The Book of Nothing Page 20
Einstein’s development of the theories of special and general relativity was one half of the story of the development of modern physics. The other half is the story of quantum physics, pioneered by Einstein, Max Planck, Erwin Schrödinger, Werner Heisenberg, Niels Bohr and Paul Dirac. Whereas the new theory of gravity was a single-handed creation by Einstein, needing no revision or interpretation, the quantum theory of the microworld was the work of many hands which had a tortuous path to clarity and utility. The task of unravelling what it meant combined challenging problems of mathematics with subtleties of interpretation and meaning, some of which are far from resolved even today. Each year several popular science books will appear which seek to explain the mysteries of quantum mechanics in a manner that readers who are not physicists will be able to understand.3 Each of these authors is motivated to try yet another explanation of how it works by some of the unnerving words of warning from the founding fathers. From Niels Bohr, its principal architect,
“Anyone who is not shocked by quantum theory has not understood it”;4
or from Einstein,
“The quantum theory gives me a feeling very much like yours. One really ought to be ashamed of its success, because it has been obtained in accordance with the Jesuit maxim: ‘Let not thy left hand know what thy right hand doeth’”;5
or Richard Feynman,
“I think I can safely say that nobody understands quantum mechanics”;6
or Werner Heisenberg,
“Quantum theory provides us with a striking illustration of the fact that we can fully understand a connection though we can only speak of it in images and parables”;7
or Hendrick Kramers,
“The theory of quanta is similar to other victories in science; for some months you smile at it, and then for years you weep.”8
Yet for all this ambivalence, the quantum theory is fabulously accurate in all its predictions about the workings of the atomic and subatomic worlds. Our computers and labour-saving electronic devices are built upon the things it has revealed to us about the workings of the microworld. Even the light-detectors that enable astronomers to see supernovae near the edge of the visible universe rely upon its strange properties.
The quantum picture of the world grew out of the conflicting pieces of evidence for the wavelike and particlelike behaviour of light. In some experiments it behaved as if it were composed of ‘particles’ possessing momentum and energy; in others it displayed some of the known properties of waves, like interference and diffraction. These schizophrenic behaviours were only explicable if energy possessed some revolutionary properties. First, energy is quantised: in atoms it does not take on all possible values but only a ladder of specific values whose separation is fixed by the value of a new constant of Nature, dubbed Planck’s constant and represented by the letter h. An intuitive picture of how the wavelike character of the orbital behaviour leads to quantisation can be seen in Figure 7.1, where we can see how only a whole number of wave cycles can fit into an orbit.
Second, all particles possess a wavelike aspect. They behave as waves with a wavelength that is inversely proportional to their mass and velocity. When that quantum wavelength is much smaller than the physical size of the particle it will behave like a simple particle, but when its quantum wavelength becomes at least as large as the particle’s size then wavelike quantum aspects will start to be significant and dominate the particle’s behaviour, producing novel behaviour. Typically, as objects increase in mass, their quantum wavelengths shrink to become far smaller than their physical size, and they behave in a non-quantum or ‘classical’ way, like simple particles.
Figure 7.1 Only a whole number of wavelengths will fit around a circular orbit, as in (a) but not in (b).
The wavelike aspect of particles turned out to be extremely subtle. The Austrian physicist Erwin Schrödinger proposed a simple equation to predict how a wavelike attribute of any particle changes in time and over space when subjected to forces or other influences. But Schrödinger did not have a clear idea of what this attribute was that his equation could so accurately calculate. Max Born was the physicist who saw what it must be. Curiously, Schrödinger’s equation describes the change in the probability that we will obtain a particular result if we conduct an experiment. It is telling us something about what we can know about the world. Thus, when we say that a particle is behaving like a wave, we should not think of this wave as if it were a water wave or a sound wave. It is more appropriate to regard it as a wave of information or probability, like a crime wave or a wave of hysteria. For, if a wave of hysteria passes through a population, it means that we are more likely to find hysterical behaviour there. Likewise, if an electron wave passes through your laboratory it means that you are more likely to detect an electron there. There is complete determinism in quantum theory, but not at the level of appearances or the things that are measured. Schrödinger’s amazing equation gives a completely deterministic description of the change of the quantity (called the ‘wave function’) which captures the wavelike aspect of a given situation. But the wave function is not observable. It allows you only to calculate the result of a measurement in terms of the probabilities of different outcomes. It might tell you that fifty per cent of the time you will find the atom to have one state, and fifty per cent of the time, another. And, remarkably, in the microscopic realm, this is exactly what the results of successive measurements tell you: not the same result every time but a pattern of outcomes in which some are more likely than others.
These simple ideas laid the foundations for a precise understanding of the behaviour of heat radiation and of all atoms and molecules. At first, they seem far removed from the definite picture of particle motion that Newton prescribed; but remarkably, if we consider the limiting situation where the particles are much larger than their quantum wavelengths, the quantum theory just reduces to the conclusion that the average values of the things we measure obey Newton’s laws. Again, we see this important feature of effective scientific progress, that when a successful theory is superseded it is generally replaced by a new theory with an enlarged domain of applicability which reduces to the old theory in an appropriate limiting situation.
At first, the quantum theory seems to usher in a picture of a world that is founded upon chance and indeterminism, and indeed it was Einstein’s belief that this was so, and it led him to spurn the theory he had helped to create as something that could not be part of the ultimate account of how things were. He could not believe that ‘God plays dice’. Yet, on reflection, something like the quantum theory is needed for the stability of the world. If atoms were like little solar systems in which a single electron could orbit around a single proton with any possible energy, then that electron could reside at any radius at all. The slightest buffeting of the electron by light or distant magnetic fields would cause tiny shifts in its energy and its orbit to new values because all possible values are permitted. The result of this democratic state of affairs would be that every hydrogen atom (made of a proton and an electron) would be different: there would be no regularity and stability of matter. Even if all atoms of the same element started off identical, each atom in Nature would undergo its own succession of external influences which would cause a random drift in its size and energy. All would be different.
The quantum saves us from this. The electron can only occupy particular orbits around the proton, with fixed energies: hydrogen atoms can only have a small number of particular energies. In order to change the structure of the atom it must be hit by a whole quantum of energy. It cannot just drift into a new energy state that is arbitrarily close to the old one. Thus we see that the quantisation of atomic energies into a ladder of separate values, rather than allowing them to take on the entire continuum of possible values, lies at the heart of the life-supporting stability and uniformity of the world around us.
One of the most dramatic consequences of the wavelike character of all mass and energy is what it does for our idea of a vacuum.
If matter is ultimately composed of tiny particles, like bullets, then we can say unambiguously whether the particle is in one half of a box or the other. In the case of a wave, the answer to the question ‘Where is it?’ is not so clear. The wave spans the whole box.
The first application of the quantum idea was made in 1900 by the great German physicist Max Planck, who sought to understand the way in which energy is distributed amongst photons of different wavelengths in a box of heat radiation – what is sometimes called ‘black-body’ radiation.9 Observations showed that the heat energy apportioned itself over different wavelengths in a characteristic way. Our heat and daylight are provided by the Sun. Its surface behaves like a black-body radiator with a temperature of about 6000 degrees Kelvin.10 There is little energy at short wavelengths. The peak is in the green part of the spectrum of visible light but most of the energy is emitted in the infrared region which we feel as heat (see Figure 7.2). The shapes of the curves change as the temperature increases in the manner shown in Figure 7.3. As the temperature increases, so more energy is radiated at every wavelength, but the peak of the emission shifts to shorter wavelengths.
Figure 7.2 The spectrum of a ‘black-body’ radiation source with a temperature of 6000 degrees Kelvin, similar to that of the Sun.
Tantalisingly, before Planck’s work, it was not possible to explain the overall shape of this curve. The long-wavelength region where the energy steadily falls could be explained, as could the location of the peak, but not the fall towards short wavelengths. Planck was first able to ‘explain’ the curves by proposing a formula of a particular type. But this was not really explaining what was going on, merely describing it succinctly. Planck wanted a theory which predicted a formula like the one that fitted so well. He was impressed by the fact that the black-body energy distribution had a universal character. It did not matter what the emitter was made of; whether it was a flame, or a star, or a piece of hot iron, the same rule applied. Only the temperature mattered. It was a bit like Newton’s law of gravity: the material out of which things are made does not seem to matter to gravity, they can be cabbages or kings; it is just their mass that determines their gravitational pull.
Figure 7.3 The changing shape of the Planck curves as temperature, in degrees Kelvin, increases.
Planck wanted to describe the behaviour of black-body radiation by the action of a collection of tiny oscillators, gaining energy by collisions with each other as heat is added, and losing energy by sending out electromagnetic waves at a frequency determined by that of the oscillations. Here, Planck had his most brilliant insight. In the past it had always been assumed that the oscillators in a system like this could emit any fraction of their energy, no matter how small. Planck proposed instead that the energy emission can only occur in particular quotas, or quanta, proportional to the frequency, f. Thus the energies emitted can only take the values 0, hf, 2hf, 3hf, and so on, where h is a new constant of Nature,11 which we now call Planck’s constant. Planck modelled the whole glowing body as a collection of many of these quantised oscillators, each emitting light of the same frequency as it vibrates. Their energies can only change by a whole quantum step. At any moment it is more likely that the hot body contains more oscillators with low energies than those with high energies because the former are easier to excite. From these simple assumptions, Planck was able to show that the radiation emitted at each wavelength was given by a formula that precisely followed the experimental curves. The ‘temperature’ is a measure of the average value of the energy. Better still, the energy that his formula predicted would be emitted at wavelengths not yet observed subsequently proved to be correct.
Ever since these successful predictions, Planck’s black-body law has been one of the cornerstones of physics. Most dramatically, in the last twelve years, astronomers have managed to measure the heat radiation left over from the hot early stages of the expanding Universe with unprecedented precision using satellite-based receivers, observing far above the interfering effects of the Earth’s atmosphere. What they found was spectacular: the most perfect black-body heat spectrum ever observed in Nature, with a temperature of 2.73 degrees Kelvin.12 This famous image is shown in Figure 7.4.
Figure 7.4 The spectrum of the heat radiation left over from the early stages of the Universe and measured by NASA’s Cosmic Background Explorer satellite (COBE). No deviations from a perfect Planck curve have been found.
Planck’s deductions about the nature of thermal equilibrium between matter and radiation at a given temperature were widely explored and ultimately led to the creation of a full quantum theory of all atomic interactions. This picture turned out to have one mysterious aspect to it. It described the intuitive idea of an equilibrium of radiation in a container. If the radiation started hotter than the walls of the container then the walls would absorb heat until they attained the same temperature as the radiation. Conversely, if the walls were initially hotter than the radiation they enclosed then they would emit energy that would be absorbed by the radiation until the temperatures were equalised. If you tried to set up an empty box whose walls possessed a finite temperature then the walls of the box would radiate particles to fill the vacuum.
As the implications of the quantum picture of matter were explored more fully, a further radically new consequence appeared that was to impinge upon the concept of the vacuum. Werner Heisenberg showed that there were complementary pairs of attributes of things which could not be measured simultaneously with arbitrary precision, even with perfect instruments. This restriction on measurement became known as the Uncertainty Principle. One pair of complementary attributes limited by the Uncertainty Principle is the combination of position and momentum. Thus we cannot know at once where something is and how it is moving with arbitrary precision. The uncertainty involved is only significant for very small things with a size comparable to their quantum wavelength. One way of seeing why such an uncertainty arises is to recognise that the act of making a measurement always disturbs the thing being measured in some way. This was always ignored in pre-quantum physics. Instead, the experimenter was treated like a bird-watcher in a perfect hide. In reality, the observer is part of the system as a whole and the perturbation created by an act of measurement (say light bouncing off a molecule and then being registered by a light detector) will change the system in some way. Another, more sophisticated and more accurate, way to view the Uncertainty Principle is as a limit on the applicability of classical notions like position and momentum in the description of a quantum state. It is not that the state has a definite position and momentum which we are prevented from ascertaining because we change its situation when we measure it. Rather, it is that classical concepts like position and velocity cannot coexist when one enters the quantum regime. In some ways this is not entirely surprising. It would be a very simple world if all the quantities that describe the behaviour of very big things were exactly those that were needed to describe very small things. The world would need to be the same all the way down to nothing.
THE NEW VACUUM
“The vacuum is that which is left in a vessel after we have removed everything which we can remove from it.”
James Clerk Maxwell13
The Uncertainty Principle and the quantum theory revolutionised our conception of the vacuum. We can no longer sustain the simple idea that a vacuum is just an empty box. If we could say that there were no particles in a box, that it was completely empty of all mass and energy, then we would have to violate the Uncertainty Principle because we would require perfect information about motion at every point and about the energy of the system at a given instant of time. As physicists investigated more and more physical systems in the light of quantum theory, they found that the last stand mounted by the Uncertainty Principle manifested itself in the form of what became known as the zero-point energy. If one looked at the impact of quantisation on systems like the oscillators that lay at the heart of Planck’s description of heat radiation equilibrium, it emerged that there
was always a basic irreducible energy present that could never be removed. The system would not permit all its energy to be extracted by any possible cooling process governed by the known laws of physics. In the case of the oscillator, the zero level was equal to one-half of hf, the quantum of energy.14 This limit respects and reflects the reality of the Uncertainty Principle in that if we know the location of a particle oscillator then its motion, and hence its energy, will be uncertain, and the amount is the zero-point motion.
This discovery at the heart of the quantum description of matter means that the concept of the vacuum must be somewhat realigned. It is no longer to be associated with the idea of the void and of nothingness or empty space. Rather, it is merely the emptiest possible state in the sense of the state that possesses the lowest possible energy: the state from which no further energy can be removed. We call this the ground state or the vacuum state.