Talk:Hermite polynomials

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[edit] Applications of the Hermite polynomials

Can anyone provide more concrete examples of when the Hermite polynomials arise in real-life science or engineering applications? 171.64.133.56 22:50, 24 February 2006 (UTC)


I really have absolutely no idea how to edit pages and i am far too busy to learn, but shouldn't there be something on the self-duality of hermite polynomials. (self-duality means that a function is its own fourier transform). This property is inherrently important in the use of Hermite poly's and is one of the physical reasons that they are the solution to the Schrodinger Wave equation for the x^2 potential. A reference that shows this is http://www.math.rutgers.edu/~sdmiller/math574/math574notes.pdf just above equation 4.5. Again sorry about me not knowing how to edit pages. - Chris 21st May 2007

[edit] Conventions

This article does not follow the standard abramowitz & stegun conventions. Grrrrr >:( Most of the article seems to discuss He_n not H_n and this is confusing w.r.t. standard quantum mechanics usage. I'd like to see consistent notation usage, but don't want to embark on this cleanup now. linas 04:56, 26 Mar 2005 (UTC)

Standard?? I'm tempted to say: nobody uses that convention; only physicists use it. The convention used here is standard. Among probabilists. The convention used among physicists seems to be a result of the idiotic practice of defining the "error function" as
\operatorname{erf}(x)=\int_0^x e^{-u^2}\,du\,
rather than as
\operatorname{erf}(x)=\int_0^x e^{-u^2/2}\,du\,
which is rather obviously the right way to do it. Michael Hardy 21:29, 26 Mar 2005 (UTC)
I'm glad that you were only tempted, and didn't actually say any of those things :) A&S is more of a standard than any other text on special functions I can think of. As to "nobody uses that convention"; of the half-dozen books I have that might mention it, I'm pretty sure none of them used the convention in this article. I don't doubt the probabilists use a different convention; the goal here is to give two different functions two different unique names so that we can reliably know which one is being talked about. My pet peeve is that the Gamma function is clearly off by one, but it seems that only number theorists are aware of this fact. There are actually plenty of stories like this in math. What can one do? We now live in a global, connected world, the virtual distance between different topics is much much smaller. linas 01:57, 2 Apr 2005 (UTC)
Well should at least some weight be given to the fact that the convention followed by probabilists and statisticians makes sense and the one followed by physicists and special-functions theorists does not? And also the fact that the convention followed by non-statisticians can be regarded as at least a little bit non-standard, since there aren't very many non-statisticians (except among those who never think about Hermite polynomials at all)? Michael Hardy 02:33, 2 Apr 2005 (UTC)
Uh, well, this bit about one convention making sense, and the other not, is at best debatable and at worst trolling. In the harmonic oscillator (if I recall correctly) using the statistician's definition would lead to ugly factors of square-root two hanging out in ugly places (if I remember correctly). And vice-versa. I presume that this is the reason for the divergent definitions. No-one wants to use the "ugly" form in their work. And as to the suggestion to turn this into a popularity contest, I don't think this solves any problems. Never mind the fact that the statisticians would loose, being vastly outnumbered by undergrads studying the harmonic oscillator. I was fishing for constructive suggestions, not attacks. linas 04:15, 3 Apr 2005 (UTC)
Well, the number of undergrads studying statistics is far larger. But definitely the article should be explicit about divergent conventions. Michael Hardy 23:26, 3 Apr 2005 (UTC)
A constructive suggestion might be: "lets go thorough our collective references and see who uses which definition". linas 04:22, 3 Apr 2005 (UTC)
And it appears that I have only one book (besides A&S) that covers Hermite polys, and the convention that book uses is in the none-of-the-above category. (Its Ugo Fano's Physics of Atoms and Molecules). Oh well. linas 04:35, 3 Apr 2005 (UTC)

[edit] Comprise

A usage note at comprise in the American Heritage Dictionary says:

USAGE NOTE: The traditional rule states that the whole comprises the parts and the parts compose the whole. In strict usage: The Union comprises 50 states. Fifty states compose (or constitute or make up) the Union. Even though careful writers often maintain this distinction, comprise is increasingly used in place of compose, especially in the passive: The Union is comprised of 50 states. Our surveys show that opposition to this usage is abating. In the 1960s, 53 percent of the Usage Panel found this usage unacceptable; in 1996, only 35 percent objected. See Usage Note at include.

So I will not order Lethe to feel embarrassed for more than one minute, but at least for now I'm going to be a conservative on this issue, so I've changed it. Michael Hardy 01:12, 9 Jul 2004 (UTC)

D'oh!!! Lethe 14:33, Jul 9, 2004 (UTC)

[edit] Harmonic oscillator?

This article needs to have some discussion of the relationship of Hermite polynomials to the eigenfunctions of Schrodinger's equation for the quantum harmonic oscillator, since that is one of the major problems in which they arise. (I didn't want to insert it offhand...have to do some checking to make sure the conventions match.)

More generally, I would recommend discussing the history and applications of the Hermite polynomials before giving their mathematical definition, to make this article more broadly accessible.

—Steven G. Johnson 03:35, 14 October 2005 (UTC)

  1. The conventions don't match, theres a factor of 2.
  2. I don't understand why this can't be done in the article quantum harmonic oscillator, with this article providing no more than a sentance or two pointing the reader there.
  3. Agree about history: this article should get a proper introduction, including a 1-5 sentance history/uses/applications that is a part of the introduction. The formal definition following next. If there is a longer historical commentary, it goes in its own separate section.

linas 00:16, 15 October 2005 (UTC)

Which came first? Study of the polynomials, or study of quantum mechanics? --HappyCamper 00:19, 15 October 2005 (UTC)
Well, I thought Charles Hermite himself introduced the Hermite polynomials, and he died in 1901, when quantum mechanics was just getting started. I wouldn't be surprised if he never heard of quantum physics. Michael Hardy 00:26, 15 October 2005 (UTC)
Ah, yes you are right. The solutions to the harmonic oscillator do use Hermite polynomials, but perhaps a mentioning in passing is sufficient. --HappyCamper 01:18, 15 October 2005 (UTC)

Linas, aren't Hermite polynomials used independently of quantum physics, in probability theory? Michael Hardy 00:29, 15 October 2005 (UTC)

Hi Micheal, you may recall our earlier conversation, why, at the very top of this very talk page! Do I need to report you to Wikipedia Esperanza as having too high a stress level? What can I do to ease your stress and make you feel better about things? linas 02:14, 16 October 2005 (UTC)
Yup. See this rich paper: [1] --HappyCamper 01:18, 15 October 2005 (UTC)
Ohh, I really like the structure constants cklm defined on page two of that ref. Care to add that to this article? Not too often that one sees such nice, cleanly defined coeffs outside of Lie algebras. linas 02:25, 16 October 2005 (UTC)

[edit] Eigenfunctions

I'm going to follow Abramowitz & Stegun conventions in this comment, since I agree with Linas: it's as close to a standard as exists for special functions. It looks to me like the short section Eigenfunctions of the Fourier transform is incorrect. The functions H_n(x)e^{-\frac{x^2}{2}} are indeed eigenfunctions of the Fourier transform when the physicist's definition is used. However the eigenvalues are − ( − i)n for the forward transform and − in for the inverse transform. (The product of the forward and inverse eigenvalues must be unity; if both eigenvalues were − in, as the entry states, their product would be ( − 1)n).

Additionally, the probabilist's Hermite polynomials do not appear to me to yield eigenfunctions of the Fourier transform when multiplied by e^{-\frac{x^2}{2}}. Since H_{e_n}(x) = 2^{-\frac{n}{2}} H_n \left( \frac{x}{\sqrt{2}} \right), the scaling theorem can be used to show that the functions H_{e_n}(x) e^{-\frac{x^2}{4}} are eigenfunctions of the 'Fourier-like' transform T(f(x)) = \frac{1}{2\sqrt{\pi}} \int_{-\infty}^{\infty}f(x) e^{-\frac{itx}{2}} dx and it's inverse. The eigenvalues are once again − ( − i)n and − in respectively. --jmh 01:13, 15 January 2006 (UTC)

The eigenvalues given here and in the external site at wolfrom.com are off by -1. The correct eigenvalues (physicist form) are ( − i)n for the forward transform and in for the inverse transform. This is readily checked for n=0. This may have been the cause of much confusion. It can be further verified by checking equation 43 of Gaussian integral page of Wolfrom.com. Note that the formula holds for the punctured complex plane. Ustad NY 18:58, 23 September 2007 (UTC)

[edit] Plurals

Disclaimer: I do not speak or read French.

I have just reverted the French grammatical change made by Eskimbot. I assume it was enforcing a WP style guideline that says things in this context should be singular. But orthogonal polynomials are special. There's no such thing as an orthogonal polynomial -- it has nothing to be orthogonal to. I believe this issue has been discussed in the past by one of the regular gang, probably Michael Hardy.

If there is a style guideline that is enforced by bots and that we need an exception for, there may be an ongoing problem. Is this going to continue to happen? Can we flag this so that it doesn't? William Ackerman 04:07, 21 January 2006 (UTC)

[edit] Bibliography: Self-references

Here, as in many other math-related articles, User:Rea5, and other anonymous IPs (probably a dynamic IP) have been adding references to a book authored by Refaat El Ataar. This is not a notable math book (specially because it was edited in 2006!), so many users have been reverting those reference inclusions. Probably, it's a self-reference. (this may be coincidence but the user name Rea coincides with the initials of the author).

If you are the user who includes this references, please discuss it here first and explain why you think that book should be listed here. Otherwise, references to Refaat El Ataar books in this article will keep being removed.

--John C PI 14:37, 31 January 2006 (UTC)

[edit] physicists vs. probabilists redux; partial revert

User:Genjix has changed the definitions to match the "physicists" polynomials, from the previous use of the "probabilists" polynomials. This is a very controversial point, being discussed at length on this page. I happen to be in the "physicist" camp and sympathize with the change, but, unfortunately, that makes a number of things on the page (weight function, graphs, etc.) incorrect.

So I have temporarily patched things by showing both conventions in some of the most prominent places, and labeled the graph as showing the probabilists' polynomials. This issue needs to be cleared up, and I'm not sure how to proceed. William Ackerman 22:43, 27 February 2006 (UTC)

Including everything on both conventions, and the context within which each is preferable, would seem to be the ideal way to write this. I'll probably be back. Michael Hardy 23:48, 27 February 2006 (UTC)

I'd like to propose that, to help clear up the physicist/probabilist dichotomy, we change the symbol for the probabilists' polynomials to "He". This is the symbol used by Abramowitz and Stegun. I think we should continue to have the tags "(physicist)" and "(probabilist)" next to the equations. Having the same symbol letter for both functions is really awkward. Opinions? William Ackerman 17:02, 21 March 2006 (UTC)

  • Agree w/ Will above. linas 15:45, 18 November 2006 (UTC)