bbradn.htm(Ó R. Egerton)

Quantum mechanics represents the second major revolution in physics in the 20th century, at least equal to Relativity in its impact. Although Einstein played an important role in its early development, the basic idea developed late in the 19th century from observations of the electromagnetic radiation emitted by hot objects.

Electromagnetic radiation can be described in terms of wavelength (lsmbda) or frequency (f = c/lambda), ranging from long-wavelength radio waves (lambda many km) to gamma rays (lambda as short as 1 fm = 10^-15m). The "radiant heat" which is emitted from heated objects (e.g. a light-bulb filament or the sun) has most of its energy in the infrared region (lambda ~ a few micrometers). We can also talk about the intensity I of this radiation, defined as the energy per unit area (J/s/m^2 = W/m^2) emitted from the heated surface. Stefan's law relates I to the absolute temperature T , the Stefan-Bolzmann constant (sigma) and the emissivity (epsilon) of the surface: I = (epsilon)(sigma) T^4 . By measuring I , instruments called infrared pyrometers allow the temperature T to be remotely measured.

A more complete description of the radiation is in terms of its spectrum: the intensity per unit wavelength dI/d(lambda), plotted against wavelength (see Fig. 2.3). The total area under this curve (its integral) is the intensity I . In general, the spectrum exhibits a maximum at a wavelength which shifts to smaller values as the temperature T is increased (Fig. 2.3). However, the exact form of the spectrum depends on epsilon, which is in general a function of wavelength.

The simplest case is that of a black body, for which e = 1 independent of l . Since e is also equal to the absorptivity of the surface (the fraction of electromagnetic radiation of a given wavelength which would be absorbed if the surface were illuminated by an external source) a black body absorbs all radiation, even that in the visible region of the spectrum, and therefore appears black.

It is not easy to produce a perfect black body; painting its surface with carbon paint provides the easiest approximation. However, any cavity (a hollow object such as a furnace) containing a small hole will emit blackbody radiation through the hole, whose spectrum is the same as that of an object with epsilon = 1 . As justification, consider that externally applied radiation would enter the hole and be totally absorbed (perhaps after many reflections) if the hole were small enough (see Fig.2.4) so the hole behaves like a black body. In other words, cavity radiation = blackbody radiation. A cavity is essentially a system in equilibrium, internally emitting and re-absorbing radiation; only a small part of this radiation escapes through the hole, if the latter is sufficiently small.

A challenge to 19th-century physicists (in the field of thermodynamics) was to produce a theory which would predict the spectrum of cavity radiation. One such attempt was made by the physicist W. Wien, treating the radiation inside the cavity as loosely analogous to gas molecules. His theory predicted the correct overall form of the spectrum, including a maximum value, but disagreed with experimental data (obtained by the German spectroscopists) at long wavelengths; see Fig. 2.7.

Another attempt was made by Lord Rayleigh and James Jeans, considering the radiation inside a cavity to be made up of a number of standing waves. Their formula approximated the data at long wavelengths but failed dramatically at short wavelengths; instead of passing through a maximum, the spectrum was predicted to rise to infinity as the wavelength decreased to zero, a behaviour referred to as the ultraviolet catastrophe; see Fig. 2.12.

The German scientist Max Planck deduced a formula which approximates to the Rayleigh-Jeans law at long wavelengths and to the Wien's formula at short wavelengths, and therefore provides a good fit to the measured spectrum; see Fig.2.12. He then spent many years trying to derive this equation from basic principles of classical thermodynamics. Basically his idea was that the walls of a cavity (or any radiation-emitting surface) contain oscillators which emit electromagnetic radiation. Nowadays, we would identify these oscillators with oscillating electrons of the atoms present at the surface (an oscillating charge radiates energy, according to Maxwell's principles of electromagnetism). According to classical physics, these oscillators could have any energy, so energy is radiated (and re-absorbed, in a cavity) in randomly-varying amounts. But for mathematical convenience, Planck assumed the radiated energy to occur in discrete (quantised) amounts (rather than a continuous range of values) and found that he was then able to derive his radiation formula. When he repeated the calculations without the quantisation approximation (integrating over a continuous energy distribution) his result was the Rayleigh-Jeans formula, with its attendant ultraviolet catastrophe!

Planck therefore realised, somewhat reluctantly, that quantisation must be a property of nature. Using modern notation, energy corresponding to electromagnetic radiation of frequency f can only be released into the cavity in multiples of the basic quantum of energy given by E = hf , where h = 6.626 x 10^-34 Js is Planck's constant, one of the fundamental physical constants.

At high f , corresponding to short wavelength, the average thermal energy of an emitting atom (approximately kT where k is the Boltzmann constant) is less than hf ; therefore the chance of energy being emitted at this frequency is greatly reduced and the ultraviolet catastrophe is avoided.

Many physicists regarded this idea as just a mathematical trick for achieving the right answer; Planck himself was never completely happy with his discovery, although he realized its importance. But Einstein extended the quantum concept to cover not only the emission of energy from a surface but also the nature of electromagnetic radiation itself.