1. The Complete Expression of Quantum

In the previous article, we emphasized that quantum is primarily not a concept of "minimum unit," but rather a "set of states" that can be completely expressed through mathematical structures. However, this expression is still incomplete. Since quantum is a wave, it cannot merely be a collection of static possibilities—it must naturally also contain information about how it evolves over time. In other words, a truly complete quantum description must include not only the set of probability amplitudes corresponding to each measurement result, but also the phase information for how each probability amplitude changes over time.

Matter waves are also a fundamental assumption of quantum mechanics (de Broglie, 1924), with wavelength and momentum satisfying λ=h/p. However, the previous definition of quantum did not explicitly reflect wave nature. Wave nature is precisely realized through phase evolution.

Mathematically, this description is typically expressed using the following summation expression with phases:

|ψ(t)⟩ = Σₙ cₙ e^(-iEₙt/ℏ) |n⟩

Here, |n⟩ represents a set of eigenstates of the system, cₙ is the initial probability amplitude of each eigenstate (typically complex), Eₙ is the corresponding energy eigenvalue, t is time, and e^(-iEₙt/ℏ) is the phase factor evolving with time. The entire quantum state is not a superposition of a single eigenstate, but rather the sum of all eigenstates with their complex coefficients and respective phases "co-evolving."

This "superposition with phase" is the key to quantum physics. Each probability amplitude doesn't merely mean "the possibility of this outcome"—its phase relationship determines how the quantum state produces interference and how it unfolds over time.

• When the phases of various components are synchronized, they can construct enhancement at specific positions (constructive interference)

• When phases are opposite, they can completely cancel out (destructive interference)

What we typically call "quantum individuals"—whether electrons, photons, or qubits—all their properties are rooted in this probability amplitude superposition and phase evolution. This is why quantum can exhibit phenomena beyond classical intuition such as interference, entanglement, and tunneling: it's not about "what is," but rather the result of "what could be" and "how they evolve together" acting in concert.

A quantum description without phase is incomplete. In all experiments and theoretical calculations, considering the temporal and spatial phases of probability amplitudes is the prerequisite and foundation for understanding all quantum behaviors (including interference, entanglement, quantum information processing, etc.).

2. Why Quantize?

Quantization means transforming the objects studied in the physical world from classical particles or classical fields into mathematically tractable collections of "quantum states." Of course, this is not merely changing the mathematical description, but rather acknowledging that physical objects themselves are fundamentally quantum—these abstract structures containing probability amplitudes, phases, superposition, interference, and other peculiar properties. Given this, we must develop and adopt an entirely new mathematical language and system of formulas to handle the evolution, interaction, and measurement results of quantum entities.

Classical physical theory can only handle physical realities such as point masses, fluids, and classical fields. Their mathematical structures are limited to intuitive categories like finite-dimensional phase space, traceable trajectories, and causality. The "quantum entity" of quantum mechanics, however, differs from classical entities—it exists in probability superpositions, phase interference, and indivisible collections of probability amplitudes. Only by establishing and utilizing the mathematical formula system of quantum (such as operator algebra, wave functions, Hilbert spaces, etc.) can we correctly describe and calculate the behavior of such physical objects.

Quantization means using an entirely new mathematical system to describe physical objects, breaking free from the limitations of classical physics' mathematical methods.

All "objects" in physical theory—whether an electron, a light field, or a solid lattice—must abandon the intuition of deterministic trajectories or continuous values, and instead be described using quantum tools like wave functions, operators, or abstract vectors.

Specifically, quantization means elevating the degrees of freedom of classical systems to symbolic "operators," or viewing them as wave-like states—that is, "objects" no longer have just definite positions or momenta, but are quantum states determined jointly by "probability amplitudes" and "phase evolution."

This concept is not just a change in mathematical tools, but expresses a completely new description of the "essence of physical objects": objects themselves must also be "quantized"—classical pictures no longer apply.

In operational details, quantization includes replacing canonical variables (q, p become operators q̂, p̂), introducing the uncertainty principle, allowing state superposition and coherence, and endowing the system with entirely new statistical laws (such as Fermi-Bose distributions). Essentially, any "entity" in classical theory—from single particles to continuous fields, from atoms to molecules, from physical quantities to interactions—must be elevated after quantization to "state vectors in wave function space" or "elements in operator algebra."

What quantization requires is not merely a change in mathematical treatment methods, but a fundamental transformation in the ontological understanding of physical objects. Classical "particles" or "fields" as entities give way to collections of "quantum states" at the microscopic level. After quantization, physical objects themselves lose the reality and determinacy of the classical world, no longer being particles with precisely trackable trajectories, but becoming "waves," "operators," or "state vectors" in state space. This means that "objects" that were fixed and concretely described in classical physics become collections of superposition, interference, entanglement, and indivisible probability amplitudes under quantum theory. After undergoing quantization, physical objects themselves, along with all their properties and evolution laws, must be endowed with quantum attributes. This thorough "quantization of physical objects" is the most fundamental method of quantum mechanics, requiring us to "first quantize the object, then describe the object"—in other words, first acknowledge that the world is "quantum," then understand it through the mathematical tools of quantum mechanics.

3. The Essence of Quantization: Wave-ification, Functionalization, and Functionalization

The essence of quantization is not merely giving classical systems a different mathematical expression, but a fundamental reconception of the structure of the physical world. The first step of quantization is "wave-ification": elevating everything we once viewed as "solid particles" with "definite trajectories" to wave existence. This "wave" is not a traditional water wave or sound wave, but a matter wave in the broad sense—its mathematical essence is a function or mathematical object that evolves according to definite laws in space and time.

Formally, quantization is functionalization. Classical physics describes objects of motion with a finite number of variables (such as a particle's position x and momentum p), with these variables having definite values at each moment. In quantum theory, each particle's state must be described by a function (wave function ψ(x)), indicating the "probability amplitude of appearing" at all possible positions. Once field theory is involved, the state of the entire physical system becomes a functional—that is, a "function defined on function space," such as Ψ[φ(r)], describing all possible distribution forms of the entire field.

Functionalization is the further generalization of the quantization process in many-body systems and quantum field theory. It means not only must individual particles be described by waves, but every degree of freedom of the field must "wave" together, with the quantum state of the entire system becoming an indivisible whole, uniformly expressed using functionals of functions.

Some intuitive examples:

• In atomic physics, electron orbitals are no longer circular trajectories, but "probability clouds" described by wave functions

• In solid-state physics, phonon excitations in a lattice are superpositions of the entire lattice's motion modes, with each mode being a quantum state in functional space

• In quantum field theory, each "particle" is an excitation of the field, each excitation corresponding to "the result of an operator's action"—this is the essential manifestation of functional quantization

Quantization means representing all classical physical objects using waves, functions, and functionals, making physical systems no longer single rigid entities composed of point masses, but distributions of probability amplitudes and phases throughout space. This new foundational framework is completely different from humanity's long-held traditional understanding of "how objects exist" and "causality."