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.

Department(s)

Physics

Publication Status

Open Access

Comments

National Science Foundation, Grant PHY-1403973

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

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

Publication Date

01 Jan 2018

Included in

Physics Commons

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