Alright, so you have made up your mind to learn about quantum mechanics. Maybe you have already skimmed through one of the textbooks, learned about the existence of Schrödinger equation or even determined the energy spectrum of your very first quantum system. But if you still ask yourself: What exactly means that a system is being “quantum”? And what is this thing about “quantization”? Then this article is for you!

Actually, quantization might refer to different yet related concepts, depending on whom you ask and which level of understanding you assume. So we’ll start with the most tangible one, the notion of physical quanta as particles. Next, we continue on a little more abstract level with the quantization of physical quantities such as energy and finally we’ll have a glimpse at the process of canonical quantization. Let’s dive in!

#### a) The Notion of Particles

Quantum physics is the science of atomic and sub-atomic particles at about the length scale of one millionth of a hair’s diameter. While some of these particles are formed by smaller, more fundamental particles, like a proton is made up of quarks, some of these particles are – to the best of our current knowledge – non-divisible and thus constitute the quanta of matter: the smallest, indivisible building blocks of our universe, such as quarks, electrons or photons.

On this level, quantization means that matter and light only come in certain portions and that larger portions are always a multiple of these single portions. Think about a laser beam having a number of say \(n\) photons. As we know from Einstein’s explanation of the photoelectric effect, a single photon has an energy \(E\) proportional to its frequency \(f\) by \(E = h \cdot f\). And the entire beam will have an energy of $$ E_{\mathrm{tot}} = n \cdot h \cdot f $$

So the energy of this laser beam is quantized, as only multiples of a single photons’ energy will occur, never one half of it and neither twenty-three thirds. It basically the same with the quantization of paper money, there is a bank note for one dollar, five dollars or ten dollars, but there are no three dollar bank notes.

#### b) Quanta of Physical Variables

We are now going one step further and assume you have heard about the existence of Schrödinger equation: $$\hat{H} \, \psi(x) = E \cdot \psi(x)$$

In the previous section, the energy of that laser beam was quantized, because the beam itself consisted of a discrete number of particles. Yet the energy of the single photon could take any value, only depending on it’s frequency, and that was the case because these photons were free particle, i.e. they were free to float around in space going anywhere.

But what if we consider now a particle of mass \(m\) being subject to a potential \(V(\vec{r})\), e.g. placed inside a finite potential well? If the particle’s energy is greater than the potential well’s depth \(V_0\), we get scattering states and the particle’s energy still can assume any value: we get a continuous energy spectrum.

The whole story changes, if our particle has less energy and is bound by the potential. In order to figure out, what is the particle’s energy you would a) solve Schrödinger’s equation: $$ \hat{H} \, \psi = \left( \frac{-\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V \right) \psi(x) = E \cdot \psi(x) $$

and b) take into account the boundary conditions the particle’s wave function needs to fulfill when hitting the well’s boundary. But the point is, that these boundary conditions can only be fulfilled for certain, discrete energies.

The example gets more memorable once you consider an infinite potential well, where the particle’s wave function has to assume exactly zero value at the well’s boundary and beyond. (That well is infinitely high, so the particle can’t be at the well’s boundary or even go beyond – quantum tunneling is only possible for potential barriers of finite extent).

The boundary conditions for the wave function is analogous the string of a guitar being fixed at both ends and thus having standing waves as its eigenmodes. Only those modes who leave both ends fixed comply with the boundary conditions and thus occur as eigenmodes or particle wave function. And therefore only the energy of those modes will pop up in the energy spectrum, forming a set of discrete energy values.

A similar story can be told about the energy levels of a hydrogen atom: in the simplest, semi-classical picture an ‘electron wave’ given the De Broglie wavelength \(\lambda = \frac{h}{p} \) is orbiting the core proton. As the electron wave is going around, it eventually reaches the starting point of its motion, beginning to interfere with itself. Thus, only those electron waves survive, which form standing waves, i.e. whose orbits covers a multiple of the De Broglie wavelength. And only the energy of those electron waves will show up as discrete energy levels in the energy spectrum.

So it’s all about boundary conditions. If a particles motion is bound by a potential, it will feature discrete energy levels. Besides that also other quantities such as spin, angular momentum or parity can be quantized and it is characteristic for quantum mechanics that physical quantities assume under some circumstance only discrete values or be a multiple of some smallest, indivisible quantum portion.

#### c) Canonical Quantization

Finally we are going one more step further and assume you have heard about quantum mechanical operators, which we will label by a hat, like \(\hat{p}\) for the momentum operator. In quantum mechanics, physical observable such as position, momentum or energy are described by Hermitian operators. Their eigenvalues constitute possible measurement results of the respective observable and the absolute value squared of their eigenstates tells us about the likelihood that the corresponding eigenvalue is measured.

On an abstract level, quantization now refers to the process of canonical quantization, that is making the transition from classical physics to quantum physics. Or more specifically constructing a quantum (field) theory out of a classical theory by replacing classical variables like position \(x\) or momentum \(p\) by quantum mechanical operators \(\hat{x}\) or \(\hat{p}\), while keeping the formal structure of the theory. So what does ‘keeping the formal structure’ mean?

In classical physics, a system is governed by a Hamiltonian \(H(q,p)\), which depends on the position \(q\) and the canonical momentum \(p\). The relation between these two quantities is manifested by the so-called Poisson bracket $$ \left\{ A, B \right\} = \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} – \frac{\partial A}{\partial p} \frac{\partial q}{\partial q}$$

capturing the canonical (also called symplectic) structure of the theory by \(\{q, p\} = 1\). ‘Preserving the structure’ now means, that in analogy to the Poisson bracket one introduces the so-called commutator for quantum operators \(\hat{A}, \hat{B}\) $$ \left[ \hat{A}, \hat{B} \right] = \hat{A} \hat{B} – \hat{B} \hat{A}$$

which checks if you are allowed to interchange these two operators. If so, i.e. \(\hat{A} \hat{B} = \hat{B} \hat{A}\), the commutator assumes a value of zero. This has major implications for the relation between the two operators \(\hat{A}, \hat{B}\) representing physical observables, for example that measuring the value of \(\hat{A}\) does not influence the measurement of the value of \(\hat{B}\).

Vice versa, a non-zero commutator implies, that these two operators may not be interchanged, which is the case for example for the position-momentum-commutator $$\left[ \hat{x}, \hat{p} \right] = i \hbar$$

which tells us, that position and momentum of particle cannot be measured exactly at the same time (the measurement of either influences the measurement of the other) as expressed by Heisenberg’s uncertainty relation.

To cut a long story short: Canonical quantization is the process of going from classical physics to quantum physics by replacing classical variables (i.e. numbers) by operators (i.e. linear mappings) and Poisson brackets by commutators

$$x \to \hat{x}, \quad p \to \hat{p}$$

$$\{x,p\} = 1 \to \frac{1}{i \hbar}[\hat{x},\hat{p}] = 1$$

which reproduces the familiar canonical structure of classical mechanics and allows for the description of all those quantum mechanics effects. However, this mapping is not unique, in a sense that not all combinations of \(x\) and \(p\) can be mapped exactly to their quantum analogs (you get problems for polynomials of degree four and higher) and also the way this mapping is performed in detail can be chosen according to different ‘quantization schemes’. While quantization usually refers to canonical quantization, there are also alternative approaches such as path integral quantization and more exotic ones.

**TL;DR **Quantization might refer either to something coming in fundamental, indivisible portions, like a particle or a discrete energy spectrum, or might refer to the process of going from classical physics to quantum physics called canonical quantization.

#### Further Reading

Jerry Schirmer, Energy is quantized, URL (version: 2012-09-27): https://physics.stackexchange.com/q/38438

Physics 582 General Field Theory, Fall Semester 2019, Eduardo Fradkin

http://eduardo.physics.illinois.edu/phys582/582-chapter4.pdf