1.1 Quantum Physics Foundations: Max Planck's Discovery of Particle / Quantum Properties of Light (1900)

In 1900 Max Planck made a profound discovery. He showed (from purely formal / mathematical foundations) that light must be emitted and absorbed in discrete amounts if it was to correctly describe observed phenomena (i.e. Blackbody radiation).
Prior to then light had been considered as a continuous electromagnetic wave, thus the discrete nature of light was completely unexpected, as Albert Einstein explains;

About fifteen years ago (1899) nobody had yet doubted that a correct account of the electrical, optical, and thermal properties of matter was possible on the basis of Galileo-Newtonian mechanics applied to molecular motion and of Maxwell's theory of the electromagnetic field. (Albert Einstein, 1915)

Then Planck showed that in order to establish a law of heat radiation (Infra red light waves) consonant with experience, it was necessary to employ a method of calculation whose incompatibility with the principles of classical physics became clearer and clearer. For with this method of calculation, Planck introduced into physics the quantum hypothesis, which has since received brilliant confirmation. (Albert Einstein, 1914)

In the year nineteen hundred, in the course of purely theoretical (mathematical) investigation, Max Planck made a very remarkable discovery: the law of radiation of bodies as a function of temperature could not be derived solely from the Laws of Maxwellian electrodynamics. To arrive at results consistent with the relevant experiments, radiation of a given frequency f had to be treated as though it consisted of energy atoms (photons) of the individual energy hf, where h is Planck's universal constant. During the years following, it was shown that light was everywhere produced and absorbed in such energy quanta. In particular, Niels Bohr was able to largely understand the structure of the atom, on the assumption that the atoms can only have discrete energy values, and that the discontinuous transitions between them are connected with the emission or absorption of energy quantum. This threw some light on the fact that in their gaseous state elements and their compounds radiate and absorb only light of certain sharply defined frequencies. (Albert Einstein, 1940)

Even the Greeks had already conceived the atomistic nature of matter and the concept was raised to a high degree of probability by the scientists of the nineteenth century. But it was Planck's law of radiation that yielded the first exact determination - independent of other assumptions - of the absolute magnitudes of atoms. More than that, he showed convincingly that in addition to the atomistic structure of matter there is a kind of atomistic structure to energy, governed by the universal constant h, which was introduced by Planck. This discovery became the basis of all twentieth-century research in physics and has almost entirely conditioned its development ever since. Without this discovery it would not have been possible to establish a workable theory of molecules and atoms and the energy processes that govern their transformations. Moreover, it has shattered the whole framework of classical mechanics and electrodynamics and set science a fresh task: that of finding a new conceptual basis for all physics. Despite remarkable partial gains, the problem is still far from a satisfactory solution. (Albert Einstein, 1950) Quantum-Physics