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IEEE journal on multiscale and multiphysics computational techniques

Fourier Transform, Dirac Commutator, Energy Conservation, and Correspondence Principle for Electrical Engineers

Christopher Jayun Ryu, Erhan Kudeki, Dong-Yeop Na, Thomas E. Roth, Weng Cho Chew

Published Jan 1, 2022• 3 citations
Publisher

Abstract

The canonical commutation relation is a fundamental postulate of the quantum theory regarding the operators needed in quantum description of physical observables. It is shown that the Fourier transform is derivable from this seemingly simple postulate along with the basic properties of the position and momentum operators. Further discussions on the canonical commutation relation reveal its connection to a more fundamental notion that energy must be conserved. This discussion also unveils the mathematical homomorphism between the classical and quantum theories for systems represented by sum separable Hamiltonians. Another link between the classical and quantum theories is established by the correspondence principle which states that the classical theory emerges from quantum theory in the limit of vanishingly small Planck constant. Finally, the quantum Maxwell’s equations, which have been derived in our previous works, are presented and briefly discussed, and the 3-D mode transform is derived that can be interpreted as a generalization of the Fourier transform. We present both the details and meanings of the 3-D mode transform which will serve as a foundation for a full 3-D quantum finite-difference time-domain method.

DOI
10.1109/jmmct.2022.3148215

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•Jul 9, 2026•0 citations
The canonical commutation relation is a fundamental postulate of the quantum theory regarding the operators needed in quantum description of physical observables. It is shown that the Fourier transform is derivable from this seemingly simple postulate along with the basic properties of the position and momentum operators. Further discussions on the canonical commutation relation reveal its connection to a more fundamental notion that energy must be conserved. This discussion also unveils the mathematical homomorphism between the classical and quantum theories for systems represented by sum separable Hamiltonians. Another link between the classical and quantum theories is established by the correspondence principle which states that the classical theory emerges from quantum theory in the limit of vanishingly small Planck constant. Finally, the quantum Maxwell’s equations, which have been derived in our previous works, are presented and briefly discussed, and the 3-D mode transform is derived that can be interpreted as a generalization of the Fourier transform. We present both the details and meanings of the 3-D mode transform which will serve as a foundation for a full 3-D quantum finite-difference time-domain method.
Based on the hypothesis that the position-representation of a physical state <x|α\alphaα> is the Fourier transform of its momentum-representation <P|α\alphaα> and that the time-representation <t|α\alphaα> is the inverse Fourier transform of its energy-representation <E|α\alphaα>, we are able to find the Planck relation E = hv, the de Broglie relation P = hλ\frac{h}{\lambda}λh​. Afterward, utilizing the Dirac delta function δ\deltaδ(x) and the property FTxf(x) = -iℏ\hbarℏ∂\partial∂, pFTxf(x) = -iℏ\hbarℏ∂\partial∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^\hat{p}, \hat{x}p^​,x^] = -iℏ\hbarℏI^\hat{I}I^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.
From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.
Based on the Bohr's correspondence principle it is shown that relativistic mechanics and quantum mechanics may be considered as generalizations of classical mechanics. A comparative description of relativistic and classical mechanics is given using three main principles: the definition of momentum, the main law of dynamics (Newton's second law) and the law of conservation of energy. The difference of relativistic mechanics from classical mechanics is due to the new definition of momentum that is proportional to energy and velocity. It is shown that new relativistic laws of dynamics make it necessary to change kinematical relations of classical mechanics such as the law of velocity composition, coordinate transformation etc. It is demonstrated that quantum mechanics may be considered as a rational statistical generalization of classical mechanics. Statistical conformities in quantum mechanics are fundamental and are not due to the incomplete information about the system. From all possible multi-parametric statistical models the root model is most remarkable. Constructing of a root multi-parametric statistical model leads to obtaining such frequencies and base functions in Fourier decomposition so that classical laws of motion are fulfilled in average. Root model leads to a consistent condition that connects eigenvectors and eigenvalues of a mechanical system that is described by a matrix equation of Heisenberg. Matrix equation of Heisenberg leads to an operator equation. The solution of the operator equation may be considered as the construction of Hamiltonian of the system and a move to the Schrodinger representation. The considered approach naturally leads to the notion of momentum operator, fundamental commutation relations, density matrix and Liouville equation construction etc.
Some connections between quantum mechanics and classical physics are explored. The Planck–Einstein and De Broglie relations, the wavefunction and its probabilistic interpretation, the canonical commutation relations and the Maxwell–Lorentz equation may be understood in a simple way by comparing classical electromagnetism and the photonic description of light provided by classical relativistic kinematics. The method used may be described as 'inverse correspondence' since quantum phenomena become apparent on considering the low photon number density limit of classical electromagnetism. Generalization to massive particles leads to the Klein–Gordon and Schrödinger equations. The difference between the quantum wavefunction of the photon and a classical electromagnetic wave is discussed in some detail.
Reconstructing the Canonical Commutation Relation from a Correlation Matrix: From Three Technical Assumptions to [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ
Abstract The canonical commutation relation [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ is a fundamental postulate of quantum mechanics. This paper presents a rigorous reconstruction of this relation from an alternative algebraic starting point: a correlation matrix defined on an abstract index set, supplemented by three technical assumptions regarding its spectral inverse and an emergent distance. The system relaxes to a translation-invariant state, where the correlation matrix acquires a Toeplitz structure. The equidistance constraint enforced by uniform correlations selects a uniform lattice in the continuum limit via Γ\GammaΓ-convergence. On this lattice, the Toeplitz matrix is diagonalized by the discrete Fourier transform, yielding a plane-wave basis. Translation symmetry yields the momentum operator; Fourier duality yields the position operator. The commutator [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ is recovered as an algebraic corollary of the Toeplitz geometry and Fourier duality. The existence of a minimal nonzero action scale ℏ\hbarℏ arises from the topology of the compactification core (S3S^3S3, Hopf invariant c1=1c_1=1c1​=1); its precise numerical value is not required for this derivation. The Hilbert space structure itself—the inner product, completeness of the basis, and the dual representation of observables—emerges naturally from the same Toeplitz–Fourier geometry. This paper does not claim a derivation from deeper principles—rather, it demonstrates that, given the three stated assumptions, the CCR can be rigorously recovered without postulating it as an independent axiom of quantum mechanics. ORCID: 0009-0007-1999-0293 Email: 1352533302@qq.com