Abstract
We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation ∇→2 g(x→ - x→') = δ(D) (x→ - x→') where x→ and x→' are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D = 2), we find a subtle interplay of the necessarily introduced scale L with the radial component of zero magnetic quantum number. For D = 3, the well-known expressions are briefly recalled; this is done in order to highlight the analogy with the four-dimensional case, where we uncover analogies of the four-dimensional spherical harmonics with the familiar three-dimensional case. Remarks on the SO (4) symmetry of the hydrogen atom complete the investigations.
Recommended Citation
U. D. Jentschura and J. Sapirstein, "Green Function of the Poisson Equation: D = 2, 3, 4," Journal of Physics Communications, vol. 2, no. 1, article no. 015026, IOP Publishing, Jan 2018.
The definitive version is available at https://doi.org/10.1088/2399-6528/aaa3bd
Department(s)
Physics
Publication Status
Open Access
Keywords and Phrases
associated Gegenbauer polynomials; cusp condition; Green functions; Poisson equation; special functions
International Standard Serial Number (ISSN)
2399-6528
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2024 The Authors, All rights reserved.
Creative Commons Licensing
This work is licensed under a Creative Commons Attribution 4.0 License.
Publication Date
01 Jan 2018
Comments
National Science Foundation, Grant PHY-1403973