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Heidelberg 2022 – wissenschaftliches Programm

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 11: Thermodynamics and fundamental aspects of field theory

MP 11.1: Vortrag

Donnerstag, 24. März 2022, 16:35–16:55, MP-H6

Relation between the Cartesian multipole expansion and the spherical harmonic expansion — •Nils Walter Schween and Brian Reville — Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg 69117, Germany

The multipole expansion, which is, for example, used to approximate an electrostatic potential (or a gravitational potential), has two equivalent forms. First, it is a Taylor expansion, i.e.

    4πє0 φ(r) = 
ρ(r′)
|r − r′|
  dr3  = 
l = 0
 
1
l!
 
ri1 ⋯ ril Qi1 ⋯ il
r2l + 1
   .

Note the Einstein summation convention. Secondly, it is a spherical harmonic expansion, i.e.

  4πє0φ(r) = 
l = 0
l
m = −l
2l + 1
 qlm
rlYlm(θ, ϕ)
r2l + 1
   .

We show that the relation between these two expansions can be formalised as a series of basis transformations of spaces of homogeneous polynomials of increasing degree l. These basis transformations allow us to derive an algorithm to express the components of the multipole tensors, i.e. Qi1il, as linear combinations of the spherical multipole moments qlm for an arbitrary degree l. Since the spherical multipole moments are

  qlm := Ylm*(θ, ϕ) rl ρ(r)  dr3   ,

this opens the opportunity to compute Qi1il solving the above integral instead of performing the derivatives and integrations needed to compute the multipole tensor components directly.

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