Abstract
The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves, commonly known as Fourier decomposition after Jean-Baptiste Fourier’s Théorie analytique de la Chaleur (1822), plays an important role in applications of the theory. Dirac’s δ function, on the other hand, gained prominence primarily through its use in quantum mechanics, although today it is also commonly used in mechanics and electrodynamics to describe sudden impulses, mass points, or point charges. Both concepts are intimately connected to the completeness of eigenfunctions of self-adjoint operators. From the quantum mechanics perspective, the problem of completeness of sets of functions concerns the problem of enumeration of all possible states of a quantum system.
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Notes
- 1.
Yu.V. Sokhotsky, Ph.D. thesis, University of St. Petersburg, 1873; J. Plemelj, Monatshefte Math. Phys. 19, 205 (1908). The “physics” version (2.11) of the Sokhotsky-Plemelj relations is of course more recent than the original references because the δ distribution was only introduced much later.
- 2.
We are not addressing matters of definition of domains of operators in function spaces, see e.g. [21] or Problem 2.6. If the operators \(A_{\boldsymbol{x}}^{+}\) and \(A_{\boldsymbol{x}}\) can be defined on different classes of functions, and \(A_{\boldsymbol{x}}^{+} = A_{\boldsymbol{x}}\) holds on the intersections of their domains, then \(A_{\boldsymbol{x}}\) is usually denoted as a symmetric operator. The notion of self-adjoint operator requires identical domains for both \(A_{\boldsymbol{x}}\) and \(A_{\boldsymbol{x}}^{+}\) such that the domain of neither operator can be extended. If the conditions on the domains are violated, we can e.g. have a situation where \(A_{\boldsymbol{x}}\) has no eigenfunctions at all, or where the eigenvalues of \(A_{\boldsymbol{x}}\) are complex and the set of eigenfunctions is overcomplete. Hermiticity is sometimes defined as equivalent to symmetry or as equivalent to the more restrictive notion of self-adjointness of operators. We define Hermiticity as self-adjointness.
Bibliography
R. Courant, D. Hilbert, Methods of Mathematical Physics, vols. 1 & 2 (Interscience Publ., New York, 1953, 1962)
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966)
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Dick, R. (2016). Self-adjoint Operators and Eigenfunction Expansions. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_2
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DOI: https://doi.org/10.1007/978-3-319-25675-7_2
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