1.2. Waves or Particles? 3
the photoelectric effect, an effect in which a photon having the right amount of
energy knocks off a bound electron from the atom.
Max Planck proposed that the electromagnetic energy is emitted and absorbed
in the form of discrete bundles. The energy carried by such a bundle (that is, a
photon) is proportional to the frequency of the radiation.
E=hν (1.2.1)Hereh=6. 626 × 10 −^34 Jsis the Planck’s constant that was initially determined by
Max Planck to solve the black body spectrum anomaly. It is now considered to be
a universal constant. Planck’s constant has a very important position in quantum
mechanics. The frequencyν and wavelengthλof electromagnetic radiation are
related to its velocity of propagation in vacuumcbyc=νλ. If radiation is traveling
through another medium, its velocity should be calculated from
v=nνλ, (1.2.2)wherenis the refractive index of the medium. The refractive index of most materials
has a nonlinear dependence on the radiation frequency.
These experiments and the consequent theoretical models confirmed that some-
times radiation behaves as particles and not continuous waves. On the other hand
there were effects like interference, which could only be explained if light was con-
sidered to have continuous wave characteristics.
To add to this confusion, de Broglie in 1920 introduced the idea that sometimes
particles (such as electrons) behave like waves. He proposed that one could associate
a wavelength to any particle having momentum p through the relation
λ=h
p. (1.2.3)
For a particle moving close to the speed of light (the so called relativistic particle)
and rest massm 0 (mass of the particle when it is not moving), the above equation
can be written as
λ=
h
m 0 v√
1 −
v^2
c^2. (1.2.4)
For slow moving particles withv<<c, the de Broglie relation reduces to
λ=
h
mv. (1.2.5)
De Broglie’s theory was experimentally confirmed at Bell Labs where electrons
diffraction patterns consistent with the wave picture were observed. Based on these
experiments and their theoretical explanations, it is now believed that all the enti-
ties in the Universe simultaneously possess localized (particle-like) and distributed
(wave-like) properties. In simple terms, particles can behave as waves and waves
can behave as particles^2. This principle, known as wave-particle duality, has played
a central role in the development of quantum physics.
(^2) A more correct statement would be:all particles, regardless of being massless or massive, carry both
wave and particle properties.