Quantum theory of light

Blackbody radiation

Blackbody radiation refers to the spectrum of light emitted by any heated object; common examples include the heating element of a toaster and the filament of a light bulb. The spectral intensity of blackbody radiation peaks at a frequency that increases with the temperature of the emitting body: room temperature objects (about 300 K) emit radiation with a peak intensity in the far infrared; radiation from toaster filaments and light bulb filaments (about 700 K and 2,000 K, respectively) also peak in the infrared, though their spectra extend progressively into the visible; while the 6,000 K surface of the Sun emits blackbody radiation that peaks in the centre of the visible range. In the late 1890s, calculations of the spectrum of blackbody radiation based on classical electromagnetic theory and thermodynamics could not duplicate the results of careful measurements. In fact, the calculations predicted the absurd result that, at any temperature, the spectral intensity increases without limit as a function of frequency.

In 1900 the German physicist Max Planck succeeded in calculating a blackbody spectrum that matched experimental results by proposing that the elementary oscillators at the surface of any object (the detailed structure of the oscillators was not relevant) could emit and absorb electromagnetic radiation only in discrete packets, with the energy of a packet being directly proportional to the frequency of the radiation, E = hf. The constant of proportionality, h, which Planck determined by comparing his theoretical results with the existing experimental data, is now called Planck’s constant and has the approximate value 6.626 × 10⁻³⁴ joule∙second.

Photons

Planck did not offer a physical basis for his proposal; it was largely a mathematical construct needed to match the calculated blackbody spectrum to the observed spectrum. In 1905 Albert Einstein gave a ground-breaking physical interpretation to Planck’s mathematics when he proposed that electromagnetic radiation itself is granular, consisting of quanta, each with an energy hf. He based his conclusion on thermodynamic arguments applied to a radiation field that obeys Planck’s radiation law. The term photon, which is now applied to the energy quantum of light, was later coined by the American chemist Gilbert N. Lewis.

Einstein supported his photon hypothesis with an analysis of the photoelectric effect, a process, discovered by Hertz in 1887, in which electrons are ejected from a metallic surface illuminated by light. Detailed measurements showed that the onset of the effect is determined solely by the frequency of the light and the makeup of the surface and is independent of the light intensity. This behaviour was puzzling in the context of classical electromagnetic waves, whose energies are proportional to intensity and independent of frequency. Einstein supposed that a minimum amount of energy is required to liberate an electron from a surface—only photons with energies greater than this minimum can induce electron emission. This requires a minimum light frequency, in agreement with experiment. Einstein’s prediction of the dependence of the kinetic energy of the ejected electrons on the light frequency, based on his photon model, was experimentally verified by the American physicist Robert Millikan in 1916.

In 1922 American Nobelist Arthur Compton treated the scattering of X-rays from electrons as a set of collisions between photons and electrons. Adapting the relation between momentum and energy for a classical electromagnetic wave to an individual photon, p = E/c = hf/c = h/λ, Compton used the conservation laws of momentum and energy to derive an expression for the wavelength shift of scattered X-rays as a function of their scattering angle. His formula matched his experimental findings, and the Compton effect, as it became known, was considered further convincing evidence for the existence of particles of electromagnetic radiation.

The energy of a photon of visible light is very small, being on the order of 4 × 10⁻¹⁹ joule. A more convenient energy unit in this regime is the electron volt (eV). One electron volt equals the energy gained by an electron when its electric potential is changed by one volt: 1 eV = 1.6 × 10⁻¹⁹ joule. The spectrum of visible light includes photons with energies ranging from about 1.8 eV (red light) to about 3.1 eV (violet light). Human vision cannot detect individual photons, although, at the peak of its spectral response (about 510 nm, in the green), the dark-adapted eye comes close. Under normal daylight conditions, the discrete nature of the light entering the human eye is completely obscured by the very large number of photons involved. For example, a standard 100-watt light bulb emits on the order of 10²⁰ photons per second; at a distance of 10 metres from the bulb, perhaps 10¹¹ photons per second will enter a normally adjusted pupil of a diameter of 2 mm.

Photons of visible light are energetic enough to initiate some critically important chemical reactions, most notably photosynthesis through absorption by chlorophyll molecules. Photovoltaic systems are engineered to convert light energy to electric energy through the absorption of visible photons by semiconductor materials. More-energetic ultraviolet photons (4 to 10 eV) can initiate photochemical reactions such as molecular dissociation and atomic and molecular ionization. Modern methods for detecting light are based on the response of materials to individual photons. Photoemissive detectors, such as photomultiplier tubes, collect electrons emitted by the photoelectric effect; in photoconductive detectors the absorption of a photon causes a change in the conductivity of a semiconductor material.

A number of subtle influences of gravity on light, predicted by Einstein’s general theory of relativity, are most easily understood in the context of a photon model of light and are presented here. (However, note that general relativity is not itself a theory of quantum physics.)

Through the famous relativity equation E = mc², a photon of frequency f and energy E = hf can be considered to have an effective mass of m = hf/c². Note that this effective mass is distinct from the “rest mass” of a photon, which is zero. General relativity predicts that the path of light is deflected in the gravitational field of a massive object; this can be somewhat simplistically understood as resulting from a gravitational attraction proportional to the effective mass of the photons. In addition, when light travels toward a massive object, its energy increases, and its frequency thus increases (gravitational blueshift). Gravitational redshift describes the converse situation where light traveling away from a massive object loses energy and its frequency decreases.