Chapter 4 Duality

  The two different views of light, as a particle and as a wave, both contain insight and value. They have each enabled both new understanding of the natural world and the development and design of new technologies. Yet they appear to be vastly different in their conception of what light actually is. On the one hand, the particle model views light as a localized entity, a bundle of energy,that moves along a well-defined trajectory. On the other hand, the wave model describes light as a diffuse entity, permeating through space with no connection to the motion of solid things. How can these two pictures possibly refer to the same thing? This dilemma was recognized early on by Huygens and his contemporaries,but the two views remained in tension, as alternative descriptions of light, until the 19th century.

  When Maxwell developed his theory of electromagnetic fields,he was able to use this to explain the properties of light as wave motion of those fields, as we saw in Chapter 3. This triumph of reasoning appeared to confirm the experiments of Thomas Young and Auguste Fresnel (described in Chapter 3) by providing an explanation of two fundamental phenomena, interference and diffraction, that did not easily fit within the particle model.Yet the concept of trajectories remained, and still remains, an extraordinarily powerful one for the analysis and design of optical systems. So there’s an uneasy truce of these two pictures—a dualism within classical physics—that requires some consideration.How can they be reconciled?

  Looking at trajectories again

  In the 17th century the Frenchman Pierre de Fermat proposed an ingenious formulation of refraction that was very different from that of Snell. Recall that Snell’s law deals with the change of direction of a ray of light at an interface between two transparent media. The ray, defined by the direction in which it is travelling towards the interface and the point at which it hits the interface,has its direction altered by an amount proportional to the ratio of the refractive indices of the two materials. It is only the local properties of the ray and interface that are important. Snell’s law applies at each point along the trajectory, as if the ray is『feeling』 its way along, adjusting direction when it encounters a new interface.

  Fermat’s conception was radically different. He argued that one should define the trajectory in terms of starting and ending points, as shown in Figure 21. He suggested that the question to ask is: what is the path that the light takes to traverse the space between the two points? He proposed that it should take the path that minimizes the time of flight between the two points. That this gives the same answer as Snell is remarkable and profound.Fermat’s 『principle of least time』 suggests that the light considers the overall picture of the situation, and that the notion of a ray is one that takes into account both the initial and final positions and directions as well as everything in between. The contrast with the local model of a particle reacting to its immediate environment is telling.

  21. Fermat’s conception of a light ray as a path of least time connecting the start and end of the trajectory. The ray crosses an interface between two optical media in which light moves at diferent speeds.

  This idea was taken up by the German natural philosopher Gottfried Wilhelm von Leibniz, who was Newton’s contemporary and antagonist. Leibniz was impressed by the holistic picture of the process described by Fermat, and the concept of『optimization』 that it implied: a ray explores the whole of space and picks just that path that will minimize its transit time between the specified beginning and end points. He developed the mathematical tools for analysing this idea—the calculus of variations—by which the effects small changes on a trajectory would have on the time taken to traverse the modified trajectory could be calculated. Leibniz recognized the importance of the notion that Fermat’s principle provided: the movement of light from one point to another defines an 『optimal’trajectory.

  Indeed, so taken was Leibniz by this concept of optimization that he elevated it to a teleological principle: that the world itself, in all its aspects, was on the optimal trajectory between a starting point and a finishing point. The contradictions inherent in such a position, when applied outside of the realm of science, were ably lampooned by Voltaire in his novel Candide, where Leibniz’s ideas are put into the mouth of Dr Pangloss, who insists disasters both natural and man-made were nonetheless evidence that this is the『best of all possible worlds』.

  Connecting waves and rays

  記住全網最快小説站𝙗𝙖𝙣𝙭𝙞𝙖𝙗𝙖.𝙘𝙤𝙢

  Nonetheless, Leibniz’s mathematical ideas proved to be very fruitful. They were taken up by the renowned Irish mathematician William Rowan Hamilton in the 19th century. He showed formally how the idea of a wave can be allied to that of a collection of particles. Waves can be defined by their wavelength, amplitude,and phase (see Figure 15). Particles are defined by their position and direction of travel (see Figure 5), and a collection of particles by their density (i.e. the number of them at a given position) and range of directions. The media in which the light moves are characterized by their refractive indices. This can vary across space. For example, at the interface shown in Figure 20 there is a step change in the refractive index across the boundary between the two media.

  Hamilton showed that what was important was how rapidly the refractive index changed in space compared with the length of an optical wave. That is, if the changes in index took place on a scale of close to a wavelength, then the wave character of light was evident. If it varied more smoothly and very slowly in space then the particle picture provided an adequate description.He showed how the simpler ray picture emerges from the more complex wave picture in certain commonly encountered situations. The appearance of wave-like phenomena, such as diffraction and interference, occurs when the size scales of the wavelength of light and the structures in which it propagates are similar. Thus you see diffraction patterns arising when the object that the light hits is a few microns in diameter, or has a very sharp edge, such as the delicate structure in a bird’s feather, or a butterfly wing. Otherwise, as in the case of a camera lens, the trajectory provides a sufficient description, since the refractive index is uniform throughout the glass of the lens itself.

  22. Hamilton’s idea of rays as connecting wavefronts-thus joining the two primary conceptions of light.

  Further, Hamilton showed that Fermat’s trajectories related directly to a property of the wave—the wavefronts. These are the loci of points at which the wave has the same phase at each point in space. For instance, when you see the ripples on the surface of a pond after a stone has been cast into it, the circular patterns are just these wavefronts. They are the places on the surface where the wave 『peaks』 (or troughs) at a given instant of time. Now, what Hamilton noted was that rays could be considered as lines that intersected the wavefronts at right angles,as shown in Figure 22, thus connecting adjacent wavefronts by a well-defined trajectory.

  Hamilton’s 『optical analogy』

  This remarkable result suggested another profound comparison—Hamilton’s so-called 『optical analogy』. What he noted was that the well-known formulation of mechanics—the motion and position of solid bodies of matter—was based on the idea of trajectories.The idea that these may also be in some sense 『optimal』 had been considered by Pierre Louis Maupertuis in the 18th century.

  Maupertuis had formulated a way to evaluate the optimal value of a quantity called the 『action』—essentially the velocity of the body multiplied by the distance it moves (and times its mass)—along the body’s trajectory.

  He argued that the action should be minimal for an actual trajectory between two points, just as in Fermat’s argument that the time taken to traverse a light ray should be minimal. Maupertuis』『principle of least action』 is very similar in concept to Fermat’s『principle of least time』. Indeed, Leonhard Euler, a Swiss mathematician, showed how to use Leibniz’s calculus to derive Newton’s famous equations of motion from Maupertuis』 principle.Thus Euler connected a description of a trajectory in terms of a particle sensing its way through its environment to one in which the whole of space between the specified starting and finishing points influences the path.

  What Hamilton did was to find equations that encapsulated the variations in action in terms of a simple description of the specific environment in which the body was moving. And this equation turns out to have a very similar form to the one he found for describing the trajectories of light rays (for which the environmental description is just how the refractive index changes with position in the medium). So there is a hint of a latent analogy between the trajectories of solid objects and a fictive wavefront: perhaps all bodies might have both particle-like trajectories and wave-like properties? Indeed, Hamilton’s equation, and his eponymous function, turns out to be very important in thinking about the next big step in understanding light—quantum mechanics.

  Unsolved puzzles

  This was by no means the only hint of a new opportunity for science. About this time, towards the end of the 19th century, light still offered a few puzzles that were unexplainable in terms of the prevailing models of its properties, even with the reconciliation that Hamilton had provided. Two of the most important of these were: the colour of hot objects (including the Sun), and the colour of different atoms in a flame.

  When things are heated up, they change colour. Take a lump of metal. As it gets hotter and hotter it first glows red, then orange,and then white. Why does this happen? This question stumped many of the great scientists of the time, including Maxwell himself.The problem was that Maxwell’s theory of light, when applied to this problem, indicated that the colour should get bluer and bluer as the temperature increased, without a limit, eventually moving out of the range of human vision into the ultraviolet—beyond blue—region of the spectrum. But this does not happen in practice.

  The second example arose from the study of light emitted by atoms, pioneered by a Swiss schoolmaster, Johannes Balmer. We』ll look at this mechanism in more detail in Chapter 5, but the important feature of the light is in the distribution of colours in its spectrum, shown in Figure 23a. In this respect it is very different from sunlight (the Sun is a good example of a hot body), which has the familiar 『rainbow』 spectrum, shown in Figure 23b, consisting of all colours continuously from red to violet (and beyond at each end—just not visible to us). By contrast, a collection of atoms emits a discrete set of colours—a set of spectral 『lines』 of particular wavelengths—specifically associated with the internal structure of the particular atom involved.

  Both of these phenomena required a radical revision of thinking about light, because they could not be explained within the contemporary models of wave motion and atomic structure.

  Max Planck, working at Humboldt University in Berlin in the late19th century, first came up with an idea to explain the spectrum emitted by hot objects—so-called 『black bodies』. He conjectured that when light and matter interact, they do so only by exchanging discrete 『packets』, or quanta, or energy. Planck recognized the revolutionary nature of his idea, and was therefore reluctant to infer too much about light itself from it, though it would eventually dramatically change our view of what light is. His notion revived the idea of light as a stream of particles—discrete objects that carried a fixed amount of energy that could be absorbed by atoms or emitted by them.

  23. Spectrum of light emitted from a. the Sun (a 『black body』), andb. a neon lamp. The former has continuous band of colours, whereaslatter shows the discrete lines of particular colours that are a『fingerprint』 of the neon atoms.

  This seemed like a retrograde step: the wave model of light could explain all of the hitherto observed effects, and it was clear from Hamilton’s work that even trajectory-like behaviour, previously the most obvious evidence of a particle-like entity, emerged from a wave model of light in certain common situations. So the idea of a particle of light appeared not to be necessary. Surely it was simply a calculational 『fix』, thought up to get out of a tight spot,and would eventually be replaced by a more reasonably consistent picture of light. However, combined with Balmer’s observations,this conjecture was set to radically change physics.

  In the years immediately after Planck’s suggestion, Albert Einstein used the idea of discrete exchanges of energy between light and matter to expose another piece of physics that had eluded explanation—the photoelectric effect. This effect is seen when light shines on a piece of metal. Some electric charges—electrons,in fact—are ejected from the metal. The speed with which the electrons are ejected depends on the wavelength of the light. The light must be sufficiently 『blue』, that is have a short enough wavelength, in order to see any electrons emitted at all. As it gets bluer and bluer, the electrons are ejected with more and more energy, and thus higher and higher speed.

  Einstein explained this by noting that the electrons required a minimum amount of energy to escape the clutches of the metal,and by assigning discrete amounts of energy to a particle of light—the photon—proportional to the frequency of the light (the constant of proportionality being known as Planck’s constant, h),so that when the frequency of the light is high enough (and thus the wavelength short enough) the light, when it is absorbed, can provide enough energy for an electron to escape. His model suggested that the origin of the discrete exchange of energy between light and matter that is central to Planck’s notion arises from the actual discrete character of light—a full revival of the particle model.

  This idea chimes nicely with Balmer’s observations of discrete line spectra of light emission from atoms. But a full explanation of this phenomenon clearly requires a bit more thought about why atoms would deliver light in such packets. The key idea came from Niels Bohr, a Danish physicist working in Manchester. He suggested that the reason light was emitted as discrete packets of energy was that the atoms themselves could only exist in certain configurations. He imagined atoms as analogous to tiny planetary systems: electrons in orbit around a central nucleus. The electron can 『jump』 between two stable orbits, emitting or absorbing light as it does so (depending on whether the jump is to an orbit of lower or higher energy). The character of these orbits—or quantum states—depends on the details of the atom: how many electrons and the size of the nucleus. Therefore the energy given or taken up when an electron moves between two quantum states is a signature of the particular atom itself. So when the energy of a light particle—or photon—matches that of the difference in energy of two quantum states of the electron in an atom, then absorption or emission is possible. Bohr’s ideas explained Balmer’s observations neatly, and added some weight to the idea of light beams as a collection of discrete particles.

  All these developments had the potential to undermine the wave model of light that had been so strongly affirmed by Maxwell’s theory. They went beyond even Hamilton’s reconciliation of trajectories and wave motion, since they appeared to be fundamental,not simply the result of an approximation about size and scale. Thus they reopened the question of the nature of light once again.

  In 1908, George Taylor, working in Cambridge, performed Young’s double-slit experiment with exceptionally feeble light—so weak that on average there was less than one photon in the apparatus at any time. Yet he still saw interference fringes. That outcome is very strange. If we think of one path from the light source to the detector as being via one slit, and a second path via the other slit, then there are two ways that the photon can get from source to detector. The evidence of interference fringes posed a dilemma that caught the attention of the leading scientists of the day. Bohr captured the difficulty, noting that we would 『be obliged to say, on the one hand, that the photon always chooses one of the two ways, and on the other that it behaves as if it passed through both』. Even single particles can exhibit wave-like behaviour.

  Waving altogether

  As you might imagine, it took a truly radical thought to figure a way out of this conundrum. That thought occurred to Paul Dirac,a physicist working at Cambridge in the 1920s. Dirac suggested that the fundamental property of light was that it was both a particle and a wave at the same time. Now, this might appear to you as simply sophistry, a logical trick that resolves nothing. But there is much more behind it than a slogan. What Dirac did was to develop a quantum mechanical version of Maxwell’s theory of electromagnetic fields. Using this he was able to show that, if you measured these 『quantum fields』 using a set-up like Young’s double-slit interferometer, you would see interference effects characteristic of wave-like behaviour. Whereas, if you simply measured the intensity of the light, you would be effectively counting the number of photons in the beam.

  This turned out to be a very profound step. It set the quantum field up as the fundamental entity on which the universe is built—neither particle nor wave, but both at once; complete wave–particle duality. It is a beautiful reconciliation of all the phenomena that light exhibits, and provides a framework in which to understand all optical effects, both those from the classical world of Newton, Maxwell, and Hamilton and those of the quantum world of Planck, Einstein, and Bohr. But the cost is a perplexing and non-intuitive entity at the heart of nature—a quantum field—of which light is but one example.

  The radical idea that light was both wave and particle stimulated some major new ideas. For instance, Louis de Broglie suggested that if this kind of dualism existed for light, surely it should for all other things too. So, material bodies that we normally had considered only as collections of particles should also have『wave-like』 properties, taking the next step beyond what Hamilton had considered. He even defined what the wavelength should be(now called the de Broglie wavelength λdB)—proportional to the inverse of the particle’s momentum (i.e. its mass and velocity,the constant of proportionality again being Planck’s constant):

  λdB =h/mv

  This suggests that to look for such effects you should use either very light particles or very cold ones that are moving very, very slowly. Such effects can be observed. Figure 24 shows an interference pattern made using a double-slit-like interferometer,but using molecules instead of light. The implications of this are mind-boggling. If you think of a molecule as just a very light particle, then you cannot explain the pattern, because you consider that it must have passed through one slit or the other.However, the idea that a particle with mass could be so delocalized as to have passed effectively through both slits to interfere with itself is astounding and beggars belief.

  24. An interference pattern made using molecules passing one at atime through a tiny version of Young’s apparatus—two very small slitsseparated by billionths of a metre.

  An important quantity for light is its intensity, proportional to the square of the amplitude of the field. This is related directly to the density of photons in the light beam. Similarly, the square of the wave function is related to the density of particles at a particular point in space at a particular time. But it is impossible to say for certain that a particle occupies a specific position at a specific instant. This indeterminacy seems to be a very fundamental property of the world, and it is related deeply to the fact that quantum fields lie at the heart of things.

  Nothing is something

  Another consequence of this fact is that 『nothing』 is actually something. That is, the complete absence of matter (e.g. electrons or atoms) or light (i.e. photons) still has measurable properties.This void is called the 『electromagnetic quantum vacuum』, and is the state of the universe from which all extractable energy has been removed. Yet it is a seething mass of activity, consisting of fluctuating fields but containing no photons whatsoever. What is even more surprising is that the quantum vacuum has observable consequences. How can this 『nothing』 give rise to an effect we can see?

  We』ve seen that light can be thought of as a wave motion of the electromagnetic field. Imagine this field as ripples on the surface of the sea. These would buffet about any boat, but would not move the boat up and down or push it along as a well-defined wave might. On average, the boat does not move, but it nonetheless rocks back and forth. Now imagine the same thing for a charged particle like an electron. It 『feels』 the random changes in the electromagnetic vacuum, being buffeted by these. If the electron is bound in an atom, this buffeting is revealed as a shift of the energy of the quantum states that the electron may occupy. Since the frequency of the photon that the atom may absorb depends on the difference in energy of two such states, these changes can be seen in a change of the colour of light that the atom may absorb. The changes are tiny—a shift of less than one billionth of the wavelength of the light—but measurement techniques for frequencies are so precise that such changes can indeed be determined. The first person to do this, Willis Lamb, working in New York in the 1950s, won a Nobel Prize for showing this frequency shift, which is now named for him.

  The dual identity of light has numerous facets. Even in the pre-quantum world, the dichotomy of ray and wave demanded a resolution. That came about by understanding the nature of the wave motion that light embodied, and the scale and nature of objects with which it interacted. Particle-like behaviour—motion along a well-defined trajectory—is sufficient to describe the situation when all objects are much bigger than the wavelength of light, and have no sharp edges. Quantum mechanics puts a new twist on this duality. Light acts as a particle of more or less well-defined energy when it interacts with matter. Yet it retains its ability to exhibit wave-like phenomena at the same time. The resolution is a new concept: the quantum field.Light particles—photons—are excitations of this field, which propagates according to quantum versions of Maxwell’s equations for light waves.

  Quantum fields, of which light is perhaps the simplest example,are now regarded as being the fundamental entities of the universe, underpinning all types of material and non-material things. The only explanation is that the stuff of the world is neither particle nor wave but both. This is the nature of reality.