Abstract
We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation , where and 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.
Highlights
Solutions of the equation∇2g(x − x′) = δ(D)(x − x′) (1)enter a myriad of physical problems, from the elementary Coulomb problem in electrostatics (D = 3), to the attraction among vortices in two-dimensional systems (D = 2), and on to the four-dimensional formulation of the hydrogen Green function (D = 4, see Ref. [1])
Plm(cos θ) = −l (l + 1) Plm(cos θ). These formulas are recalled with the notion of clarifying the analogies with the four-dimensional case, as will be done in the following
The most important formulas of this brief paper can be found in Eqs. (20), (26) and (37): We derive [and in the case of Eq (26), just recall] the decomposition of the two, three- and four-dimensional Green functions of the Poisson equation into radial and angular parts
Summary
Enter a myriad of physical problems, from the elementary Coulomb problem in electrostatics (D = 3), to the attraction among vortices in two-dimensional systems (D = 2), and on to the four-dimensional formulation of the hydrogen Green function We shall attempt to provide a unified treatment of the radial and angular decompositions of the two-, three- and four-dimensional Green functions, which are solutions to Eq (1). In D = 2, a scale has to be introduced, which corresponds to a physically irrelevant overall constant term, while in D = 3, the formulas are very familiar In D = 4, we attempt to reveal a structure of (re-)defined associated ultraspherical polynomials (Gegenbauer polynomials), which highlights analogies to the associated Legendre functions that enter the case D = 3
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