2.2 Interaction of radiation with matter 37
2.2.4 Wave–particle duality
As a result of thorough experimental and theoretical analysis of the properties of
light its dual nature was established. In some processes light behaves as a wave
(interference, diffraction, etc.), whereas in others it behaves as a flux of particles
(blackbody radiation, the photoelectric effect, etc.). The equations
ε
ph
= h
-
ω, (2.76)
p
ph
= h
-
k (2.77)
relate the particle and wave properties of light. The left-hand sides of these
equations (with ε
ph
and p
ph
) characterize the photon as a particle and the right-
hand sides (with ω and k) characterize the photon as a wave.
There is an important trend in the observations of the dual nature of light. For
long-wavelength radiation (e.g., infrared radiation) its quantum properties are
not so obvious and mainly its wave properties are detected. However, on going to
the shorter wavelengths the quantum properties of light become more apparent.
Wave and quantum properties of light are connected, and they supplement each
other. The quantum properties of light become apparent by virtue of the fact
that the energy, momentum, and mass of radiation are concentrated in particles –
photons. The probability of finding photons at particular points of space is defined
by the amplitude of the light wave, i.e., by the wave properties of light.
The wave properties are inherent not only to large ensembles of photons,
but also to each individual photon. This is evident because it is not possible
to specify the location of a photon and the direction which it will have after a
collision with an obstacle. We can talk only about the probability of finding an
individual photon in one place or another. The description of the behavior of such
an object on the basis of classical laws is impossible. Nevertheless, experimental
facts allow us to state that this duality in light’s behavior is a law of nature. Light
was the first object that allowed physicists to observe and to interpret the wave–
particle duality of matter. The further development of physics greatly enhanced
the class of such objects. As will be discussed later, other particles, such as the
electron, proton, neutron, etc., may have wave properties too, which broadens
our knowledge about matter.
Example 2.5. An X-ray photon with frequency ν = 6 ×10
18
Hz scatters on a
free electron at angle θ = 90
◦
. Find the frequency of the scattered photon, as well
as the momentum, velocity, and energy of the electron after its collision with the
photon.
Reasoning. After the collision of a photon with a motionless electron, the wave-
length of the scattered photon increases, in accordance with Eq. (2.66), by the
magnitude
λ = (1 − cos θ), (2.78)