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Quantization of energy and particle-wave duality 213

Related topics Diffraction by solids (A6) The wave nature of matter (G4)

The failures of classical physics

In the everyday world of macroscopic objects the Newtonian laws of classical physics
account extremely well for the motion of particles along defined trajectories. These laws
assume that the position and velocity of a particle can be defined at every instant, from
which it is possible, at least in theory, to calculate the precise position and velocity of the
particle at every other instant. Classical physics further assumes that any type of motion
can be supplied with any arbitrary amount of energy. Thus, for example, the range of an
artillery shell is, in principle, continuously variable according to the amount of energy
supplied at the initial firing.
However, it turns out that the laws of classical mechanics are an approximate
description of the motion of particles, accurate only in the limit of large objects travelling
at velocities much less than the speed of light. To account for the behavior of very small
particles such as molecules, atoms or electrons requires the application of a more
fundamental set of laws, the laws of quantum mechanics. Quantum theory was
formulated in the early years of the 20th century when classical physics failed to account
for many sets of experimental observations arising from atomic-scale phenomena, for
example, the ultraviolet catastrophe and the photoelectric effect. The resolution of these
failures incorporated the postulate that energy was quantized and introduced the concept
of particle-wave duality for radiation and matter.

Quantization

A fundamental outcome of the theory of quantum mechanics is that properties such as
energy are no longer permitted to assume any value within a continuum but are confined
to a series of discrete values only. This outcome is called quantization and the discrete
values are called quanta. The values of the quanta depend on the specific boundary
conditions of the system under consideration (Topic G4). Other properties to which
quantization applies include position and angular momentum.

The Planck constant
The Planck constant, h, is a fundamental constant of quantum theory and appears in very
many equations describing quantum mechanical phenomena. It is the constant of
proportionality between the energy, E, of a photon and the frequency, v, of the associated
electromagnetic radiation:
E=hv

Planck’s constant can be determined from an analysis of the photoelectric effect. The
value is 6.626×10 −34 J s. So, for example, an ultraviolet photon of wavelength 300 nm