Quantum mechanics owes its name from the hypothesis that originated it, initially proposed by Max Planck, and successively extended by Niels Bohr and others.
The hypothesis of quantization can be formally expressed through Bohr-Sommerfeld conditions, which constitute the core of what is presently called the Old Quantum Theory. Their most immediate consequence is that the energy of an individual particle is in some cases quantized, i.e., only allowed to take on specific, well-defined values, arranged into a regular pattern, describable mathematically by simple formulae.
In these cases, an experimental measurement of the energy yields no other outcome than one of these “allowed” values — nothing in-between is ever observed. This is what is meant by “quantization”.
In order to illustrate the basic idea in simple language, let us make use of the same analogy proposed in this documentary (minute 9:00 onward), in the context of Bohr’s model of the hydrogen atom. A hydrogen atom consists of two electrically charged particles, a proton and an electron, the electron being some 2,000 times lighter than the proton. In the simple (actually, simplistic, and now abandoned) model of Bohr, the electron is on a circular orbit around the proton much like a planet around the Sun, the only difference being that the attraction is electric, not gravitational in character (electron and proton attract one another due to their equal and opposite electric charges).
The twist here, is that while planets may be found on circular orbits of any radius, an electron is seemingly forced to be on orbits at specific distances from the proton, other orbits being forbidden by Nature. As a result, because the energy of an electron depends on the radius of its orbit, it is quantized, as the radius itself is quantized .
Where does the quantization (of the radius of the orbit or of the energy) come from ?
“Think of your daily life… Do you think that you have to take a certain amount of minimum food ? … Food is not quantized ! [...] but the energy of the electron in an atom is quantized… that is very mysterious… why that is”.
Thus states a prominent physicist in the above-mentioned documentary (minute 11:00), his words at least implying that there is something inherently inexplicable about the fact that energy is (sometimes) quantized, supposedly hopelessly irreconcilable with our daily experience.
That is the only one, truly unfortunate passage of an otherwise remarkable piece of scientific divulgation. At this point in the documentary (and at this point only), misleading and inaccurate information is given out to the general public. I come back to this point below .
Otherwise, I really doubt if one could do a better job than what theoretical physicist Brian Greene did with “Quantum Leap” (part of its celebrated series of documentaries titled “The Fabric of the Cosmos“), at illustrating some of the most stunning, jaw-dropping aspects of this fascinating subject, too intriguing for me not to decide, many years ago, to try to make it my profession.
Particularly effective, in my opinion (shared by a number of friends who are not scientists and watched it), is the section of the documentary that follows the above discussion, in which Greene attempts to convey to a non-specialist viewership, by means of simple but clever analogies, at least a sense of one of the most profound, truly mind-boggling aspect of quantum mechanics, namely wave-particle duality. Wave-particle duality is observed in the landmark double-slit experiment, and is formally embodied in the Schroedinger’s Equation (or in the equivalent Feynman’s Path Integral formulation of quantum mechanics). I am not going to try to explain wave-particle duality here. If you are interested, watch the documentary, it really is well done.
OK, what are you bitchin’ about, then ?
The fact that molecules, atoms, sub-atomic particles, can display wave-like behaviour, manifesting itself through phenomena such as interference — that is the true mystery of quantum mechanics, a fact around which not even its founding father Paul A. M. Dirac could wrap his head . On this core assumption, it is possible to formulate a mathematical theory affording quantitative predictions for phenomena occurring at the atomic or molecular level — predictions of spectacular accuracy, confirmed by every single experiment carried out so far, making quantum mechanics inarguably one of the most successful scientific theories, ever.
What about quantization of energy ? Is it not also a crucial aspect of quantum mechanics ?
First of all, energy is not the only quantity known in Nature to be quantized. Electric charge is quantized as well, and this was known before quantum theory was even introduced. Angular momentum is also quantized. And, quantization of both electric charge and angular momentum is in many respects more fundamental than quantum mechanics itself .
Secondly, and more importantly, while electric charge and angular momentum are inherently, always quantized, there is something funny about energy quantization, in that it only takes place in specific circumstances, namely when the motion of a particle is restricted in space.
Indeed, while the energy of an electron in an atom is quantized, that of an electron that breaks free of its tie to an atom, for example following the adsorption of light, and is allowed to wander around in free space, is not quantized, i.e., it can take on any value (as long as it is allowed by Einstein’s theory of relativity, but that is another story). In other words, unlike charge or angular momentum, there is nothing intrinsic to energy, that requires that it be quantized.
So, what you sayin’ … ?
As it turns out, quantization of energy, somewhat misleadingly presented at the beginning of the documentary as one of its cornerstones, is not a fundamental result of quantum mechanics; in fact, once one manages to come to terms, digest so to speak, the idea of wave-particle duality, energy quantization immediately ensues as a fairly mundane, straightforward, in many respect almost spurious, accidental consequence of it.
Whether or not food is quantized may be a matter of taste (no pun intended), but that “quantization” is a concept foreign to physical phenomena to which we are exposed in our daily life, seems really a bizarre notion.
Any freshman student who has the (mis)fortune to take introductory physics learns that the string of a guitar vibrates at frequencies that are multiple integers of a fundamental frequency, which is inversely proportional to the length L of the string. The discretization (sort of another word for quantization) of the vibration spectrum of a string, is a simple consequence of the fact that the string has a fixed, finite length. The frequencies of the allowed oscillations are determined by the condition that the end points remain fixed, i.e., that the oscillation vanish at those points.
If the length of the string is increased, the spacing between consecutive vibration frequencies (also known as harmonics) becomes smaller, i.e., harmonics become arranged on an increasingly finer grid. At some point, when L becomes very large, the spacing between two consecutive harmonics becomes smaller than experimental resolution, and quantization may be ignored altogether, both conceptually and practically.
The same, identical considerations explain the quantization of the energy spectrum of a quantum-mechanical particle confined to moving in one dimension between two perfectly reflecting walls. Wave-particle duality tells us that there is a connection between the particle’s linear momentum, and the wavelength of the (standing) wave associated to the particle. Here too, the wave must vanish at the two impenetrable walls, because particle cannot proceed any further once it hits either — much like a wave propagating on a guitar string must bounce back, once it arrives to either fixed end. Because the wavelength is proportional to the reciprocal of the frequency, and because the energy of the particle is proportional to the square of the linear momentum, the energy is quantized.
This conclusion is quite general, in that every time the motion of a particle is confined in space, every time there exist “walls” beyond which a particle cannot escape, its energy is quantized. It is, essentially, a boundary effect. The mathematics may be more or less involved, but conceptually there is nothing more than understanding the vibration of a string. Once one accepts (never mind understanding, no one does) wave-particle duality, energy quantization seems relatively simple a consequence of it.
Wave-particle duality, the probabilistic interpretation of a particle’s wave function, the spooky, intellectually disturbing idea of entanglement of particles — all of that is mysterious. I would not say that of energy quantization. I do not think that it is anywhere near as difficult to understand, profound, stunning or mind-boggling as wave-particle duality.
This is the only (mild) criticism that I have on “Quantum leap”. I think that Greene, after painstakingly introducing viewers to the notion of wave-particle duality, could easily have attempted to explain energy quantization along the above lines — obviously doing a much better job than me. I think a lot of viewers of his wonderful documentary would have understood and appreciated.
Instead, energy quantization is initially advertised as two things that it is not, i.e., a “mystery”, and one of the hallmarks of quantum mechanics, and then dropped from the discussion. I suspect that some viewers have thought, at the end of the show “Yeah but… why is energy quantized ? He did not really explain that… he just talked about waves… it must be still a mystery”.
 Or vice versa, if one wishes. In fact, the basic assumption of quantization in the Bohr model is that of the orbital angular momentum of the electron, whence that of both the radius of the orbit and of the energy ensue.
 Incidentally, if there is one thing that really is quantized, pretty much no matter in which context and at what scale one looks at it, is food.
 This is how, in his “Principles of Quantum Mechanics” masterpiece, he comments on the apparent impossibility of providing a satisfactory, intuitive picture of the most striking implications of the new theory:
“[...] the main object of physical science is not the provision of pictures, but is the formulation of laws governing phenomena and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance. In the case of atomic phenomena no picture can be expected to exist in the usual sense of the word ‘picture’, by which is meant a model functioning essentially on classical lines.”
Could not have said it better myself
 Quantum Mechanics offers no insight as to why electric charge is quantized. And even though it tells us what the fundamental quantum of angular momentum is, the statement that angular momentum should be quantized in the first place can be made based on considerations of symmetry of physical laws under arbitrary rotation of space, with the aid of an elegant mathematical formalism called group theory. See, for instance, M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley (1962).