate by the University of Vienna
1914–18
Serves as an artillery officer in the Austrian Army during the First World War
1921
Appointed Professor of Theoretical Physics at Zurich, Switzerland
1926
Schrödinger’s equation lays the foundations of wave mechanics
1927
Moves to Berlin, taking up the professorship vacated by Max Planck
Voluntarily leaves Nazi Germany and moves to Oxford; in the same year shares the Nobel Prize in Physics with Englishman Paul Dirac
1935
A paper entitled ‘The Present Situation in Quantum Mechanics’ presents the paradox of Schrödinger’s Cat
1939
Appointed Director of Theoretical Physics at the Institute for Advanced Studies, Dublin, Ireland
1944
Cambridge University Press publishes What is Life?
1956
Retires his post in Dublin and returns to Vienna
4 January 1961
Dies in Vienna
The most famous illustration of how quantum theory defies intuition is a thought experiment proposed in 1935 by Erwin Schrödinger. He suggested that it should be possible to make the state of a macroscopic object – whether a cat enclosed in a box is alive or dead, say – dependent on a microscopic quantum event, such as the decay of an atom. He imagined a device in which the radioactive decay of an atom – a quantum event governed by chance – releases a hammer that breaks a vial of poison, killing the cat. The problem is that the decaying atom can be in a mixture of states, called a superposition, implying that the cat may be simultaneously both killed and not killed. Quantum superpositions are generally destroyed by making a measurement on the quantum object, so immediately we open the box to look, the cat will be in one state or the other. But that doesn’t tell us the cat’s condition before we looked. Some scientists feel that something will intervene to put the cat in one state or the other, whether we look or not. Others are content to imagine a live-dead superposition for the cat.
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The Schrödinger’s Cat thought experiment shows how counterintuitive quantum theory is: a quantum system that is simultaneously in two states determines whether a cat is alive or dead.
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Could we test experimentally whether Schrödinger’s cat is alive or dead? Sustaining a delicate quantum superposition of states in a system big enough to contain a real cat would be almost impossible, but a microscopic ‘cat’ – a bacterium or virus, say – could be isolated from disturbances more easily. Researchers in Germany have proposed an experiment in which a virus trapped by laser light could be coaxed into a quantum superposition.
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1902–95
Hungarian-born physicist who linked Schrödinger’s cat to the problem of consciousness by postulating a friend who performed the experiment in Wigner’s absence
1965–
Spanish physicist who proposed a Schrödinger-cat experiment using microscopic living organisms
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Philip Ball
In 1927 German theoretical physicist Werner Heisenberg formulated the Uncertainty Principle – a fundamental property of quantum systems. It states that it is impossible to measure simultaneously with perfect accuracy certain pairs of physical properties (so-called complementary variables) of an atom or a particle – for example, both its position and momentum – or to be certain of its energy at some specific instant in time. The more precise the measurement of one quantity, the less precisely can the other be measured or controlled. The effect of this phenomenon is so small that it can be ignored in everyday affairs, but it is dramatic for subatomic particles and underpins quantum mechanics, which describes the motion and dynamics of atoms. This uncertainty is an intrinsic limit on our ability to measure natural phenomena at small distances; it is a fundamental property of quantum theory and not simply a failure in the measuring apparatus. One consequence is that the total energy of a particle can fluctuate by some amount, E, for a short time, t, as long as the product of E times t does not exceed Planck’s constant divided by 4 pi. This in turn means that the law of conservation of energy can be evaded for very short time spans.
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Particles are like politicians: the more you attempt to pin them down, the faster they change their position.
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The Uncertainty Principle is a reason why particle accelerators, such as the Large Hadron Collider, are so huge. In order to handle distances a thousand times smaller than the size of a proton, we require beams of particles whose energies are trillions of times larger than those of particles at room temperature. This requires huge accelerators to energize the beams to such extremes.
RELATED THEORIES
3-SECOND BIOGRAPHIES
ERWIN SCHRÖDINGER
1887-1961
Austrian physicist who created a non-relativistic equation for quantum mechanics and the equation including relativity
WERNER HEISENBERG
1901–76
German theoretical physicist who proposed the Uncertainty Principle
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Frank Close
The closer we pin down the location of a quantum particle, the less we can know about its momentum (and vice versa).
The Schrödinger equation, which encapsulates all we can know about a given quantum system in the form of the wavefunction, can only predict the various probabilities of finding it in a particular state, whereas a measurement made on the system gives a unique answer: we find it in this state or that. It appears that the very act of observing the system winnows the possibilities to a single outcome. This is called collapsing the wavefunction. Exactly how collapse occurs may depend on how the measurement is made. This represented one of the key philosophical shocks of the theory when it was first developed, because it seemed to undermine science’s supposed objectivity: the observer, apparently, could not help but influence the result. This is now known as the ‘measurement problem’. But is this collapse mere mathematical formality, or a real physical process? The conventional ‘Copenhagen Interpretation’ of quantum theory only insists that the state cannot be known until it is observed. Some think wavefunction collapse is an illusion, because all possible outcomes are realized in different worlds. For others collapse is a process like radioactive decay, with a definite timescale, which might involve the gravitational force, thus achieving the long-sought link between gravity and quantum theory.
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Wavefunction collapse reduces a superposition of quantum states to a single state, which generally happens when a measurement is made of the system’s state.
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Another interpretation of wavefunction collapse was developed by American physicist David Bohm, based on de Broglie’s ‘pilot-wave’ theory in which quantum particles are accompanied and guided by waves. It assumes that there is a single wavefunction governing the entire universe (so that everything depends on everything else), which never in fact collapses but just appears to do so locally, due to decoherence acting between the local wavefunction and that of the rest of the universe.
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3-SECOND BIOGRAPHY
ROGER PENROSE
1931–
British mathematical physicist who proposes that wavefunction collapse is the result of spacetime curvature in general relativity
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Philip Ball
In a quantum system, like the two slit process, particles exist as a probability wave, causing interference and other effects: on observation, this collapses leaving a single value.
The microscopic world is governed by quantum rules, but in the everyday world of billiard balls and teapots, classical physics applies. How does quantum become classical – where does the quantum weirdness go? A common view is that quantum effects such as the wave behaviour of particles get ‘washed out’ by interactions between the quantum particles and their environment, a phenomenon known as decoherence. These interactions mean that a particle and its environment become ‘entangled’: the properties of the particle are no longer intrinsic to it but depend on the environment. To see quantum behaviour in a system, decoherence must be suppressed, by making the system as isolated as possible from its environment. That is why quantum effects such as superpositions are usually observed only in the laboratory; they are fragile and easily destroyed by decoherence. Decoherence is a one-way affair: once it has washed away quantum-ness, you cannot get it back. The rate of decoherence – the speed at which quantum superpositions vanish – increases exponentially as the number of particles in the system increases, so big objects become classical almost instantly. Decoherence, then, makes the quantum-to-classical switch a well-defined process that depends on the precise environmental conditions.
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Decoherence is the loss of quantum behaviour owing to interactions of its constituent particles with their environment.
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How big can a quantum system be before decoherence becomes impossible to suppress, and the system starts to behave classically? Some large molecules, such as 60-atom C60, can show wavelike quantum interference effects, but these can be washed away by passing the molecules through a gas so that collisions induce decoherence. Soon it should be possible to detect quantum superpositions of vibrating states in tiny oscillating beams visible in the electron microscope.
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HEINZ-DIETER ZEH
1932–
German physicist who in 1970 identified the origin of decoherence
WOJCIECH ZUREK
1951–
Polish-American physicist who explained how decoherence ‘selects’ a few classical properties from the palette of possible quantum states
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Philip Ball
