where the Boltzmann constant k = 1.381 × 10−23 [J/K]. This result indicated that the total energy radiated by a blackbody according to this “classical” model would increase with ν2 as shown by the dashed curve in Figure 1.2b. If this equation were correct, any temperature of a material above absolute zero would be impossible, since any material above 0 K would emit radiation according to Eq. (1.5), and the total energy emitted would be unrestricted and approach infinity. Particularly, toward higher frequency, more and more radiation would be emitted, and the blackbody would cool instantaneously to 0 K. Thus, any temperature above 0 K would be impossible. (For a more detailed discussion on this “ultraviolet catastrophe,” see Engel and Reid [2].)
This is, of course, in contradiction with experimental results and was addressed by M. Planck (1901) who solved this conundrum by introducing the term 1/(ehν/kT − 1) into the blackbody equation, where h is Planck's constant:
The shape of the modified blackbody emission profile given by Eq. (1.6) is in agreement with experimental results. The new term introduced by Planck is basically an exponential decay function, which forces the overall response profile to approach zero at high frequency. The numerator of the exponential expression contains the quantity hν, where h is Planck's constant (h = 6.626 × 10−34 Js). This numerator implies that light exists as “quanta” of light, or light particles (photons) with energy E:
This, in itself, was a revolutionary thought since the wave properties of light had been established more than two centuries earlier and had been described in the late 1800s by Maxwell's equations in terms of electric and magnetic field contributions. Here arose for the first time the realization that two different descriptions of light, in terms of waves and particles, were appropriate depending on what questions were asked. A similar “particle–wave duality” was later postulated and confirmed for matter as well (see below). Thus, the work by Planck very early in the twentieth century is truly the birth of the ideas resulting in the formulation of quantum mechanics.
Incidentally, the form of the expression
Figure 1.2 shows that the overall emitted energy increases with increasing temperature and that the peak wavelength of maximum intensity shifts toward lower wavelength (Wien's law). The total energy W radiated by a blackbody per unit area and unit time into a solid angle (the irradiance), integrated over all wavelengths, is proportional to the absolute temperature to the fourth power:
(1.8)
(Stefan–Boltzmann law)
The irradiance is expressed in units of
The implication of the aforementioned wave–particle duality will be discussed in the next section.
1.3 The Photoelectric Effect
In 1905, Einstein reported experimental results that further demonstrated the energy quantization of light. In the photoelectric experiment, light of variable color (frequency) illuminated a photocathode contained in an evacuated tube. An anode in the same tube was connected externally to the cathode through a current meter and a source of electric potential (such as a battery). Since the cathode and anode were separated by vacuum, no current was observed, unless light with a frequency above a threshold frequency was illuminating the photocathode. Einstein correctly concluded that light particles, or photons, with a frequency above this threshold value had sufficient kinetic energy to knock out electrons from the metal atoms of the photocathode. These “photoelectrons” left the metal surface with a kinetic energy given by
where ϕ is the work function, or the energy required to remove an electron from metal atoms. This energy basically is the atoms' ionization energy multiplied by Avogadro's number. Furthermore, Einstein reported that the photocurrent produced by the irradiation of the photocathode was proportional to the intensity of light, or the number of photons, but that increasing the intensity of light that had a frequency below the threshold did not produce any photocurrent. This provided further proof of Eq. (1.9).
This experiment further demonstrated that light has particle character with the kinetic energy of the photons given by Eq. (1.7), which led to the concept of wave–particle duality of light. Later, de Broglie theorized that the momentum p of a photon was given by