The Goal of Quantization

The development of early quantum theory (1900-1925) was filled with ad-hoc solutions to specific problems. Planck's energy quantization applied only to blackbody radiation; the Bohr model (1913), while successfully explaining hydrogen atomic spectra, had orbital quantization rules that couldn't be generalized to complex atoms or molecules. These models were all "custom-made," lacking the ability to handle general problems.

Thus quantum mechanics needed a method or mathematical framework for describing and solving general physical problems. As discussed above, quantization means transforming physical objects—whether particles, fields, or complex systems—into time-evolving waves, functions, or functionals. Only by completing this transformation can we establish systematic methods for solving general physical problems.

The Schrödinger Equation

The wave equation proposed by Schrödinger (1926) was the first capable of handling general quantum systems:

$i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$

The revolutionary aspects of this equation:

• Applicable to particles in arbitrary potential fields

• Can be generalized to many-particle systems

• Naturally yields observables like energy levels and transition probabilities

However, Schrödinger never clearly explained how he obtained this equation. His original paper mentioned:

• Inspiration from de Broglie's matter waves

• Analogy with classical optics

• Verification against known hydrogen energy levels

• Some kind of "mathematical intuition"

This ambiguity in derivation troubled contemporary physicists. Schrödinger had "guessed" the correct equation rather than deriving it from fundamental principles.

The Canonical Quantization Method

Just months after Schrödinger's equation appeared, Dirac (September 1926) proposed a systematic quantization method in "The General Theory of Quantum Mechanics," first using the term "canonical quantization."

"Canonical": The term originates from the Latin "canon," which in Western tradition and everyday usage means "standard" or "authoritative." Dirac used this word to emphasize the standard transformation from classical to quantum. However, "正则" (zhèngzé) in Chinese is a neologism introduced through physics translation, not existing in everyday language, making it sound arcane. English readers can associate "canonical" with "standard, regular," while the Chinese "正则" is somewhat obscure—translating it directly as "standard" would also work.

Dirac noticed that Heisenberg's matrix mechanics and Schrödinger's wave mechanics, though formally different, implied the same mathematical structure. The key was finding the bridge connecting classical and quantum. Dirac discovered:

• Heisenberg's matrix commutator: $[A,B]=AB-BA$

• Classical mechanics' Poisson bracket: $\{A,B{\}}_{P.B.}$

• These are related through: $[A,B]=i\hslash \{A,B{\}}_{P.B.}$

The canonical quantization procedure:

1. Identify the classical system's canonical variables $(q,p)$

2. Promote them to operators satisfying $[q,p]=i\hslash$

3. Any classical quantity $f(q,p)$ corresponds to quantum operator $\hat{f}(\hat{q},\hat{p})$

4. Classical Poisson brackets correspond to quantum commutators

Canonical quantization transformed quantum theory from "inspired discovery" to "systematic method."

The Canonical Form of Classical Mechanics

Classical mechanics' canonical form didn't exist from the beginning. Newtonian mechanics (1687) directly dealt with forces and accelerations; Lagrange (1788) introduced generalized coordinates and the principle of least action; Hamilton (1834) finally established the canonical form. This evolution took nearly 150 years.

Hamilton's canonical form describes systems using generalized coordinates $q_i$ and conjugate momenta $p_i$, introducing the Hamiltonian $H(q,p)$ as total energy, with dynamics determined by Hamilton's equations:

${\dot{q}}_i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q_i}$

Physical quantities relate through Poisson brackets:

$\{f,g{\}}_{P.B.}={\sum }_i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$

Impact on understanding physical reality:

• Newtonian mechanics: Physical reality is concrete forces and trajectories

• Lagrangian mechanics: Physical reality is energy and action

• Hamiltonian mechanics: Physical reality is evolution flow in phase space

The canonical form revealed classical mechanics' abstract structure—no longer focusing on specific force forms, but on the system's geometric evolution in abstract phase space. This abstraction paved the way for quantum mechanics: when phase space points become state vectors in Hilbert space and Poisson brackets become commutators, the classical canonical form naturally transitions to quantum theory.

The emergence of canonical form shows that even in classical physics, the meaning of "reality" was constantly evolving—from intuitive mechanical pictures to abstract mathematical structures.

Hamilton first called $(q_i,p_i)$ "canonical variables" (or canonical coordinates), emphasizing they are "the most natural, most symmetric variable pairs" for describing system states. This terminology came from mathematical studies of "canonical transformations"—transformations preserving phase space's geometric structure (like symplectic forms), and Hamilton's equations were derived from Lagrange's equations precisely through canonical transformation. This is the source of Dirac's use of "canonical."

The New Mathematical Framework

After canonical quantization, we obtained:

Wave functions (state functions): $\psi (x,t)$ or state vectors $|\psi (t)⟩$, describing the system's state evolution over time. These functions contain all information about the system.

Operator system:

• Position operator: $\hat{x}=x$ (in coordinate representation)

• Momentum operator: $\hat{p}=-i\hslash \frac{\partial }{\partial x}$

• Hamiltonian operator: $\hat{H}=\frac{{\hat{p}}^2}{2m}+V(\hat{x})$

Commutation relations: $[\hat{x},\hat{p}]=i\hslash$, determining quantum measurement uncertainty and operator algebraic structure.

Dynamical equations:

• Schrödinger equation: $i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$

• Heisenberg equation: $\frac{d\hat{A}}{dt}=\frac{i}{\hslash }[\hat{H},\hat{A}]$

This mathematical structure's problem-solving capabilities:

• Energy levels: Solving the time-independent Schrödinger equation $\hat{H}{\psi }_n=E_n{\psi }_n$ yields system energy levels and eigenstates

• Time evolution: Given initial state $\psi (0)$, find any time state through evolution operator $U(t)=e^{-i\hat{H}t/\hslash }$

• Transition probabilities: Calculate transition matrix elements $⟨f|\hat{O}|i⟩$ between different states, predicting experimental observations

• Expectation value evolution: Physical quantity expectation values $⟨\hat{A}⟩=⟨\psi |\hat{A}|\psi ⟩$ change over time

Specific applications:

• Hydrogen atom: Exact solution of energy levels, explaining spectral lines

• Harmonic oscillator: Describing molecular vibrations, lattice vibrations

• Tunneling effect: Calculating barrier penetration probability

• Scattering problems: Predicting particle collision cross-sections

Canonical quantization transformed "unsolvable" quantum problems into "computable" mathematical problems, moving microscopic physics from qualitative description to quantitative science.

The Deep Meaning of Canonical Quantization: Continuity with Classical Physics

The success of canonical quantization reveals a profound fact: though the microscopic world exhibits quantum properties, it's still governed by classical forces and interactions at bottom. Canonical quantization relies on the classical Hamiltonian system—including classical concepts like potential energy, kinetic energy, and interactions. This shows the quantum world isn't a completely alien realm, but rather the manifestation of classical physical concepts and laws in the microscopic domain.

Equivalent Lagrangian formulation: Quantum mechanics can equally be expressed in Lagrangian form. Feynman's path integral is precisely based on classical action:

$\int L(q,\dot{q})dt$

Quantum evolution is obtained by summing the phase $e^{iS/\hslash }$ over all possible paths. This proves the principle of least action remains valid in the quantum realm, just changing from "selecting a unique path" to "weighted sum over all paths."

Globally correlated physical system: The validity of the least action principle and variational methods shows quantum systems are globally correlated:

• Classical particle: Moves along the path minimizing action

• Quantum particle: Simultaneously "senses" all possible paths, selecting probability distribution through phase interference

This global correlation explains quantum non-locality: particle behavior isn't determined solely by local forces, but by the total contribution of all possible evolutions throughout spacetime. In the double-slit experiment, the electron "knows" both slits exist—this is precisely a manifestation of this global correlation.

The success of canonical quantization isn't accidental. It shows quantum mechanics doesn't overthrow classical physics but generalizes classical local, deterministic descriptions to global, probabilistic ones. Classical concepts like force, energy, and action remain fundamental to understanding nature, just requiring consideration of coherent superposition of all possibilities at microscopic scales.

The Price of Abstraction: Information Completeness Issues in Canonical Quantization

The necessity from abstract to concrete: While the canonical form is elegant, canonical coordinates $(q,p)$ must correspond to specific physical quantities to have meaning. For example:

• Particle motion: $q$ is position, $p$ is linear momentum

• Rotating systems: $q$ is angle, $p$ is angular momentum

• Field theory: $q$ is field configuration, $p$ is field conjugate momentum

Fundamental problems brought by abstraction: "Abstraction" means detachment from the ontological essence. When we simplify complex physical systems to $(q,p)$, we inevitably lose certain information, such as:

• Internal structure information: Treating composite particles as point particles ignores internal degrees of freedom

• Environmental interactions: The isolated system assumption ignores entanglement with environment

• Constraint conditions: Certain physical constraints may not be obvious in canonical form

• Topological properties: Global structure of phase space may be lost in local coordinates

Specific examples:

• Spin: Cannot be described by classical canonical variables, must be additionally introduced

• Identical particles: Canonical quantization doesn't automatically give Fermi/Bose statistics

• Gauge fields: Non-physical degrees of freedom exist, requiring additional constraints

Criteria for information completeness: Whether an abstract model preserves all necessary information needs verification through:

• Completeness of experimental predictions

• Absence of non-physical results

• Correct representation of symmetries

• Reasonable limiting cases

Therefore, successful application of canonical quantization requires caution. Each abstraction may lose information, and when extrapolating from abstract models we must constantly be vigilant: does our mathematical framework truly capture the essence of the physical system? This critical thinking is key to avoiding theoretical misdirection.