Light's particle-like traits are best explained by the photoelectric effect, the theory that Albert Einstein won his Nobel Prize for.
The photoelectric effect reffers to the emission (or ejection) of electrons from the surface of a metal in response to incident light. Energy contained by the incident light is absorbed by the electrons in the metal; this gives the electrons enough energy to be emmitted from the surface of the metal.
An interesting thing was dicovered when trying to apply Maxwel's wave theory of light to this experiment. According to the classical wave approach. the energy of the electrons should increase with the intensity of the incident light - the more intense the light, the bigger the average energy carried by an emmitted electron.
However, experimental data showed that the energies of the emmitted electrons were independent of the intensity of the light; these energies were proportional to the frequencies of the incident light.
Here is where Einstein's theory provided an explanaition. Einstein argued that this could be explained if light was composed of "bundles", or photons; when a photon strikes the metal's surface, its energy was transferred to the electron - much like when two billiard balls collide. This was the ultimate evidence for light's particle nature.
The energy carried by a photon is mathematically described as
The wave theory of light was proven in 1801 by English physicist Thomas Young, who designed and conducted the famous double-slit experiment.
The experiment was fairly simple and required a light source, a thin card with two holes cut side by side, and a screen.
The expected result of the experiment was the presence of two bright spots on the screen, corresponding to the two slits made in the thin card. However, the observed result showed something very different.
Instead of two bright spots, a bar code pattern of bright bands and dark bands was visible on the screen; this was in agreement with the idea that light traveled in the form of waves.
Each of the two slits created a separate wave front, where interference between the waves occured. The bright bands were formed on the screen when two wave crests overlapped and added together (constructive interference), while the dark bands were formed when crests and troughs overlapped and cancelled each other out (destructive interference).
In addition to the conclusions drawn from this experiment, it is also known that light reflects, refracts, diffracts, and exhibits the Doppler effect in the same way any wave would.