The Classical Spectral Analysis Origins of Quantum Mechanics' Two Postulates

Commutation Relations and Hilbert Spaces—Not Water Without a Source

In previous discussions, we pointed out that quantum mechanics adds only two elements atop the complete dynamical skeleton of classical mechanics: the commutation relation [q^,p^]=iℏ[q^,p^]=iℏ and the Hilbert space structure. Textbooks typically present these two postulates as entirely new starting points for quantum mechanics, representing a rupture with classical physics. However, if we expand our视野 from classical mechanics to classical mathematical physics—specifically spectral analysis methods—we discover that every element of these two postulates has deep classical roots.

I. The Classical Precursor to Commutation Relations: Conjugate Uncertainty in Fourier Analysis

The concrete realization of the commutation relation [q^,p^]=iℏ[q^,p^]=iℏ in the coordinate representation is:

p^=−iℏ∂∂qp^=−iℏ∂q∂

This means the momentum operator is simply the differential operator with respect to position (multiplied by a constant), while the position operator q^q^ is the multiplication operator. Their commutation relation:

[q,−iℏ∂∂q]f(q)=iℏf(q)[q,−iℏ∂q∂]f(q)=iℏf(q)

is not an invention of quantum mechanics. In classical Fourier analysis, time tt and frequency ωω are a pair of conjugate variables. The Fourier transform maps a function from the time domain to the frequency domain:

f~(ω)=∫−∞∞f(t)e−iωtdtf~(ω)=∫−∞∞f(t)e−iωtdt

In this framework, the relationship between the multiplication operator t^t^ in the time domain and the differential operator −id/dω−id/dω in the frequency domain is mathematically isomorphic to the relationship between q^q^ and p^p^ . The time-frequency uncertainty principle, well-known in classical signal processing:

Δt⋅Δω≥12Δt⋅Δω≥21

is a pure mathematical theorem (sometimes called the Gabor limit, explicitly formulated in 1946, but its mathematical content traces back much earlier). It involves no ℏℏ , no quantum mechanics, no particles, and no wavefunctions. It is simply an intrinsic property of the Fourier transform: a function cannot be arbitrarily localized simultaneously in both the original space and the dual space.

Heisenberg's uncertainty relation in quantum mechanics:

Δq⋅Δp≥ℏ2Δq⋅Δp≥2ℏ

is merely the same mathematical theorem after substituting ω→p/ℏω→p/ℏ . The appearance of ℏℏ simply determines the metric conversion factor between coordinate space and momentum space; the mathematical structure of the uncertainty relation was already complete within Fourier analysis.

In other words, the deep content of the commutation relation [q^,p^]=iℏ[q^,p^]=iℏ is this: Position and momentum are a pair of Fourier conjugate variables. This is not a discovery of the quantum revolution, but a fact established long ago in the works of Fourier (1807/1822), Plancherel (1910), Wiener, and others in classical mathematical physics. Quantum mechanics merely assigned a physical constant ℏℏ to this mathematical structure as a scaling factor.

II. Hilbert Space and Spectral Decomposition: A Legacy of Classical Mathematical Physics

Sturm-Liouville Theory (1836–1837)

The core mathematical operation in quantum mechanics is: given a Hermitian operator H^H^ , find its eigenvalues and eigenfunctions:

H^ψn=EnψnH^ψn=Enψn

and then expand any state as a superposition of these eigenfunctions:

ψ=∑ncnψn,∣cn∣2=Probabilityψ=n∑cnψn,∣cn∣2=Probability

This structure—eigenvalue problems, orthogonal complete bases, and function expansion—is precisely the content of Sturm-Liouville theory. Between 1836 and 1837, Sturm and Liouville studied boundary value problems in classical mathematical physics:

ddx[p(x)dydx]+[λw(x)−r(x)]y=0dxd[p(x)dxdy]+[λw(x)−r(x)]y=0

They proved that under appropriate boundary conditions, the eigenvalues λnλn of such equations form a discrete spectrum, the eigenfunctions yn(x)yn(x) form an orthogonal complete system, and any function (satisfying certain regularity conditions) can be expanded as a series of these eigenfunctions.

The time-independent Schrödinger equation in quantum mechanics:

−ℏ22md2ψdx2+V(x)ψ=Eψ−2mℏ2dx2d2ψ+V(x)ψ=Eψ

is exactly a Sturm-Liouville problem. The discretization of energy, the orthonormality of wavefunctions, and the superposition principle—these so-called "fundamental features of quantum mechanics"—were mathematically fully established nearly ninety years before the birth of quantum mechanics.

Hilbert's Spectral Theory (1904–1910)

Between 1904 and 1910, David Hilbert developed the spectral theory of operators in infinite-dimensional function spaces. He introduced the concept of what is now named after him: the Hilbert space—a complete linear space equipped with an inner product structure. He proved that compact operators can be diagonalized into a discrete set of eigenvalues and established the correspondence between integral equations and infinite matrices.

The mathematical framework of quantum mechanics—state vectors, inner products, Hermitian operators, spectral decomposition, and probability amplitudes—is a direct application of Hilbert's spectral theory, not an independent creation. Von Neumann explicitly acknowledged this in his 1932 work Mathematical Foundations of Quantum Mechanics, reformulating quantum mechanics rigorously as an operator algebra theory within Hilbert space. What he performed was a translation: translating the physicists' intuitive formulations into an existing mathematical language. The mathematical language itself was already in place.

Fourier Series and Analysis (1807–1822)

The earliest and most intuitive example is the Fourier series. Any periodic function can be expanded as a superposition of trigonometric functions (sines and cosines):

f(x)=a02+∑n=1∞(ancos⁡nπxL+bnsin⁡nπxL)f(x)=2a0+n=1∑∞(ancosLnπx+bnsinLnπx)

Sine and cosine functions are precisely the eigenfunctions of the Laplacian operator −d2/dx2−d2/dx2 on the interval [0,L][0,L] , corresponding to eigenvalues n2π2/L2n2π2/L2 . The expansion coefficients an,bnan,bn are completely analogous to the probability amplitudes cncn in quantum mechanics. The "discrete spectrum" arises from the constraints of boundary conditions—a fact already perfectly clear in classical string vibration theory.

A string fixed at both ends can only vibrate at specific frequencies— νn=nν1νn=nν1 . This is the oldest and most intuitive classical example of "quantization" (discretization).

III. The Classical Soil of Probability Interpretation

Even the probability interpretation ∣cn∣2=Pn∣cn∣2=Pn did not appear out of thin air. In classical spectral analysis, Parseval's Theorem (Parseval, 1799) states:

∫∣f(x)∣2dx=∑n∣cn∣2∫∣f(x)∣2dx=n∑∣cn∣2

The left side represents the total "energy" (or total "power") of the signal, while the right side is the sum of the energy contributions of each frequency component. ∣cn∣2∣cn∣2 represents the energy proportion of the signal in the nn -th mode. In classical physics, this is an allocation of energy; in quantum mechanics, it is reinterpreted as an allocation of probability.

Born's probability interpretation (1926) certainly possesses profound physical significance—it reinterprets ∣cn∣2∣cn∣2 from "energy proportion" to "the probability of measuring EnEn ." However, the very possibility of this reinterpretation exists precisely because the mathematical structure (Parseval's identity, orthogonal decomposition, modulus-squared normalization) was already fully ready in classical spectral analysis. Born did not need to invent new mathematics; he only needed to assign new physical meaning to existing mathematics.

IV. The Panorama: Mapping Classical Spectral Analysis to Quantum Mechanics

表格

Classical Spectral Analysis Concept	Era	Corresponding Concept in Quantum Mechanics	Era
Fourier Series Expansion	1807–1822	Expansion of states in eigenfunctions	1926
Sturm-Liouville Eigenvalue Problem	1836–1837	Time-independent Schrödinger Equation	1926
Parseval's Theorem ( ∑∣cn∣2=∣f∣2∑∣cn∣2=∣f∣2 )	1799	Probability Conservation / Normalization	1926
Uncertainty of Fourier Conjugate Variables	Known in 19th Century	Heisenberg Uncertainty Relation	1927
Hilbert Space & Operator Spectral Theory	1904–1910	Mathematical Foundations of QM (von Neumann)	1932
Discrete Spectrum of String Vibrations	18th Century	Energy Quantization	1900–1926
Superposition of Normal Modes	18th–19th Century	Quantum Superposition Principle	1926

This mapping table clearly demonstrates that the mathematical framework of quantum mechanics is almost a item-by-item translation of classical spectral analysis. Every "fundamental postulate" of quantum mechanics has a classical mathematical precursor that predates it by at least half a century, if not a full century.

V. What Does This Mean?

If the mathematical structure of commutation relations stems from the theory of Fourier conjugate variables, the Hilbert space structure from Sturm-Liouville and Hilbert's spectral analysis, and the probability interpretation from Parseval's energy allocation, then the two "postulates" of quantum mechanics are not a revolutionary rupture in physics, but a natural extension of classical mathematical physics.

The truly physical content that needs explanation is compressed into an extremely small core: Why did nature choose a specific Fourier conjugate scaling factor where ℏ≠0ℏ=0 ? Or equivalently: Why does the behavior of physical systems resemble a spectral analysis process involving coherent superposition over all paths, rather than the classical process of selecting only the extremal path?

If particles are field structures with finite spatial scales, then the intrinsic vibration of the field itself is a Fourier decomposition, and the spatial extension of the field naturally leads to a "full coverage" of paths rather than the selection of a single trajectory. In this scenario, ℏℏ may not be an unaskable fundamental constant, but rather a characteristic quantity of the particle's field structure—just as the tension and linear density of a string determine its fundamental frequency. Spectral analysis methods are not merely an analogy for quantum mechanics; they may be its ontology.

Ultimately, the two postulates of quantum mechanics are not water without a source. Their origins are clearly visible in the tradition of classical mathematical physics' spectral analysis. Acknowledging this does not diminish the achievements of quantum mechanics; rather, it more accurately positions quantum mechanics within physics—not as a break from the classical world, but as a deepening of the classical world's spectral analysis.