Quantum
From Academic Kids

The word quantum, pl. "quanta", comes from the Latin "quantus", for "how much". In general, it refers to an "amount of something". But, the term is often used in the more specific sense which it has in physics, where a quantum refers to an indivisible, and perhaps elementary entity. For instance, a "light quantum", being a unit of light (that is, a photon). In combinations like "quantum mechanics", "quantum optics", etc., it distinguishes a more specialized field of study.
Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "quantization". This means that the magnitude can take on only certain numerical values, rather than any value, at least within a range. For example, the energy of an electron bound to an atom (at rest) is quantized. This accounts for the stability of atoms, and matter in general.
An entirely new conceptual framework was developed around this idea, during the first half of the 1900s. Usually referred to as quantum "mechanics", it is regarded by virtually every professional physicst as the most fundamental framework we have for understanding and describing nature. For the very practical reason that it works. It is "in the nature of things", not a more or less arbitrary human preference.
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How quantization was discovered
Quantum physics, being the branch of physics based on quantization, was founded in the year 1900, with the theory, due to Max Planck, explaining the socalled black body radiation. His work incorporated quantization in essentially the same way that it is used today. But, the break with classical mechanics was so significant that it took another 30 years of investigation, before quantum theory became correctly formulated and understood. Some would maintain that it remains not fully understood until the present day. There is much to be learned about the nature of science from seeing how this came about.
Planck himself felt disturbed by the new idea of quantization, and with good reason. But, finding no alternative, he nevertheless ended up using it, and the work was well received. After "a few weeks of the most strenuous work of his life", as he recalled it in the lecture on the occasion of receiving the Nobel Prize for physics, 18 years later. In the course of those weeks he even had to discard much of his own theoretical work from the preceding years. Quantization turned out to be the only way to describe the new, and detailed experiments which were just then being performed. He did this practically overnight, openly reporting his change of mind to his scientific colleages, in the October, November, and December meetings of the German Physical Society, in Berlin, where the black body work was being intensely discussed. In this way, careful experimentalists (including F. Paschen, O.R. Lummer, E. Pringsheim, H.L. Rubens, and F. Kurlbaum), and a reluctant theorist, ushered in the greatest revolution science has ever seen.
The quantum blackbody radiation formula
When a body is heated, it emits heat radiation, which is infrared radiation, being a form of electromagnetic waves. All of this was well understood at the time, and of considerable practical importance. When the body becomes redhot, the red wavelength parts start to become visible. This had been studied over the previous years, as the instruments were being developed. However, most of the heat radiation remains infrared, until the body becomes as hot as the surface of the Sun (about 6000 °C, where most of the light is green in color). This was not achievable in the laboratory at that time. What is more, to measure specific infrared wavelengths was only then becoming feasible, due to newly developed experimental techniques. Until then, most of the black body spectrum was not measurable, and therefore not mapped out in detail.
The quantum blackbody radiation formula, being the very first piece of quantum mechanics, appeared sunday evening October 7, 1900, in a socalled backoftheenvelope calculation by Planck. It was based on a report by Rubens (visiting with his wife) of the very latest experimental findings in the infrared. Later that evening, Planck sent the formula on a postcard, which Rubens had the following morning. A couple of days later, he could tell Planck that it worked perfectly. As it does to this day.
At first, it was just a fit to the data. Only weeks later did it turn out to enforce quantization.
That the latter became possible involved a certain amount of luck (or skill, even though Planck himself called it "a fortuitous guess at an interpolation formula"). It only had that drastic "side effect" because the formula happened to become fundamentally correct, in regard to the as yet nonexistent quantum theory. And normally, that much is not at all expected. The skill lay in simplifying the mathematics, so that this could happen. And here Planck used hard won experience from the previous years. Briefly stated, he had two mathematical expressions:
 (i) from the previous work on the red parts of the spectrum, he had x;
 (ii) now, from the new infrared data, he got x².
Combining these as x(a+x), he still has x, approximately, when x is much smaller than a ( the red end of the spectrum). But now also x², again approximately, when x is much larger than a (in the infrared). The luck part is that, this procedure turned out to actually give something completely right, far beyond what could reasonably be expected. The formula for the energy E, in a single mode of radiation at frequency f, and temperature T, can be written
 <math>E = \frac{h f}{e^{\frac{h f}{k T}}  1} <math>
This is (essentially) what is being compared with the experimental measurements. There are two parameters to determine from the data, written in the present form by the symbols used today: h is the new Planck's constant, and k is Boltzmann's constant. Both have now become fundamental in physics, but that was by no means the case at the time. The "elementary quantum of energy" is hf. But such a unit does not normally exist, and is not required for quantization.
The birthday of quantum mechanics
From the experiments, Planck deduced the numerical values of h and k. Thus he could report, in the German Physical Society meeting on December 14, 1900, where quantization (of energy) was revealed for the first time, values of the AvogadroLoschmidt number, the number of real molecules in a mole, and the unit of electrical charge, which were more accurate than those known until then. This event has been referred to as "the birthday of quantum mechanics".
Quantization in antiquity
In a sense, it can be said that the quantization idea is very old. A string under tension, and fixed at both ends, will vibrate at certain quantized frequencies, corresponding to various standing waves. This, of course, is the basis of music. The basic idea was regarded as essential by the Pythagoreans, who are reported to have held numbers in high esteem.
It is a curious fact that, the famous formula, named after Pythagoras, for the side lengths of a right triangle, today serves as a cornerstone of quantum mechanics as well.
The very existence of atoms, molecules, solids, and so on, can be ascribed to various forms of quantization. Contrary to notions of matter as some form of continuous medium. This was also understood already in antiquity, particularly by Leucippos and Democritos, although not generally appreciated, even by physicists, really, until the invention of quantum mechanics.
It should be mentioned, though, that later works within the Epicurean school of thought played a significant role in forming the physics and chemistry of the Renaissance period in Europe. In particular the famous tutorial poem "De rerum natura" by the Roman author Titus Lucretius Carus.
References
 M. Planck, A Survey of Physical Theory, transl. by R. Jones and D.H. Williams, Methuen & Co., Ltd., London 1925 (Dover editions 1960 and 1993) including the Nobel lecture.
 J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol.1, Part 1, SpringerVerlag New York Inc., New York 1982.
 Lucretius, "On the Nature of the Universe", transl. from the Latin by R.E. Latham, Penguin Books Ltd., Harmondsworth 1951. There are, of course, many translations, and the translation's title varies. Some put emphasis on how things work, others on what things are found in nature.