One of the first people to have studied the nature of light and reached to a conclusion was Sir Isaac Newton, who in the 17th century declared that light was corpuscular, i.e., it is made up of particles. Although there were some outstanding questions, Newton's view was accepted by many scientists and popularized by his followers for over 100 years. But when early in the 19th century, Thomas Young came up with an interference pattern of light produced when light passed through a narrow slit, these views received a blow. Fransesco Grimaldi had, in the middle of the 17th century already observed the bending of light through a narrow slit, which is today known as diffraction. These two observations were a distinctive indicator of light being a wave, and the wave-theory, first proposed by Huygens came up. A subsequent work by Fresnel soon convinced scientists that light must be a wave.
In the middle of the 19th century, James Clerk Maxwell came up with some brilliant equations, today known as Maxwell's equations, and proved that electricity and magnetism were not two different phenomena, but integrally inter-related. His equations revealed that there should be a kind of wave, consisting of a special interlocked pattern of oscillating electric and magnetic fields. Such an electromagnetic wave was actually find to exist, and Maxwell's calculations could even yield a result for its speed, which was found to be exactly the same as the speed of light, that had been calculated by some other experiments. These experiments, for some time, demolished the views in the minds of some scientists about the corpuscular theory proposed by Newton, and wave-theory of light, based on solid evidence came to be universally accepted.
However, just about twenty years later, it was discovered that when light is made to shine on the surface of a metal, beams of negatively-charged particles are ejected from the surface of the metal. This was a really stunning discovery, as this observation could not be explained using the then widely accepted Maxwell's electromagnetic explanation of light. Just before the turn of the century, in 1899, Philip Lenard explained this observation by saying that the negatively charged beam consisted of electrons, which had already been discovered by J. J. Thompson in 1897. However the final explanation did not come before the contributions of two more people - Max Plank and Albert Einstein.
Another problem that came up in those days was about 'black-body radiation'. The first person to have attempted to give a solution to this problem was Max Planck, who in 1900 suggested a form of quantization that accurately predicted the results observed. However, he himself felt that the mathematics he had used was only to make the answers come out right and fitting the observations.
The photoelectric effect, however, was given an explanation by the German scientist, Albert Einstein, for which he was awarded a Nobel Prize in 1921. He used the principle of quantization that was introduced by Max Planck. He suggested that the quantization used by Planck reflects a basic aspect of the reality. His interpretation, contrary to the then popular wave-theory was that light, in some real fundamental sense, as a particle and not as a wave. He named his new particle a photon. But the scientists in those days did not seem to be in a hurry to accept Einstein's views about the quantization of light.
Einstein was recommended for membership in the Prussian Academy of Sciences in 1913, through a letter of recommendation prepared by Max Planck himself. His ideas were experimentally confirmed by American scientist Robert Milikan a year later. However, his ideas were finally accepted throughout the world only a decade later when the American physicist Compton and a Dutch physicist, Debye made independent theoretical predictions for the scattering of photons from the electron, after assuming light as being composed of particles with definite energy and directed momentum, as proposed by Einstein. Their experiments produced results that indeed confirmed the particle assumption made by them.
However, the success story of Einstein received a setback, when Prince Louis deBroglie described the electron as a wave and showed that this description helped explain many features of the atomic model at that time. He examined the consequences of combining the equations of Planck and Einstein. Since both the equations are solved for energy, they should be equal to one another. This led him to the result that wavelength is equal to h/mc, where h is the Planck's constant and c is the speed of light. But since mass times speed is defined as momentum, it might be possible to define a wavelength for a particle of matter if its momentum is known. Substituting numbers into this equation revealed that for any object in our world, the deBroglie wavelength is too tiny to detect. However, when as tiny particles as the atom and the electron are considered, the dimensions of these waves become significant. In fact, the atom's size itself is the size of its electron wave. By introducing the possibility that light could be described both as a particle as well as a wave, he stimulated others to consider the consequences of this duality on the model of the atom. He realized that the simplest way to introduce quantum ideas into an atomic model was to give wave properties to the electron. A standing wave must have integral wavelengths with no possibility of fractional wavelengths. deBroglie, for his work received a Nobel Prize in 1929.
Physicists, in those days developed a way to describe the behavior of sub-atomic phenomena in terms of both waves and particles by using mathematics, specifically through the use of Max Planck's constant, which revealed that energy actually traveled in discrete bundles, and not as a continuos flow. In those days, many people believed that it was possible to determine the future of the universe, and that events in the universe were occurring absolutely independent of the observer's frame of reference. But, the dual wave-particle nature of electrons demolished such beliefs. While Erwin Schrodinger came up with a mathematical equation, which nicely described waves proposed by de Broglie, others saw definite evidence of particulate behavior. A cloudlike wave pattern of the was hypothesized by Max Born as a probability wave of finding an electron. However, the wave pattern does not speak about where the particle is at any given moment, or where is it likely to be. The rule established then says that the square of the wave amplitude at any point in space gives the probability of finding the electron at that point. But knowing where the particle is does not tell us anything about its wave function. It was revealed later by Heisenberg, in his famous uncertainly principle, that there is actually no way to simultaneously find both the position and path of the particle, which, today we know is true.