It is perhaps useful to consider how the concept of the quantisation of energy came into being. Late in the ninetieth century, physicists were attempting to theoretically describe the radiation spectrum of a 'black body' cavity. A black body, being a perfect absorber and emitter of radiation whose radiation distribution depends upon its temperature. Using classical statistics and mechanics, many theorists had attempted to mathematically describe the radiation spectrum, for a black body cavity, but with limited success. None of the classical descriptions could accurately predict the higher energy side of the black body radiation spectrum without predicting an 'ultra violet catastrophe' of infinite energies. By classical theories we mean a continuous spread of energies that the particles, within the black body cavity, could take on. However, the German physicist Max Planck adopted a different approach to the problem, Wikipedia comments,
'... In 1894 Planck turned his attention to the problem of black-body radiation. He had been commissioned by electric companies to create maximum light from light bulbs with minimum energy. The problem had been stated by Kirchhoff in 1859: how does the intensity of the electromagnetic radiation emitted by a black body (a perfect absorber, also known as a cavity radiator) depend on the frequency of the radiation (e.g., the colour of the light) and the temperature of the body? The question had been explored experimentally, but no theoretical treatment agreed with experimental values. Wilhelm Wien proposed Wien's law, which correctly predicted the behaviour at high frequencies, but failed at low frequencies. The Rayleigh-Jeans law, another approach to the problem, created what was later known as the "ultraviolet catastrophe", but contrary to many textbooks this was not a motivation for Planck.
Planck's first proposed solution to the problem in 1899 followed from what Planck called the "principle of elementary disorder", which allowed him to derive Wien's law from a number of assumptions about the entropy of an ideal oscillator, creating what was referred to as the Wien-Planck law. Soon it was found that experimental evidence did not confirm the new law at all, to Planck's frustration. Planck revised his approach, deriving the first version of the famous Planck black-body radiation law, which described the experimentally observed black-body spectrum well. It was first proposed in a meeting of the Deutsche Physikalische Gesellschaft (DPG) on 19 October 1900 and published in 1901. This first derivation did not include energy quantization, and did not use statistical mechanics, to which he held an aversion. In November 1900, Planck revised this first approach, relying on Boltzmann's statistical interpretation of the second law of thermodynamics as a way of gaining a more fundamental understanding of the principles behind his radiation law. As Planck was deeply suspicious of the philosophical and physical implications of such an interpretation of Boltzmann's approach, his recourse to them was, as he later put it, "an act of despair ... I was ready to sacrifice any of my previous convictions about physics."
The central assumption behind his new derivation, presented to the DPG on 14 December 1900, was the supposition that the electromagnetic energy could be emitted only in quantized form, in other words, the energy could only be a multiple of an elementary unit E = h?, where h is Planck's constant, also known as Planck's action quantum (introduced already in 1899), and ? is the frequency of the radiation.
Planck in 1918, the year he received the Nobel Prize in Physics for his work on quantum theory. At first Planck considered that the quantisation was only as "a purely formal assumption ... actually I did not think much about it..."; nowadays this assumption, incompatible with classical physics, is regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck's career (Ludwig Boltzmann had been discussing in a theoretical paper in 1877 the possibility that the energy states of a physical system could be discrete). The full interpretation of the radical implications of Planck's work was advanced by Albert Einstein in 1905—for this reason, the philosopher and historian of science Thomas Kuhn argued that Einstein should be given credit for quantum theory more so than Planck, since Planck did not understand in a deep sense that he was "introducing the quantum" as a real physical entity. It was in recognition of his monumental accomplishment that Planck was awarded the Nobel Prize in Physics in 1918.
The discovery of Planck's constant enabled him to define a new universal set of physical units (such as the Planck length and the Planck mass), all based on fundamental physical constants. ..'
Thus, Planck accounted for the radiation spectrum of a black body cavity with the condition that the particles within the 'body' could only oscillate with a discrete set of 'quantised' energies. Hence, quantum theory emerged as a statistical theory that dealt with large collections of particles. However, Einstein extended Planck's work when he described the photoelectric effect as a consequence of each photon having a quantised energy of E=hf (f = frequency). Thus, quantum theory could then describe individual 'photons' of energy.
During the nineteen twenties, the full mechanics of quantum theory emerged from Schrodinger and Heisenberg. These aspects of the theory could handle electron orbits about the nucleus. The ‘band theory’ of conductors and semi/non - conductors emerged with its more statistical nature, in the following decade. Wigner extended quantum theory to deal with the energetics of the nucleus and nuclear spectra. During the nineteen-fifties, quantum mechanics was used to account for super conductivity in the BCS theory.
Hence, particle quantisation deals with the microscopic world for individual particles, atoms, electrons, and nuclei or statistical collections of particles such as black body cavities or Einstein-Bose condensates.
