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.
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 hypothesis that the position-representation of a physical state <x|
α> is the Fourier transform of its momentum-representation <P|
α> and that the time-representation <t|
α> is the inverse Fourier transform of its energy-representation <E|
α>, we are able to find the Planck relation E = hv, the de Broglie relation P =
λh. Afterward, utilizing the Dirac delta function
δ(x) and the property FTxf(x) = -i
ℏ∂, pFTxf(x) = -i
ℏ∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [
p^,x^] = -i
ℏI^, 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.
Eine philosophisch und mathematisch fundierte Darstellung der Fouriertransformation im Kontext der Quantenmechanik. Der Text verbindet historische Reflexion (Parmenides, Demokrit, Fourier, Dirac) mit einer präzisen Herleitung von Fourierreihe, Fouriertransformation und diskreter Fouriertransformation und zeigt, wie Orts‑ und Impulsdarstellung der Wellenfunktion durch Fouriertransformation ineinander übergehen. Besonderer Schwerpunkt liegt auf der ontologischen Bedeutung des Kontinuums und der Spektralanalyse für die Struktur der Quantenphysik. This work presents a concise and conceptually unified introduction to Fourier analysis in the context of quantum mechanics. It traces the historical development from classical harmonic decomposition to the modern Fourier transform and shows how periodic functions, spectral components, and exponential modes form the mathematical foundation of quantum theory. The transition from Fourier series to the continuous Fourier transform is derived rigorously, leading to the representation of quantum states in position and momentum space. Using Dirac notation, the text demonstrates that the wavefunction in momentum space is the Fourier transform of the position‑space wavefunction, revealing the spectral structure of quantum motion. The discrete Fourier transform (DFT) is introduced as a numerical analogue relevant for lattice models and quantum field theory. Overall, the text connects mathematical formalism, physical interpretation, and the ontology of the continuum in a compact and accessible way.
Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.