Talk:Multipole expansion

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edit·history·watch·refresh Stock post message.svg To-do list for Multipole expansion:

Tasks for expert:

  • derive the multipole expansion of a scalar field and explain relation to Taylor series,
  • stress immediated application to both electrostatics and Newtonian gravitostatics,
  • discuss some simple examples such as field of uniform rod, disk.
  • derive analogous expansions for vector and tensor fields (or at least give citations),
  • illustrate relation of terms to monopole, dipole, quadrupole
  • discuss non-uniqueness of sources for a given potential
  • add annonated citations, e.g. 'Mathematical Methods for Physics by Arfken and Weber (alas, insufficiently readable, organization not good enough), Methods of Mathematical Physics by Jeffries and Jeffries (better, except for unfortunately nonstandard notation),
  • link to discussion of applications in gravitation, electromagnetism, etc.,
  • link to forthcoming article on relativistic multipoles (for general relativity)

Suggested improvements[edit]

The "Interaction of two non-overlapping charge distributions" section has no reference for the form of the interaction energy formula for U_{AB}. It appears similar to that in Schwinger, Classical Electrodynamics, p. 261. Does it appear anywhere in the literature?

Still attempting to get reference for the "Interaction of two non-overlapping charge distributions" section. Is this how one does it?Wilcoxw (talk) 14:42, 23 September 2014 (UTC)

This is a good start, but this is a big topic and we need to say more (see todo list for suggestions). Note that I came here while writing todo list for Talk:Linearized gravity. ---CH 00:52, 6 February 2006 (UTC)

Introduction to Electrodynamics, 3rd Edition by David Griffiths has a pretty straight forward derivation/analysis of the multipole expansion in the context of electromagnetism. Not very rigorous by a mathematician's standards, but a good start. RyanC. 17:43, 4 March 2007 (UTC)
I noticed that more than a year ago this page was much better, but that WillowW deleted quite some material without comment. Why did he do that?--P.wormer 16:07, 13 July 2007 (UTC)
That would be "she", and I forget my motivations from over a year ago when I had first arrived here at Wikipedia - forgive me? I'll try to look it over again and remember. I seem to recall that there was much overlap with multipole moments so perhaps I was eliminating redundancy? As an aside, we should all strive for simplicity, accessibility and clarity for lay-people; not everyone is going to have heard of solid harmonics by that name. The customary format nowadays at Wikipedia seems to use inline citations as in {{cite book}} or {{cite journal}}. Friendly suggestions from Willow 21:22, 13 July 2007 (UTC)
  • Dear Willow, I am sorry, I always try to write gender neutral, but somehow the nickname "Willow" triggered something masculine in my mind (perhaps because it reminded me of Willem, a Dutch boy's name). Anyway, with regard to technical level: the older article spherical multipole moments is not any more elementary than what I wrote :). In my opinion we should try to cater for all levels of proficiency (including senior/graduate level), not only for the proverbial "average reader", or "beginning student". But, of course, we should not forget the latter two categories, and I'll do my best. As you can see I embarked on some electrostatic related articles. The main reason is that I want to fix up intermolecular forces, where the electrostatic multipole expansion of the interaction is central.
I asked you before why you have hidden the complex conjugation star (*) in the definition of the spherical multipole. Now I want to add as a comment that this star conveys information, namely that the starred quantity transforms contragrediently to an unstarred one. Hence a starred/unstarred pair is an invariant [under SO(3)]. Our mutual august authority Jackson apparently wasn't aware of this, or didn't care. I also like to ask you where you found the concepts "interior" and "exterior" multipole moment? I've never heard of them. As a final comment: feel free to correct or augment my writings, particularly with didactic stuff. If you feel the urge to scratch some advanced material, then please explain this first on the talk page. Thank you. --P.wormer 09:09, 14 July 2007 (UTC) PS I understood that the use of macros is voluntary? I haven't mastered their use yet.

Rewriting the lede[edit]

The most common use of multipole decompositions is certainly in solutions of Laplace's equation in 3-d. Solutions are easily written as series of multipole decompositions in powers of r and 1/r. However, this is by no means the most basic use of multipole expansions. The correct definition of multipole expansion in the most authoritative texts I can find only defines it on an n-sphere, and makes no reference to distance. I think this article should really be referring to that basic version. I've rewritten the lede to make this clear, while also noting the common application. The rest of the article still needs a lot of cleanup. It looks like there's also a whole lot of work to be done on related pages, like quadrupole. -- (talk) 05:02, 18 November 2007 (UTC)

What are these authoritative texts? Are they the books on angular momentum that you have cited? Your definition departs from the vast majority of uses of multipole expansions and contradicts parts of this article - for example, the idea of multipoles expressed in Cartesian or cylindrical coordinates. In every use of multipole expansions that I have seen (and some of them are very authoritative references like Morse and Feshbach, Methods of Theoretical Physics), they represent the potential of a body or radiation at a distance, and their purpose is to reduce the complexity of a source covering some finite region to point sources ("poles") at the origin. Multipole expansions do not include the positive powers of r, which increase with increasing distance.--RockMagnetist (talk) 17:17, 12 October 2010 (UTC)


In particle accelerators it is common to use dipole magnets for bending, quadrupole magnets for focusing and sextupole magnets for correcting chromatic aberrations (See Storage_ring). These should probably be mentioned on this page. I suspect that they only occur in a 2d multipole expansion, is that right? -- (talk) 11:41, 29 November 2007 (UTC)

To my knowledge the naming conventions in high energy physics and the corresponding engineering differ from standard conventions. Hence, a single magnet already is a dipole and does not show a monopole. The quadrupole magnet consist of four coils such that the naming is incorrect (see origin of naming below). Within naming conventions in a multipole expansion and the standard expansion itself a sextupole, which actually consist of six coils, does not exist. So, it might be mentioned and linked to an article on synchrotron rings (or similar), but it is not part of a multipole expansion. I guess in 2D everything is slightly different, as the potential becomes logarithmic etc. (infinite line charge).(mikuszefski) — Preceding unsigned comment added by Mikuszefski (talkcontribs) 09:14, 17 August 2012 (UTC)

Reason of the name[edit]

I think that somewhere in the first paragraph it should be explained why this expansion is called multipole expnsion. What is a pole in this context by the way?

Paranoidhuman (talk) 03:45, 26 September 2009 (UTC)

For the name one might check the book from Maxwell on electrodynamics. Here the argument is actually the other way around. Maxwell does not make a series expansion of an existing potential, but claims that in the far field any potential can be generated by special sources of a specific symmetry. He then constructs these sources starting with a singe charge, a pole. In a next step two charges are combined showing no total net charge. Hence the name dipole. These two charges are brought together such that the product of distance and charge remains constant. This object obviously creates a potential with a symmetry different to the one of the monopole. As it is necessarily related to the direction at which the two charges are brought together it is represented as a vector, the dipole moment. However, in the context of the multipole expansion one should consider this vector as a first order tensor. The next object is supposed to have neither monopole not dipole moment. This is achieved by properly combining two dipoles; hence they must be of equal strength and opposite direction. However, one is free in the direction in which these two approach each other (using the same limit conditions as before, i.e. strength times distance stays constant). Note that an additional vector comes into play. At the end one gets a second order tensor, the quadrupole. In each step an object of new symmetry is created by combining two objects of the next lower symmetry. In total one starts with 1 charge then 2, 4, 8, and so on. Each object consists virtually of 2^n charges/poles such that it can be called multipole. The standard naming convention is monopole, dipole, quadrupole, octopole (from octo - eight and not octupole as it is often found), hexadecapole, dotriacontapole, etc. (mikuszefski) — Preceding unsigned comment added by Mikuszefski (talkcontribs) 09:37, 17 August 2012 (UTC)


I don't see any particular reason why there should be separate pages for multipole expansion and multipole moments. You cannot discuss the one without the other. Hence it makes sense to try to merge the two pages. TimothyRias (talk) 08:59, 22 February 2010 (UTC)

I agree. The lead section for multipole moments would make an improved lead section for multipole expansion, and the multipole moments#Molecular electrostatic multipole moments is an application of multipole expansion#Expansion in Cartesian coordinates.--RockMagnetist (talk) 22:00, 12 October 2010 (UTC)

Quantum mechanics[edit]

In the section "Interaction of two non-overlaping charge distributions" I am missing a hint that for molecules (for example) the formula is certainly only catching the electrostatic interaction of the classical Coulomb interaction, missing all the exchange-correlation energy (This point is also very obscure in Distributed multipole analysis.) R. J. Mathar (talk)

Based on my experience, it is very common for basic quantum mechanics treatments to implicitly assume that exchange and correlation, as well as relativistic perturbations, are left out. The treatment in this article is basically what you would expect to find in an advanced undergraduate textbook or a basic graduate textbook in theoretical chemistry/computational chemistry. Its not even really a true "theoretical physics" treatment. To a degree, it makes sense to mention that this treatment leaves out more complicated (mathematically or physically) interactions and/or those that are only solvable in terms of approximation. Then again, this is the standard and you will find this implicit assumption in textbooks written from 1920 to 2014. It is by no means restricted to quantum mechanics. Fluid mechanics texts typically leave out magnetohydrodynamics until chapter 27. By that point it is assumed that you are studying using software and have implemented various forms of numerical tools to solve intractable problems. (talk) 16:11, 5 April 2014 (UTC)

Multipole radation[edit]

In the "Examples of multipole expansions" it would be nice to have an additional sentence saying that all of these electrostatic potentials and charges may start varying in time such that (thanks to the linearity of the Maxwell equations) one can move from the expansions of the charges on to the multipole radiation (which perhaps deserves an article on its own because the article on Electromagnetic waves does not mention this either, and Near and far field seems to be unaware of using some fuzzy sort of dipole approximation.) R. J. Mathar (talk)