1. 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.

2. 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.

3. 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.

4. The energy associated with each photon is given by
The photoelectric effect lead to the development of the wave-particle 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 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.

5. 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.

6. 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.

7. 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

8. Bohr postulated that in the allowed orbits the angular momentum was a multiple of h/2.+ e-
Momentum = mv
F = 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.

9. If there are n waves in the circular orbit and  is the wavelength.
n = 2r
 = 2r/n but  = h/p = h/mv
h/mv = 2r/n
2rmv = nh
mvr = 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.

10. 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.