by Brian Cox
A similarly inexplicable phenomenon was the mystery of the light emitted by atoms when they are heated. As far back as 1853, the Swedish scientist Anders Jonas Ångstrom discharged a spark through a tube of hydrogen gas and analysed the emitted light. One might assume that a glowing gas would produce all the colours of the rainbow; after all, what is the Sun but a glowing ball of gas? Instead, Ångstrom observed that hydrogen emits light of three very distinct colours: red, blue-green and violet, like a rainbow with three pure, narrow arcs. It was soon discovered that each of the chemical elements behaves in this way, emitting a unique barcode of colours. By the time Rutherford’s nuclear atom came along, a scientist named Heinrich Gustav Johannes Kayser had compiled a six-volume, 5,000-page reference work entitled Handbuch der Spectroscopie, documenting all the shining coloured lines from the known elements. The question, of course, was why? Not only ‘why, Professor Kayser?’ (he must have been great fun at dinner parties), but also ‘why the profusion of coloured lines?’ For over sixty years the science of spectroscopy, as it was known, had been simultaneously an observational triumph and a theoretical wasteland.
Figure 2.1: Bohr’s model of an atom, illustrating the emission of a photon (wavy line) as an electron drops down from one orbit to another (indicated by the arrow).
In March 1912, fascinated by the problem of atomic structure, Danish physicist Niels Bohr travelled to Manchester to meet with Rutherford. He later remarked that trying to decode the inner workings of the atom from the spectroscopic data had been akin to deriving the foundations of biology from the coloured wing of a butterfly. Rutherford’s solar system atom provided the clue Bohr needed, and by 1913 he had published the first quantum theory of atomic structure. The theory certainly had its problems, but it did contain several key insights that triggered the development of modern quantum theory. Bohr concluded that electrons can only take up certain orbits around the nucleus with the lowest-energy orbit lying closest in. He also said that electrons are able to jump between these orbits. They jump out to a higher orbit when they receive energy (from a spark in a tube for example) and, in time, they will fall back down, emitting light in the process. The colour of the light is determined directly by the energy difference between the two orbits. Figure 2.1 illustrates the basic idea; the arrow represents an electron as it jumps from the third energy level down to the second energy level, emitting light (represented by the wavy line) as it does so. In Bohr’s model, the electron is only allowed to orbit the proton in one of these special, ‘quantized’, orbits; spiralling inwards is simply forbidden. In this way Bohr’s model allowed him to compute the wavelengths (i.e. colours) of light observed by Ångstrom – they were to be attributed to an electron hopping from the fifth orbit down to the second orbit (the violet light), from the fourth orbit down to the second (the blue-green light) or from the third orbit down to the second (the red light). Bohr’s model also correctly predicted that there should be light emitted as a result of electrons hopping down to the first orbit. This light is in the ultra-violet part of the spectrum, which is not visible to the human eye, and so it was not seen by Ångstrom. It had, however, been spotted in 1906 by Harvard physicist Theodore Lyman, and Bohr’s model described Lyman’s data beautifully.
Although Bohr did not manage to extend his model beyond hydrogen, the ideas he introduced could be applied to other atoms. In particular, if one supposes that the atoms of each element have a unique set of orbits then they will only ever emit light of certain colours. The colours emitted by an atom therefore act like a finger-print, and astronomers were certainly not slow to exploit the uniqueness of the spectral lines emitted by atoms as a way to determine the chemical composition of the stars.
Bohr’s model was a good start, but it was clearly unsatisfactory: just why were electrons forbidden from spiralling inwards when it was known that they should lose energy by emitting electromagnetic waves – an idea so firmly rooted in reality with the advent of radio? And why are the electron orbits quantized in the first place? And what about the heavier elements beyond hydrogen: how was one to go about understanding their structure?
Half-baked though Bohr’s theory may have been, it was a crucial step, and an example of how scientists often make progress. There is no point at all in getting completely stuck in the face of perplexing and often quite baffling evidence. In such cases, scientists often make an ansatz, an educated guess if you like, and then proceed to compute the consequences of the guess. If the guess works, in the sense that the subsequent theory agrees with experiment, then you can go back with some confidence to try to understand your initial guess in more detail. Bohr’s ansatz remained successful but unexplained for thirteen years.
We will revisit the history of these early quantum ideas as the book unfolds, but for now we leave a mass of strange results and half-answered questions, because this is what the early founders of quantum theory were faced with. In summary, following Planck, Einstein introduced the idea that light is made up of particles, but Maxwell had shown that light also behaves like waves. Rutherford and Bohr led the way in understanding the structure of atoms, but the way that electrons behave inside atoms was not in accord with any known theory. And the diverse phenomena collectively known as radioactivity, in which atoms spontaneously split apart for no discernible reason, remained a mystery, not least because it introduced a disturbingly random element into physics. There was no doubt about it: something strange was afoot in the subatomic world.
The first step towards a consistent, unified answer is widely credited to the German physicist Werner Heisenberg, and what he did represented nothing less than a completely new approach to the theory of matter and forces. In July of 1925, Heisenberg published a paper throwing out the old hotchpotch of ideas and half-theories, including Bohr’s model of the atom, and ushered in an entirely new approach to physics. He began: ‘In this paper it will be attempted to secure the foundations for a quantum theoretical mechanics which is exclusively based on relations between quantities which in principle are observable.’ This is an important step, because Heisenberg is saying that the underlying mathematics of quantum theory need not correspond to anything with which we are familiar. The job of quantum theory should be to predict directly observable things, such as the colour of the light emitted from hydrogen atoms. It should not be expected to provide some kind of satisfying mental picture for the internal workings of the atom, because this is not necessary and it may not even be possible. In one fell swoop, Heisenberg removed the conceit that the workings of Nature should necessarily accord with common sense. This is not to say that a theory of the subatomic world shouldn’t be expected to accord with our everyday experience when it comes to describing the motion of large objects, like tennis balls and aircraft. But we should be prepared to abandon the prejudice that small things behave like smaller versions of big things, if this is what our experimental observations dictate.
There is no doubt that quantum theory is tricky, and absolutely no doubt that Heisenberg’s approach is extremely tricky indeed. Nobel Laureate Steven Weinberg, one of the greatest living physicists, wrote of Heisenberg’s 1925 paper:
If the reader is mystified at what Heisenberg was doing, he or she is not alone. I have tried several times to read the paper that Heisenberg wrote on returning from Heligoland, and, although I think I understand quantum mechanics, I have never understood Heisenberg’s motivations for the mathematical steps in his paper. Theoretical physicists in their most successful work tend to play one of two roles: they are either sages or magicians … It is usually not difficult to understand the papers of sage-physicists, but the papers of magician-physicists are often incomprehensible. In that sense, Heisenberg’s 1925 paper was pure magic.
Heisenberg’s philosophy, though, is not pure magic. It is simple and it lies at the heart of our approach in this book: the job of a theory of Nature is to make predictions for quantities that can be compared to experimental results. We are not mandated to produce a theory that bears any r
elation to the way we perceive the world at large. Fortunately, although we are adopting Heisenberg’s philosophy, we shall be following Richard Feynman’s more transparent approach to the quantum world.
We’ve used the word ‘theory’ liberally in the last few pages and, before we continue to build quantum theory, it will be useful to take a look at a simpler theory in more detail. A good scientific theory specifies a set of rules that determine what can and cannot happen to some portion of the world. They must allow predictions to be made that can be tested by observation. If the predictions are shown to be false, the theory is wrong and must be replaced. If the predictions are in accord with observation, the theory survives. No theory is ‘true’ in the sense that it must always be possible to falsify it. As the biologist Thomas Huxley wrote: ‘Science is organized common sense where many a beautiful theory was killed by an ugly fact.’ Any theory that is not amenable to falsification is not a scientific theory – indeed one might go as far as to say that it has no reliable information content at all. The reliance on falsification is why scientific theories are different from matters of opinion. This scientific meaning of the word ‘theory’, by the way, is different from its ordinary usage, where it often suggests a degree of speculation. Scientific theories may be speculative if they have not yet been confronted with the evidence, but an established theory is something that is supported by a large body of evidence. Scientists strive to develop theories that encompass as wide a range of phenomena as possible, and physicists in particular tend to get very excited about the prospect of describing everything that can happen in the material world in terms of a small number of rules.
One example of a good theory that has a wide range of applicability is Isaac Newton’s theory of gravity, published on 5 July 1687 in his Philosophiæ Naturalis Principia Mathematica. It was the first modern scientific theory, and although it has subsequently been shown to be inaccurate in some circumstances, it was so good that it is still used today. Einstein developed a more precise theory of gravity, General Relativity, in 1915.
Newton’s description of gravity can be captured in a single mathematical equation:
This may look simple or complicated, depending on your mathematical background. We do occasionally make use of mathematics as this book unfolds. For those readers who find the maths difficult, our advice is to skip over the equations without worrying too much. We will always try to emphasize the key ideas in a way that does not rely on the maths. The maths is included mainly because it allows us to really explain why things are the way they are. Without it, we should have to resort to the physicist-guru mentality whereby we pluck profundities out of thin air, and neither author would be comfortable with guru status.
Now let us return to Newton’s equation. Imagine there is an apple hanging precariously from a branch. The consideration of the force of gravity triggered by a particularly ripe apple bouncing off his head one summer’s afternoon was, according to folklore, Newton’s route to his theory. Newton said that the apple is subject to the force of gravity, which pulls it towards the ground, and that force is represented in the equation by the symbol F. So, first of all, the equation allows you to calculate the force on the apple if you know what the symbols on the right-hand side of the equals sign mean. The symbol r stands for the distance between the centre of the apple and the centre of the Earth. It’s r2 because Newton discovered that the force depends on the square of the distance between the objects. In non-mathematical language, this means that if you double the distance between the apple and the centre of the Earth, the gravitational force drops by a factor of 4. If you triple the distance, it drops by a factor of 9. And so on. Physicists call this behaviour an inverse square law. The symbols m1 and m2 stand for the mass of the apple and the mass of the Earth, and their appearance encodes Newton’s recognition that the gravitational force of attraction between two objects depends on the product of their masses. That then begs the question: what is mass? This is an interesting question in itself, and for the deepest answer available today we’ll need to wait until we talk about a quantum particle known as the Higgs boson. Roughly speaking, mass is a measure of the amount of ‘stuff’ in something; the Earth is more massive than the apple. This kind of statement isn’t really good enough, though. Fortunately Newton also provided a way of measuring the mass of an object independently of his law of gravitation, and it is encapsulated in the second of his three laws of motion, the ones so beloved of every high school student of physics:
Every object remains in a state of rest or uniform motion in a straight line unless it is acted upon by a force;
An object of mass m undergoes an acceleration a when acted upon by a force F. In the form of an equation, this reads F = ma;
To every action there is an equal and opposite reaction.
Newton’s three laws provide a framework for describing the motion of things under the influence of a force. The first law describes what happens to an object when no forces act: the object either just sits still or moves in a straight line at constant speed. We shall be looking for an equivalent statement for quantum particles later on, and it’s not giving the game away too much to say that quantum particles do not just sit still – they leap around all over the place even when no forces are present. In fact, the very notion of ‘force’ is absent in the quantum theory, and so Newton’s second law is bound for the wastepaper basket too. We do mean that, by the way – Newton’s laws are heading for the bin because they have been exposed as only approximately correct. They work well in many instances but fail totally when it comes to describing quantum phenomena. The laws of quantum theory replace Newton’s laws and furnish a more accurate description of the world. Newton’s physics emerges out of the quantum description, and it is important to realize that the situation is not ‘Newton for big things and quantum for small’: it is quantum all the way.
Although we aren’t really going to be very interested in Newton’s third law here, it does deserve a comment or two for the enthusiast. The third law says that forces come in pairs; if I stand up then my feet press into the Earth and the Earth responds by pushing back. This implies that for a ‘closed’ system the net force acting on it is zero, and this in turn means that the total momentum of the system is conserved. We shall use the concept of momentum throughout this book and, for a single particle, it is defined to be the product of the particle’s mass and its speed, which we write p = mv. Interestingly, momentum conservation does have some meaning in quantum theory, even though the idea of force does not.
For now though, it is Newton’s second law that interests us. F = ma says that if you apply a known force to something and measure its acceleration then the ratio of the force to the acceleration is its mass. This in turn assumes we know how to define force, but that is not so hard. A simple but not very accurate or practical way would be to measure force in terms of the pull exerted by some standard thing; an average tortoise, let us say, walking in a straight line with a harness attaching it to the object being pulled. We could term the average tortoise the ‘SI Tortoise’ and keep it in a sealed box in the International Bureau of Weights and Measures in Sèvres, France. Two harnessed tortoises would exert twice the force, three would exert three times the force and so on. We could then always talk about any push or pull in terms of the number of average tortoises required to generate it.
Given this system, which is ridiculous enough to be agreed on by any international committee of standards,1 we can simply pull an object with a tortoise and measure its acceleration, and this will allow us to deduce its mass using Newton’s second law. We can then repeat the process for a second object to deduce its mass and then we can put both masses into the law of gravity to determine the force between the masses due to gravity. To put a tortoise-equivalent number on the gravitational force between two masses, though, we would still need to calibrate the whole system to the strength of gravity itself, and this is where the symbol G comes in.
G is a very important number, calle
d ‘Newton’s gravitational constant’, which encodes the strength of the gravitational force. If we doubled G, we would double the force, and this would make the apple accelerate at double the rate towards the ground. It therefore describes one of the fundamental properties of our Universe and we would live in a very different Universe if it took on a different value. It is currently thought that G takes the same value everywhere in the Universe, and that it has remained constant throughout all of time (it appears in Einstein’s theory of gravity too, where it is also a constant). There are other universal constants of Nature that we’ll meet in this book. In quantum mechanics, the most important is Planck’s constant, named after quantum pioneer Max Planck and given the symbol h. We shall also need the speed of light, c, which is not only the speed that light travels in a vacuum but the universal speed limit. ‘It is impossible to travel faster than the speed of light and certainly not desirable,’ Woody Allen once said, ‘as one’s hat keeps blowing off.’
Newton’s three laws of motion and the law of gravitation are all that is needed to understand motion in the presence of gravity. There are no other hidden rules that we did not state – just these few laws do the trick and allow us, for example, to understand the orbits of the planets in our solar system. Together, they severely restrict the sort of paths that objects are allowed to take when moving under the influence of gravity. It can be proved using only Newton’s laws that all of the planets, comets, asteroids and meteors in our solar system are only allowed to move along paths known as conic sections. The simplest of these, and the one that the Earth follows to a very good approximation in its orbit around the Sun, is a circle. More generally, planets and moons move along orbital paths known as ellipses, which are like stretched circles. The other two conic sections are known as the parabola and the hyperbola. A parabola is the path that a cannonball takes when fired from the cannon. The final conic section, the hyperbola, is the path that the most distant object ever constructed by human kind is now following outwards to the stars. Voyager 1 is, at the time of writing, around 17,610,000,000 km from the Earth, and travelling away from the solar system at a speed of 538,000,000 km per year. This most beautiful of engineering achievements was launched in 1977 and is still in contact with the Earth, recording measurements of the solar wind on a tape recorder and transmitting them back with a power of 20 watts. Voyager 1, and her sister ship Voyager 2, are inspiring testaments to the human desire to explore our Universe. Both spacecraft visited Jupiter and Saturn and Voyager 2 went on to visit Uranus and Neptune. They navigated the solar system with precision, using gravity to slingshot them beyond the planets and into interstellar space. Navigators here on Earth used nothing more than Newton’s laws to plot their courses between the inner and outer planets and outwards to the stars. Voyager 2 will sail close to Sirius, the brightest star in the skies, in just under 300,000 years. We did all this, and we know all this, because of Newton’s theory of gravity and his laws of motion.