## Wave and Particle Languages

We may discuss electromagnetic radiation using two languages: wave or particle (photon) language. As with all languages, we sometimes can express ideas more succinctly or clearly in the one language than in the other. We use both, separately and sometimes together in the same breath. We need fluency in both. Much ado has been made over this supposedly lamentable duality of electromagnetic radiation. But no law requires physical reality to be described by a single language. We may hope for such a language, but Nature often is indifferent to our hopes. Moreover, we accept without protest or hand-wringing the duality of sound. We describe sound waves in air as continuous while at the same time recognizing that air, and hence sound, is composed of discrete particles (molecules) in motion.

How do we choose which language to use? Simplicity. Life is short. To understand nature we take the simplest approach consistent with accuracy. Although propagation of sound in air could be described as the motions of molecules, had this approach been taken acoustics would have floundered in a mathematical morass.

In the photon language a beam of radiation is looked upon as a stream of particles called photons with the peculiar property that they carry energy, linear momentum, and angular momentum but not mass. The mass of the photon often is said to be identically zero. But given the near impossibility of measuring zero in the face of inevitable errors and uncertainties, it would be more correct to say that the upper limit of the photon mass keeps decreasing, its present value being about 10-24 times the mass of the electron. If it bothers you that a particle

Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Craig F. Bohren and Eugene E. Clothiaux Copyright © 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40503-8

without mass can carry momentum this is because you are stuck on the notion that momentum is mass times velocity. Sometimes this is true (approximately), sometimes not. Momentum is momentum, a property complete in itself and not always the product of mass and velocity.

Photons are of one kind, differing only in their energy and momenta, whereas waves are of unlimited variety and often exceedingly complex, the simplest kind a plane harmonic wave characterized by a single (circular) frequency w and direction of propagation (see Secs. 3.3 and 3.4). The dimensions of circular frequency are radians per unit time. You may be more familiar with just plain frequency, often denoted by v (sometimes f), which has the dimensions of cycles per unit time. The unit of frequency is the hertz, abbreviated Hz, one cycle per second. Because one cycle corresponds to 2n radians, the relation between frequency and circular frequency is simple:

All electromagnetic waves propagate in free space (which does not strictly exist) with the same speed c, about 3 x 108 ms"1. A plane harmonic wave in free space can just as well be characterized by its wavelength A, related to its frequency by

You sometimes hear it said that frequency is more fundamental than wavelength. In a sense, this is correct, but wavelength is often more useful. When we consider the interaction of electromagnetic waves with chunks of matter, the first question we must ask ourselves is how large the waves are. Big and small have no meaning until we specify a measuring stick. For electromagnetic radiation the measuring stick is the wavelength. The mathematical expressions describing the interaction of such radiation with matter can be quite different depending on the size of the matter relative to the measuring stick.

How do we translate from wave to photon language? A plane harmonic wave with circular frequency w corresponds to a stream of photons, each with energy where h is Planck's constant (6.625 x 10~34 Js) and h = h/2n. The frequency of visible electromagnetic radiation (light) is about 1014 Hz, and hence the photons that excite the sensation of vision have energies around 10~20 J. This isn't much energy; the kinetic energy of a golf ball as it slices through air is about 1013 times greater.

Understanding what happens when an electromagnetic wave is incident from air on the smooth surface of glass, say, is not especially difficult if one uses the wave language. The incident wave excites molecules in the glass to radiate secondary waves that combine to form (approximately) a net reflected wave given by the law of reflection and a net transmitted wave given by the law of refraction. There is no such thing as an absolutely smooth surface, so what is meant is smooth on the scale of the wavelength.

All this makes intuitive sense and causes no perplexity. But now consider what happens when we switch to photon language. If we look upon reflection as the rebound of photons at a surface and transmission as their penetration through it, then why, if all photons are identical, are some reflected and some transmitted? This is indeed puzzling; even more so is why w = 2nv.

photons should be specularly (by which is meant mirror-like) reflected, because for photons imagined as particles of vanishingly small dimensions, all surfaces are rough.

This is not to say that one couldn't describe reflection and transmission at smooth interfaces in photon language, only that to do so would be exceedingly costly in mental effort. And the reverse sometimes is true. Many years ago one of the authors attended a colloquium entitled "The photoelectric effect without photons." By the photoelectric effect is usually meant the emission of electrons by a surface (often metallic) because of illumination by radiation (often ultraviolet). In photon language the photoelectric effect is simple to describe. When a photon of energy hv is absorbed by the surface, the maximum kinetic energy E of the electrons thereby set free is

where p is the minimum energy an electron loses in breaking free of the surface. A single photon interacting with a single electron gives up its entire energy to that electron, which if sufficient enables the electron to break free of the forces binding it to the metal. According to this equation the energies of the emitted electrons are independent of the incident power whereas the photocurrent (rate and number of emitted electrons) is proportional to it, which accords with experiment. This simple equation, first written down by Einstein in 1905, is one of the keystones of the modern theory of radiation and matter. Yet the speaker at that colloquium years ago, in an effort to describe and explicate the photoelectric effect without photons, assailed the audience with dozens of complicated equations. And even at that, part way through his mathematical tour de force his mind and tongue betrayed him and he blurted out the forbidden word "photon". At that point, your author who was there leapt up from his seat and shouted, "Photons! Photons! You promised no photons."

A mirror illuminated by an incident beam gives rise to a reflected beam. Is this reflected beam redirected incident photons? Alas, we cannot do an experiment to answer this question. To determine if reflected photons are the same as incident photons would require us to be able to identify them. But photons are indistinguishable. We cannot tell one from another. We cannot tag a photon and follow its progress. Thus if you want to believe that reflected photons are the same as incident photons, you may do so. No one can prove you wrong. But you cannot prove you are right. When faced with an undecidable proposition, you may believe whatever you wish. Note that in the wave language we would not likely even ask if the reflected wave is the same as the incident wave.

It is not often acknowledged that there is a third language for talking about light, what might be called the who-gives-a-hoot-what-light-is? language. This is geometrical or ray optics, in which the nature of light isn't addressed at all. Fictitious rays are imagined to be paths along which the energy carried by light is transported, and these paths meander and bifurcate according to simple geometrical laws.

But which language is the more useful? In a letter to American Journal of Physics, M. Psi-mopoulos and T. Theocharis ask the rhetorical questions: "What new discoveries have (i) the particle or photon aspect of light, and (ii) the wave aspect of light, given rise to? Answer: (i) we are not aware of any; (ii) holography, laser, intensity interferometry, phase conjugation." To this list we add radar, all of interferometry, on which much of the science of measurement is based, and interference filters, which have many applications. The view of these authors is extreme, but they also quote the more measured words of Charles Townes, a pioneer in masers and lasers: "Physicists were somewhat diverted by an emphasis in the world of physics on the photon properties of light rather than its coherent aspects." That is, the photon language has been the more fashionable language among physicists, just as French was the fashionable language in the Imperial Russian court. When prestigious and munificent prizes began to be awarded for flushing "ons" (electron, positron, neutron, meson, and so on) from the jungle, shooting them, and mounting their stuffed heads on laboratory walls, the hunt was on, and slowed down only with the demise of the Superconducting Supercollider.

Although the wave language undoubtedly has been and continues to be more fruitful of inventions, the photon language is perhaps more soothing because photons can be incarnated, imagined to be objects we can kick or be kicked by. Waves extending through all space are not so easily incarnated. We can readily conceive of the photon as a thing. And yet an electromagnetic wave is just as much a thing as a photon: both possess energy and momentum (linear and angular) but not, it seems, mass.