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Wave-Particle Duality
Wave Nature of Light
The determining factor for a wave motion is the ability to
produce an interference pattern. Given that light can form an
interference pattern it must be considered a wave motion.
The Particulate Nature of Light
The photoelectric effect - electrons being emitted from the
surface of a metal given the incidence of radiation of a
sufficiently high frequency - can only be explained in terms of
light having a particulate nature.
The Photoelectric Effect
Results:
• Electrons are emitted from the surface of the metal when
the incident radiation reaches a threshold frequency.
• Electrons are not emitted if the frequency of the incident
radiation is below the threshold value.
• Increasing the intensity of the radiation increases the
number of photoelectrons emitted but the average kinetic
energy of these photoelectrons remains the same.
Conclusions:
• If light were a wave motion then the existence of a threshold
frequency cannot be explained. Given sufficient time the
electrons would gather enough energy from the incoming
wavefronts that they would eventually be emitted from the
metal surface.
• Increasing the intensity of the wave would increase the
energy associated with each wavefront. This would lead to
electrons being emitted with a greater average kinetic energy.
• If light is considered to be particulate in nature then the
energy of an individual particle will be transferred to the
electron. If the particle has insufficient energy then the electron
cannot be released from the surface.
•Increasing the intensity will increase the number of particles
arriving and hence the increase in the number of
photoelectrons emitted.
The photoelectric effect lead to the development of the waveparticle duality model.
Light (and all other electromagnetic radiations can be considered
to consist of tiny massless particles called photons. These
photons can be considered to be packets of energy and each
photon is called a quantum of energy.
The energy associated with each photon is given by
E = hf
h is Planck’s constant = 6.63 x 10-34 Js
f is the frequency of the incoming radiation
i.e., the energy of the photon is directly proportional to its
frequency.
This is known as the Quantum Theory of Radiation.
Radiation as Particles
For a photon:
E = hf
from v = f,
f = v/
all e-m radiations travel at the speed of light
hence v = c
E = hc/
Mass of a photon is zero at rest but its moving mass is given by:
mph = E/c2
but E = hc/
mph = (hc/)/c2
= hc/c2
mph = h/c
The momentum of a photon: p
= mphv
= mphc
but mph = h/c
= (h/c).c
p = h/ or  = h/p
I.e., the photon has both wave () and particle (p) features.
Particles as Waves
Electrons have been regarded as showing particle-like properties
- e.g., the deflection of a stream of electrons is just like the path
of a projectile.
Electrons can also exhibit wave properties - e.g., they can
demonstrate diffraction.
De Broglie proposed that an electron of mass m moving with
speed v would have a wavelength given by:
 = h/p
This equation gives the wavelength of any material particle is
given its momentum. The wavelength is known as the de Broglie
wavelength.
For a particle to exhibit wave characteristics the ‘scattering’
object must be less than 10 de Broglie wavelengths is size.
The Bohr Model of the Atom
Neils Bohr proposed a model of the atom in which electrons
revolve in dynamic equilibrium round the nucleus.
The electrons can only occupy certain allowed orbits.
The Bohr model allowed the existence of spectral lines to be
explained. Radiation is only emitted when an electron ‘jumps’
from one orbit to another of lower energy. The energy of the
radiation emitted is given by:
hf = E2 - E1
Bohr postulated that in the allowed orbits the angular
momentum was a multiple of h/2.
Momentum = mv
eF = mv2/r
+
Later de Broglie proposed that a
particle such as an electron may be
considered to behave as a wave of
wavelength  = h/p where h is
Planck’s constant and p is the
momentum of the moving particle.
If the electron can behave as a wave, it must be possible to fit a
whole number of wavelengths around the orbit. In this case a
standing wave pattern is set up and the energy in the wave is
confined to the atom. A progressive wave would imply that the
electron is moving from the atom and is not in a stationary orbit.
If there are n waves in the circular orbit and  is the
wavelength.
n
=
2r

=
2r/n
h/mv =
2r/n
2rmv =
nh
mvr =
but  = h/p = h/mv
nh/2
Now mvr is the moment of the momentum i.e., the angular
momentum of the electron about the nucleus.
mvr =
nh/2 demonstrates that the angular
momentum is a multiple of h/2 as Bohr had previously
postulated.
The Bohr model was developed in the early 20th century and
has since been superseded by more advanced theories.
In modern theory the wave-particle duality model has been
replaced by describing all matter in terms of wave functions.
These wave functions are useful in computational terms and
they allow waves to be interpreted in terms of probabilities.
This differs from Classical Physics where the aim was to
predict exactly what would happen in certain circumstances.
Quantum Mechanics provides methods to determine
probabilities for example finding the probability of finding
an electron at a particular point in space around an atom.