| Fourier transform | |
|---|---|
| Name | Fourier transform |
| Field | Mathematical physics |
| Introduced | 19th century |
| Notable figure | Joseph Fourier; Paul Dirac; Hermann Weyl |
Fourier transform The Fourier transform is an integral transform that decomposes functions into their constituent frequencies, converting a time- or position-domain description into a frequency- or momentum-domain representation. In the context of Quantum Mechanics it provides the mathematical bridge between wave function descriptions in position and momentum space, underpinning the formulation of operators, uncertainty relations, and spectral analysis of quantum systems.
The Fourier transform of a function f(x) is typically defined by the integral F(k) = ∫_{-∞}^{∞} f(x) e^{-ikx} dx (with normalization conventions varying by factor of 2π). Common alternate conventions use symmetric factors of (2π)^{-1/2}. The inverse transform reconstructs f(x) from F(k). These formulae are central in functional analysis and are rigorously treated in the theory of Schwartz space and tempered distributions to accommodate generalized functions such as the Dirac delta distribution introduced by Paul Dirac. The transform maps between the Hilbert spaces L^2(ℝ^n) used in Hilbert space formulations of quantum states, and it is a unitary operator under appropriate normalization, which relates closely to the spectral theorem for self-adjoint operators developed by John von Neumann.
In nonrelativistic quantum mechanics the Fourier transform converts the position-space wavefunction ψ(x) into the momentum-space wavefunction φ(p), implementing the representation change associated with the canonical commutation relations [x, p] = iħ. The momentum operator p̂ = -iħ∇ in position representation becomes multiplication by p in momentum representation after applying the transform. This representation change is formalized in the framework of Dirac notation and the theory of unitary representations of the Heisenberg group. Fourier analysis also appears in the path integral formulation of Richard Feynman where decomposition into modes facilitates evaluation of propagators and partition functions.
The basic transform pair used in quantum practice is ψ(x) ↔ φ(p) = (1/(2πħ))^{1/2} ∫ ψ(x) e^{-ipx/ħ} dx, with inverse φ(p) ↔ ψ(x) = (1/(2πħ))^{1/2} ∫ φ(p) e^{ipx/ħ} dp. These relations underpin textbook treatments in works such as Dirac, P. A. M., Landau and Lifshitz, and J. J. Sakurai. The transform pair explains phenomena such as momentum distributions measurable in scattering experiments at facilities like CERN and SLAC National Accelerator Laboratory and is used in quantum optics analyses by groups at institutions such as Caltech and MIT.
Important properties include linearity, unitarity (on L^2), the Plancherel theorem relating L^2 norms, and the convolution theorem linking multiplication in one domain to convolution in the other. The Riemann–Lebesgue lemma explains decay of transform amplitudes for smooth localized wavefunctions. The uncertainty principle (as formalized by Werner Heisenberg and given rigorous form by the Hirschman–Beckner and Robertson–Schrödinger inequalities) is a direct consequence of Fourier duality between conjugate variables. Spectral decompositions of observables use the Fourier transform in proofs of the spectral theorem attributed to David Hilbert and John von Neumann, and the Weyl transform connects operator ordering problems to phase-space methods developed by Hermann Weyl and Eugene Wigner.
Fourier transforms are used to solve the time-dependent and time-independent Schrödinger equation by diagonalizing translation- and momentum-related operators; plane-wave expansions provide free-particle propagators and Green's functions. In solid-state physics, Bloch waves and band-structure calculations employ discrete and crystal Fourier transforms (related to the Brillouin zone and Floquet theory). Spectral methods using transforms enable analysis of spectral lines in atomic physics and computation of scattering amplitudes in quantum field theory perturbation theory, with frequency-space Feynman integrals commonly evaluated via Fourier techniques. Experimental techniques such as angle-resolved photoemission spectroscopy (ARPES) map measured intensities to momentum-space distributions using Fourier-based reconstruction.
Numerical implementations rely heavily on the fast Fourier transform (FFT) algorithm, introduced by Cooley and Tukey, to perform discrete Fourier transforms efficiently in grid-based quantum simulations. Careful choice of sampling, windowing, and aliasing control is necessary to respect the Nyquist–Shannon sampling theorem and to preserve unitarity in discrete approximations. Pseudopotential and plane-wave methods in electronic structure packages (e.g., implementations in Quantum ESPRESSO, VASP, ABINIT) use FFTs for kinetic energy evaluation and Poisson-solver routines. Time-propagation algorithms such as split-operator methods alternate between position and momentum representations via discrete Fourier transforms, while spectral convergence properties are analyzed in numerical analysis literature by authors like L. N. Trefethen.
Category:Mathematical physics Category:Quantum mechanics