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)

Albert Einstein (1905) used Planck's relationship to explain the results of the photoelectric effect which showed that the energy E of ejected electrons was dependent upon the frequency f of incident light as described in the equation E=hf. It is ironic that in 1921 Albert Einstein was awarded the Nobel Prize for this discovery, though he never believed in particles and acknowledged that he did not know the cause of the discrete energy transfers (photons) which were contradictory to his continuous field theory of matter!
In 1954 Albert Einstein wrote to his friend Michael Besso expressing his frustration;

All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. (Albert Einstein, 1954)

Most importantly, Albert Einstein also suspected that Matter could not be described by a continuous spherical force field;

I consider it quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. (Albert Einstein, 1954)

Albert Einstein's suspicions were well justified, for he had spent a lifetime trying (and failing) to create a unified field theory of matter that explained both Quantum Theory / Light and Relativity / Gravity.
In fact Matter, as a Spherical Standing Wave which causes the 'Field' effect, interacts with other matter in discrete standing wave patterns, not with continuous force fields as he had imagined, thus his task was ultimately impossible, as he sadly came to realise towards the end of his life.

However, his work on the photoelectric effect confirmed that light energy was only emitted and absorbed by electrons in discrete amounts or quanta. This quanta of light energy soon became known as the 'photon' (i.e. discrete like a particle) and led to the paradox that light behaved both as a continuous e-m wave (Maxwell, Albert Einstein) as well as a discrete particle/photon (Planck, Albert Einstein). So we see that Albert Einstein was partly responsible for the discovery of the particle/photon concept of light, though he completely rejected the notion of discrete particles. He writes;

Since the theory of general relativity implies the representation of physical reality by a continuous field, the concept of particles or material points cannot play a fundamental part, nor can the concept of motion. (Albert Einstein)

Albert Einstein is correct that there are no discrete particles, and that the particle can only appear as a limited region in space in which the field strength or the energy density are particularly high. But it is the high Wave-Amplitude of the Wave-Center of a Spherical Standing Wave in Space (not of a continuous spherical force field) that causes the particle effect. Thus of three concepts, particles, force fields, and motion, it finally turns out that Motion, as the spherical wave motion of space, is the correct concept, as it then explains both particles and fields.
It is most important to realize though that Albert Einstein was correct in imagining matter as being spatially extended, as he explains;

I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept empty space loses its meaning. (Albert Einstein)

It is certainly true that the particle and its forces / fields are very useful mathematical concepts, unfortunately, they also cause many problems and paradoxes because they are approximations to reality and do not physically exist. We can now finally solve these problems by understanding the reason for these discrete energy states, which are due to the fact that standing waves only exist at discrete frequencies, like notes on the string of a guitar, thus while the Spherical Standing Wave Structure of Matter predicts that energy exchanges will be discrete, as observed, the continuous e-m wave does not anticipate this. Thus the Spherical Standing Wave Structure of Matter explains Max Planck's (1900) discovery that there are only certain allowed discrete energy states for electrons in molecules and atoms, and further, that light is only ever emitted and absorbed by electrons in discrete or 'quantum' amounts, as the electrons move from one stable standing wave pattern to another.