Abstract
The single most recurring theme of mathematical physics is Fourier analysis. It shows up, for example, in classical mechanics and the analysis of normal modes, in electromagnetic theory and the frequency analysis of waves, in noise considerations and thermal physics, in quantum theory and the transformation between momentum and coordinate representations, and in relativistic quantum field theory and creation and annihilation operation formalism.
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Notes
- 1.
A piecewise continuous function on a finite interval is one that has a finite number of discontinuities in its interval of definition.
- 2.
The F n are defined such that what they multiply in the expansion are orthonormal in the interval (a,b).
- 3.
In the context of the uncertainty relation, the width of the function—the so-called wave packet—measures the uncertainty in the position x of a quantum mechanical particle. Similarly, the width of the Fourier transform measures the uncertainty in k, which is related to momentum p via p=ħk.
- 4.
Alternatively, we can multiply both sides by e −ik′x and integrate over x. The result of this integration yields δ(k−k′), which collapses the k-integrations and yields the equality of the integrands.
References
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience, New York (1962)
Simmons, G.: Introduction to Topology and Modern Analysis. Krieger, Melbourne (1983)
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Hassani, S. (2013). Fourier Analysis. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_9
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DOI: https://doi.org/10.1007/978-3-319-01195-0_9
Publisher Name: Springer, Cham
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