Resource(s) for introduction to spherical harmonics with exercises?

In summary, a good book to start with is Mathematical Methods in the Physical Sciences, By Boas. It has a decent introduction to the theory and constructions, but also has plenty of examples and exercises from different fields. Another book to consider is Lebedev, Special Functions. It has a chapter of theory followed by a chapter of applications for each topic, including spherical harmonics.
  • #1
The Bill
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What combination of resources can you recommend for introducing people to spherical harmonics? Assume that the audience has the mathematical maturity of first-year grad students in mathematics, and will want a decent introduction to the theory and constructions. But also assume that this is part of an interdisciplinary applications class, so lots of concrete and calculation/prediction heavy examples and exercises are needed from different fields of study. I don't want all the exercises and examples to be pure math and quantum physics, say.
 
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Mathematical Methods in the Physical Sciences, By Boas
 
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drmalawi said:
Mathematical Methods in the Physical Sciences, By Boas
Um. I don't have a copy of Boas, but looking at the table of contents and index of the third edition, it's either irrelevant to this thread, or both the table of contents and index are very inaccurate on this topic, at least. I would find such a bad table of contents and index to be a near total disqualification unless there's literally nothing else in print or online that can substitute for this book.
 
  • #4
Amazon has numerous books on spherical harmonics, many of them specializing in a particular topic (Edmond’s Angular Momentum in Quantum Mechanics is a time-honored example). For a nice introductory treatment with various applications, take a look at Lebedev, Special Functions. He provides a chapter of theory followed by a chapter of applications for each topic, including spherical harmonics.
 
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  • #5
The Bill said:
Um. I don't have a copy of Boas, but looking at the table of contents and index of the third edition, it's either irrelevant to this thread, or both the table of contents and index are very inaccurate on this topic, at least. I would find such a bad table of contents and index to be a near total disqualification unless there's literally nothing else in print or online that can substitute for this book.
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it just has one chapter about it, it is not an entire book on spherical harmonicsHere is another, online, treatment it has some basic and some challenging problems
https://www.physics.uoguelph.ca/chapter-4-spherical-harmonics

Fourier Acoustics by Williams has applications to sound theory

Jackson should have applications of spherical harmonics in classical electrodynamics

How long you will spend dealing with Spherical Harmonics for this course? Do you have a syllabus peraps?
 
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I think the best approach is in quantum theory with bras and kets, solving the eigenvalue problem for the angular-momentum operator (best setting ##\hbar=1##, of course). You get the solution completely using algebraic methods, which you can subsequently reformulate in terms of "wave mechanics", i.e., for wave functions, defined on the unit sphere. Physically it's the "rigid rotatator" in 3D. A very good book for this is Ballentine, Quantum Mechanics. It's among the few books, which gives the correct argument that orbital angular momentum has only integer irreps (##\ell \in \{0,1,2,\ldots \}##) and not half-integer irreps, as suggested by the algebraic method.
 
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  • #7
Perhaps a book on seismic waves and earthquakes covers method of spherical harmonics in such context

Magnetosphere of the sun, would also employ spherical harmonics analysis

vanhees71 said:
Ballentine, Quantum Mechanics
My fav QM book!
 
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If I could wave a magic wand to get what I'm hoping for, what I'd like is a couple hundred pages of "student friendly" math textbook full of good exposition, tight definitions, all the commonly used theorems, and plenty of good interdisciplinary exercises. At least two meaty chapters worth.

Back to sad reality: preferably all in one book/PDF. But not more than three physical books or, say, five digital documents

drmalawi said:
View attachment 304320

it just has one chapter about it, it is not an entire book on spherical harmonics
That's an entry in the index. For one page. Looking at the table of contents, the chapter is on partial differential equations. The section containing page 649 is five pages on steady-state temperature in a sphere.

Even Arfken has more than that, and I rejected it for consideration as an answer to the topic of this thread before I posted the top post.
 
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  • #9
I am guessing that you are not interested in the group-theoretical aspects of spherical harmonics, but, if you are, you might want to look at "Group Theory In Physics: An Introduction To Symmetry Principles, Group Representations, And Special Functions In Classical And Quantum Physics" by Tung.
 
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The Bill said:
Even Arfken has more than that, and I rejected it for consideration as an answer to the topic of this thread before I posted the top post.
Would have been good to know.
 
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A good treatment in terms of the (imho cumbersome) traditional approach can be found in Jackson's electrodynamics book. Another great source for the traditional approach is, of course, also A. Sommerfeld, Lectures on Theoretical Physics, vol. 6 (Partial differential equations).
 
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The Bill said:
What combination of resources can you recommend for introducing people to spherical harmonics? Assume that the audience has the mathematical maturity of first-year grad students in mathematics, and will want a decent introduction to the theory and constructions. But also assume that this is part of an interdisciplinary applications class, so lots of concrete and calculation/prediction heavy examples and exercises are needed from different fields of study. I don't want all the exercises and examples to be pure math and quantum physics, say.
Are you restricting the treatment to scalar spherical harmonics, or do you also want to cover vector spherical harmonics?
 
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  • #13
Andy Resnick said:
Are you restricting the treatment to scalar spherical harmonics, or do you also want to cover vector spherical harmonics?
As far as I know, only scalar spherical harmonics are necessary for this, but if a source treats those well and also treats vector spherical harmonics well that would be fine, and possibly helpful in the future.
 
  • #14
The Bill said:
As far as I know, only scalar spherical harmonics are necessary for this, but if a source treats those well and also treats vector spherical harmonics well that would be fine, and possibly helpful in the future.
I find this to be a good introduction to the topic, but there aren't any prefab "homework problems". Similarly, there's a appendix in this dissertation that covers similar material and applies it to Stokes flow, but you would need to extract/generate homework problems.

Edit: I also supplement with material from the NIST Digital Library of Mathematical Functions (https://dlmf.nist.gov/).

 
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Schaum's Guide Fourier Series is pretty good
 
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Related to Resource(s) for introduction to spherical harmonics with exercises?

1. What are spherical harmonics and why are they important in science?

Spherical harmonics are a set of mathematical functions that describe the behavior of waves on a sphere. They are important in science because they can be used to model a wide range of physical phenomena, such as electromagnetic fields, quantum mechanics, and fluid dynamics.

2. How are spherical harmonics used in practical applications?

Spherical harmonics are used in a variety of practical applications, including image and sound processing, geodesy, and satellite remote sensing. They are also commonly used in computer graphics to create realistic lighting effects.

3. What is the best resource for learning about spherical harmonics?

There are many resources available for learning about spherical harmonics, but one of the most highly recommended is "Spherical Harmonics in Practice" by Michael W. Mahoney. It provides a comprehensive introduction to the topic with exercises and examples.

4. Are there any online resources for learning about spherical harmonics?

Yes, there are several online resources available for learning about spherical harmonics, such as interactive tutorials, video lectures, and online courses. Some recommended websites include Wolfram MathWorld, Khan Academy, and Coursera.

5. How difficult is it to understand spherical harmonics for someone without a strong mathematical background?

While a basic understanding of mathematics is necessary to fully grasp spherical harmonics, there are many resources available that offer a gradual and intuitive approach to learning the topic. With dedication and practice, anyone can gain a solid understanding of spherical harmonics.

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