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Quantization of energy and particle-wave duality 215

electromagnetic radiation could oscillate at any frequency and therefore that all
wavelengths, λ, of radiation were possible. The resulting equation:

works fairly well at long wavelengths (low frequencies) but fails at short wavelengths
(high frequencies) because as λ decreases the power emitted increases continuously
towards infinity, and never passes through a maximum. The equation predicts that a black
body is a strong emitter of all wavelengths, including ultraviolet, X-rays and γ-rays, even
at room temperature. This obvious absurdity is termed the ultraviolet catastrophe.
The problem is resolved by Planck’s postulate that the energy of each electromagnetic
oscillator is limited to discrete values of energy equal to an integral multiple of its
oscillation frequency, v:
E=nhv n=0, 1, 2…

The constant of proportionality, h, is Planck’s constant (6.626×10 −34 J s). The
consequence of this quantization is that oscillators can only be stimulated when energy
of value hv (or 2hv, or 3hv, etc.) is available. The relative probability of finding
oscillators of energy nhv at a temperature T is given by the factor of the
Boltzmann distribution law (see Topic G8). This factor tends to zero as the value of v/T
in the exponential increases. Since the values of hv for X-rays or γ-rays are very large
(very high frequencies of oscillation) only a negligible fraction of these oscillators are
stimulated unless the energy (or T) of the black body is itself extremely large. Planck’s
modified version of the energy density formula for a black body includes the Boltzmann
exponential term:

and reproduces the experimental curve in Fig. 1 extremely well. At large values of λ the
Planck law and Rayleigh-Jeans law are equivalent.

The photoelectric effect

The photoelectric effect is the emission of electrons from a surface (usually a metal)
when the surface is irradiated with ultraviolet light. The maximum kinetic energy of the
2
ejected electrons, 1/2m eυ , can be calculated from the threshold negative voltage required
to repel them from a detector above the surface. Three key experimental observations of
the photoelectric effect are:

(i) no electrons are ejected, regardless of the intensity of the radiation, unless the
frequency of the radiation exceeds a threshold value characteristic of the metal;
(ii) once the threshold frequency is exceeded the kinetic energy of the ejected electrons is
linearly proportional to the frequency of the incident radiation;