Applications of orthogonal polynomials

[lonewolf219]

I was wondering if anyone could provide some examples of when/where the following orthogonal polynomials are used in physics? I'm starting a research project in the math department, and my professor is trying to steer the project back to physics, asking for specific applications of the following:

Chebyshev
Hermite
Legendre
Laguerre
Jacobi
Gegenbauer

Any info would be helpful, thanks!

 

[Simon Bridge]

Have you tried looking them up?

 

[lonewolf219]

Yes, the Laguerre polynomials page on Wikipedia, for example, offers only one sentence on their application that I can see:

"The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom".

I'm not sure what that means since I haven't taken quantum mechanics, and this description is pretty vague. But maybe I'll rethink this post and try a different question instead

 

[Simon Bridge]

One-electron atoms ... like, you know, hydrogen.
So you now know that Laguerre Polynomials are used in relation to hydrogen physics.

Basically they are all important in modelling something to do with nature.

 

[lonewolf219]

Thanks for the help, Simon Bridge.

 

[Simon Bridge]

If the main thrust of your question is whether these things are purely abstract maths or if there is a point to them, there's a point. They are so annoying to deal with that nobody would bother otherwise.

Usually, though, someone being introduced to them would be at a stage where they can parse stuff like the wikipedia page, even if they have not studied the field involved. From where I'm sitting you've basically asked if there was a practical application for pi or e.

Anyway - hopefully you have a better idea how to read that stuff now: just draw from basic knowledge.

 

[lonewolf219]

Wikipedia is a good source for information, but it's not as good as the information directly provided by a physicist

 

[voko]

Orthogonal polynomials arise when solving certain boundary value problems, where they arise as eigenfunctions of some differential operator. Quantum mechanics in general and Schrödinger's equation in particular are all about finding eigenfunctions. In fact, the prefix "eigen" has entered English-language mathematics from physics, where it is a tribute to the German-language works that made the concept so important in modern physics.

 

[jtbell]

Therefore a good way to find out which orthogonal polynomials come up in QM is to browse through a reasonably comprehensive QM textbook. Look at the solutions for situations like the hydrogen atom, the simple harmonic oscillator, the circular infinite square well, etc.

You might also look in some electrodynamics textbooks, for solutions to Poisson's or Laplace's equation to find the electric field or potential function for various geometries and boundary conditions.

 

[lonewolf219]

Ahh, good idea jtbell... thanks! I think you may have also mentioned that I check out Mary Boas' Mathematical Methods book a few months ago... That was a fantastic recommendation.

 

[analogdesign]

Orthogonal polynomials are also used in coding algorithms for digital communications. Orthogonality allows the simultaneous transmission of uncorrelated messages across the same channel. These types of polynomials are also useful in cryptography.

 

[Redbelly98]

Hermite polynomials come up in studying the intensity profile of laser beams.

 

[Claude Bile]

Hermite and Laguerre polynomials are used to describe modes in optical fibres with cartesian and cylindrical symmetry respectively. Legendre polynomials are used in cases of spherical symmetry such as Mie scattering.

Claude.

 

[SteamKing]

Legendre, Laguerre, and Chebyshev polynomials can be used as a basis for figuring out quadrature methods (numerical integration). The standard Gauss quadrature method in particular depends on Legendre polynomials.

 

[lonewolf219]

Thanks analogdesign, redbelly98, Claude and SteamKing... You guys are awesome !

 

[boneh3ad]

Legendre, Laguerre, and Chebyshev polynomials can be used as a basis for figuring out quadrature methods (numerical integration). The standard Gauss quadrature method in particular depends on Legendre polynomials.

Darn, you beat me to it.

Beyond that, they are used pretty extensively as orthogonal bases in the estimation of functions that are otherwise unknown, leading to their use as SteamKing said as a tool for numerical integration. If you assume that solution takes the form of a series of orthogonal polynomials, you can transform the equations into a linear algebra problem instead of a differential equation and get the (approximate) solution to arbitrary precision limited only by the machine precision and how long you want to wait for your solver to finish. It usually gives a much more accurate solution for much less computation than brute forcing a problem with finite differences.