Best way to incorporate orthogonality/orthonormality proofs

[Daniel Mietchen]

It seems to me this is a major issue for Citizendium. If we want people to read its articles, we need to figure out how to get them higher up on the search engine result lists. Has this been discussed elsewhere on some forum? If not, I think it would be useful to bring this issue to a broader audience.

Yes, many times. Various methods have, I think, been proposed. But not much ever seems to change.

Yet another reason to get Related Articles set up properly across the site (the intrawiki links they provide count for Pagerank). Caesar and I have been testing a bot that can create Related Articles pages:

Caesar started doing that, but that's overkill. Look at [[Arizona]] (a very short stub). "Related Articles" should contain its neighboring states and its cities. They are not in that "related articles" page (for instance Utah is lacking), but, as we say in Dutch,  "the Devil and his old Mom" are in there, meaning all sorts of nonsense (even Sarah Palin!) 

OK, let's take this as an example. Currently, the bot is querying Special:WhatLinksHere/Arizona for entries in the main namespace, strips the redirects off and puts this into the bot section of the related articles subpage. That's as much automation we can get to populate Related Articles subpages. If Utah does not link to Arizona, then the bot has no way to know that Utah might be relevant, unless we use some sort of LinksFromPage (which would have found Utah) as a complement.

I hope that the bot's changes are reversible?

It currently only operates on articles for which no Related Articles subpage exist. Once the page is created, reversing this change would involve either to delete the page or to delete the inappropriate entries. Both actions require almost the same manual effort (though automated deletion could in principle be initiated by going through the whole category for this sort of bot edits), and the second one may simply be performed by moving the relevant pieces into the "parent", "sub" and "other" sections and deleting what remains of the bot section. This is actually the outcome the bot was supposed to facilitate, and once some knowledgeable human comes along to do this, he or she could easily add Utah or whatever is missing, and check whether the Arizona article has properly been related to these articles.

Related to search engine indexing, it would also help if less of CZ were "disallowed" in our robots.txt - e.g. the possibility for users to opt out of the default "disallow" for selected namespaces (since many CZ users link to CZ pages from their user page), or a separate and crawlable namespace for references (many of which link back to CZ articles).

Many more issues come into play with search engine ranking - e.g. Core Articles, Subpages, or informing the world about CZ activities via services like FriendFeed (also good for discussions) or Twitter (even Downing Street, the UK's Science Minister and many scientists do it!) - but at the end of the day, what impacts most on those rankings is increasing content, and for this, I think scaling up Eduzendium will be crucial.

Perhaps the summer break is an ideal time to make plans on how to contact universities or individual instructors about how to set up such courses, and to rethink how the subpage system can be made more flexible to accommodate things like /Proofs.

[Daniel Mietchen]

Since I am not a quantum mechanician, I have no knowledge whether the orthogonality of spherical harmonics is obvious to someone who is (I will take your word for it). However, as a student attempting to learn quantum mechanics, I can relate my experience, which is - it wasn't obvious to me. I think this raises a more general issue. Encyclopedias are used by different groups of people for different reasons. For someone who already understands a topic, they are resources to remind them of something they already know, but have a fuzzy recollection of. For someone who is investigating a topic, they are pedagogical. At least in the latter case, providing concrete arguments for a non-trivial mathematical assertion seems worthy.

You are right, an (excellent) encyclopedia should be helpful for all types and levels of readers.
So, what is obvious for quantum mechanician (and similar) should nevertheless be pointed out for those for whom it is not as obvious.

I agree, and this brings us back to references (of which a mathematical proof is just one form): CZ:Article Mechanics states "Citations are not usually needed for information that is common knowledge among experts." but if we are writing for non-experts too, there should be some way to bridge such gaps in obviousness. I am playing around, academic style, with Direct Referencing, but one could also envisage to use some mouseovering (perhaps combined with colour codes), though I have no idea where those templates are and how a mouseover display can link to a reference. 

[Dan Nessett]

You are right, an (excellent) encyclopedia should be helpful for all types and levels of readers.
So, what is obvious for quantum mechanician (and similar) should nevertheless be pointed out for those for whom it is not as obvious.
I think, it is done in the article on spherical harmonics, but it this may not be the place where you are looking for when you seek information on Sturm-Liouville. An article on Sturm-Liouville and/or on the solutions -- putting, among others, the topic into context -- is certainly a good idea. But what Paul pointed out is that orthogonality is the consequence of a general (and not very complicated) theorem on selfadjoint linear operators and their eigenvectors(-functions), so one only needs to check if this theorem can be applied.
Nevertheless, it might be useful to show it by direct calculation for those who prefer this to an abstract concept.
(Many readers will not be interested in either of these two proofs.)

I agree that there are several ways to get to the result. The orthogonality (but not orthonormality) is demonstrated by an appeal to a more general result. This still requires a derivation of the normalization constant. However, a humble student studying quantum mechanics, such as I, may not have the general result available when the specific result is required. For example, Shankar (which I believe is a fairly popular QM textbook) simply asserts the orthonormality of the spherical harmonics. At the point where this assertion stands, he has not yet discussed the solution of rotationally invariant problems (this comes in the next section). The spherically harmonic functions arise by solving part of a variable separated differential equation (if you have access to a copy of Shankar, this all takes place on page 334-335). The fact that the spherical harmonics are constructed from the associated Legendre functions doesn't appear until pg. 337 (and then Shankar only mentions Legendre polynomials, not the more general class of funtion).

Of course, I am not arguing that the proofs should be provided so I or even students using Shankar might have them available at the right time. I am attempting to make a more general argument that such proofs are valuable to a wide audience that has not yet fully comprehended the relationship between spherical harmonics, the associated Legendre functions, how these are related to the eigenfunctions of certain QM operators and how the differential equation that they solve is an example of the Sturm-Liouville equation (something about which I had no idea until my informal tutor instructed me). Students are in the unenviable position of not having every fact at their disposal (which is the reason why they are students and not experts).

[Daniel Mietchen]

informing the world about CZ activities via services like FriendFeed (also good for discussions) or Twitter

Example from earlier today.

[Howard C. Berkowitz]

Some incredibly bizarre imagery, if not pun, lurks here in an interdisciplinary manner. Would one go to a quantum mechanic if one's Turing Machine breaks down?

Recently, I wrote an article on wrenches. It really should have a section on the Heisenberg principle as applied to wrenches, especially torque wrenches. Using wrenches on hard-to-reach bolts and nuts is a Heisenberg demonstration: if you can see the nut, the wrench isn't on it. If you can't see the nut, the wrench may or may not be aligned to turn it; this may be a case of Schrodinger's Feline Technician.

[Peter Schmitt]

I agree, and this brings us back to references (of which a mathematical proof is just one form): CZ:Article Mechanics states "Citations are not usually needed for information that is common knowledge among experts." but if we are writing for non-experts too, there should be some way to bridge such gaps in obviousness. I am playing around, academic style, with Direct Referencing, but one could also envisage to use some mouseovering (perhaps combined with colour codes), though I have no idea where those templates are and how a mouseover display can link to a reference. 

I don't like such gadgets. They are distracting, as are too many footnotes or too many (Author, date) references in a paragraph.
At the first visit, encyclopedia articles should offer a good survey, not too short, but also not too long.
The reader should first check the article, and only after that follow links -- either to recommended further reading,
or to check references.
Of course, references to sources, and to additional information (internal and external) must be provided, but
in most cases the (annotated, if necessary) bibliography and the (annotated) external links should be sufficient.

To stay with the Sturm-Liouville example: The top level article should describe the equation, its origin, survey the results (and methods).
Recommended textbooks/monographs, the sources for the outstanding results should be listed.
A result like the "orthogonality" should be told in an appropriate way -- most readers will (at most) be interested in the result,
for others adequate hints should be given: e.g.,
The solutions form a system of orthogonal functions because they are (can be interpreted as) eigenvectors of the
[[linear operator]] .... to the values of the parameter ... This can also be verified directly by calculation [[if availaible]].

One may not expect, neither from a textbook nor from an encyclopedia, that all calculations are available but, of course, they can be.
(Though some teachers might not like it if all reasonable routine exercises can be found "in the net".)

[Dan Nessett]

To stay with the Sturm-Liouville example: The top level article should describe the equation, its origin, survey the results (and methods).
Recommended textbooks/monographs, the sources for the outstanding results should be listed.
A result like the "orthogonality" should be told in an appropriate way -- most readers will (at most) be interested in the result,
for others adequate hints should be given: e.g.,
The solutions form a system of orthogonal functions because they are (can be interpreted as) eigenvectors of the
[[linear operator]] .... to the values of the parameter ... This can also be verified directly by calculation [[if availaible]].

One may not expect, neither from a textbook nor from an encyclopedia, that all calculations are available but, of course, they can be.
(Though some teachers might not like it if all reasonable routine exercises can be found "in the net".)

Just a quick comment. The proof of the orthogonality of solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues is not something a professor is likely to ask his/her students to do. In a test, it would be too difficult to produce in a reasonable amount of time. As homework, there are widely available books that provide it (e.g., Churchill, which has been around at least since the 1960s). I agree that this proof should not directly appear in the main article. A small "complete proof" link in the main text near the proof outline you sketch above that takes the reader to a proof or advanced page is best.

[Paul Wormer]

Would one go to a quantum mechanic if one's Turing Machine breaks down?

No silly, quantum  mechanics are only licensed to fix quantum computers, for your broken Turing machine you must go elsewhere.

[Paul Wormer]

For a quantum mechanician the orthogonality of spherical harmonics is obvious: they are eigenfunctions of two commuting Hermitian operators L² and L_z.

I wrote http://en.citizendium.org/wiki/Hermitian_operator where I worked this out.

[Howard C. Berkowitz]

Would one go to a quantum mechanic if one's Turing Machine breaks down?

No silly, quantum  mechanics are only licensed to fix quantum computers, for your broken Turing machine you must go elsewhere.

Sigh. And there was such an interdisciplinary opportunity.

It amuses me that I'm following about 10 percent of this, but finding it informative. That, I believe, says something about the idea that every article must be understandable by the first-year undergraduate. Some articles are for ready reference; some articles are to be read and considered. This was one of the arguments I had with one of my book editors, whose previous experience was in quick reference guides while I was trying to teach how to think about a fairly abstract subject with very real applications.

[Dan Nessett]

For a quantum mechanician the orthogonality of spherical harmonics is obvious: they are eigenfunctions of two commuting Hermitian operators L² and L_z.

I wrote http://en.citizendium.org/wiki/Hermitian_operator where I worked this out.

I looked over the page you cite and understand your point. I am getting out of my field here, but I think spherical harmonics are used not only in QM, but also in other areas. So, appealing to the orthogonality of these functions on the grounds that they are eigenfunctions of the Hermitian operators L² and L_z would be confusing to students studying other application areas.

[Paul Wormer]

So, appealing to the orthogonality of these functions on the grounds that they are eigenfunctions of the Hermitian operators L² and L_z would be confusing to students studying other application areas.

I beg to disagree. The operator eigenvalue  argument predates quantum mechanics, see the 1924 book by Courant and Hilbert, the mathematics Bible for the physicists who developed QM (shortly after 1924). In my German edition  (vol I, p. 441) the spherical harmonics ("Kugelfunktionen von Laplace") are introduced as eigenfunctions of an operator  \Delta^* that is easily recognized as the \theta dependent part of L².   The only difference  with QM is that  Courant and Hilbert attach another  physical meaning to this operator (it arises in  vibrations of the surface of a sphere) .  It seems to me very plausible that,  whenever spherical harmonics appear in a any branch of physics, it is through  an eigenvalue equation of the Laplace operator expressed in spherical polar coordinates.

Further I like to add that the proof of the orthogonality of certain classes of functions almost always comes down to the proof that a certain operator is Hermitian or anti-Hermitian. As soon as you meet in a proof a partial integration  in which one term vanishes on the integration limits, you are pretty sure that you are dealing either with either an anti-Hermitian operator Q^\dagger = - Q, or with a Hermitian operator Q^\dagger = Q. (The presence of the  imaginary number i makes the difference between the two).  So, all that the usual quantum mechanical books do, is breaking up the proof into two parts.  Somewhere in the beginning of the book  the authors prove orthogonality of eigenfunctions of Hermitian operators (because the number i is essential in QM,  QM always deals with Hermitian operators, not with anti-Hermitian operators).   Much further down  in the book, when  central symmetric systems  are treated,  the Hermiticity of   L² is proved.  (This proof involves partial integration plus vanishing of a term on the integration limits). When one intertwines the two proofs,  one gets a much more complicated looking proof, but if operators and eigenvalues are not needed anywhere else,  this complication may pay off.

[Howard C. Berkowitz]

Might I suggest that in order not to lose the content here, it be redirected into an article on orthogonality/orthonormality proofs (as distinct from the articles on the specific topics?)  Perhaps it belongs on a talk page in the Mathematics and Physics workgroups -- for that matter, is there an appropriate subgroup to be created here?

[Paul Wormer]

But this is the physics workgroup.

[Dan Nessett]

So, appealing to the orthogonality of these functions on the grounds that they are eigenfunctions of the Hermitian operators L² and L_z would be confusing to students studying other application areas.

I beg to disagree. The operator eigenvalue  argument predates quantum mechanics, see the 1924 book by Courant and Hilbert, the mathematics Bible for the physicists who developed QM (shortly after 1924). In my German edition  (vol I, p. 441) the spherical harmonics ("Kugelfunktionen von Laplace") are introduced as eigenfunctions of an operator  \Delta^* that is easily recognized as the \theta dependent part of L².   The only difference  with QM is that  Courant and Hilbert attach another  physical meaning to this operator (it arises in  vibrations of the surface of a sphere) .  It seems to me very plausible that,  whenever spherical harmonics appear in a any branch of physics, it is through  an eigenvalue equation of the Laplace operator expressed in spherical polar coordinates.

Further I like to add that the proof of the orthogonality of certain classes of functions almost always comes down to the proof that a certain operator is Hermitian or anti-Hermitian. As soon as you meet in a proof a partial integration  in which one term vanishes on the integration limits, you are pretty sure that you are dealing either with either an anti-Hermitian operator Q^\dagger = - Q, or with a Hermitian operator Q^\dagger = Q. (The presence of the  imaginary number i makes the difference between the two).  So, all that the usual quantum mechanical books do, is breaking up the proof into two parts.  Somewhere in the beginning of the book  the authors prove orthogonality of eigenfunctions of Hermitian operators (because the number i is essential in QM,  QM always deals with Hermitian operators, not with anti-Hermitian operators).   Much further down  in the book, when  central symmetric systems  are treated,  the Hermiticity of   L² is proved.  (This proof involves partial integration plus vanishing of a term on the integration limits). When one intertwines the two proofs,  one gets a much more complicated looking proof, but if operators and eigenvalues are not needed anywhere else,  this complication may pay off.

As I said, I was getting out of my field, so I defer to your arguments. However, there is still a point of interest in this discussion. Not every exposition of a particular physical theory uses the same pedagogical approach. Some treat operators as first-class objects (framing the mathematics in terms of operators, eigenvalues, eigenvectors and eigenfunctions), while others just use operators as a convenient notational device. I went back and looked at my EM fields and waves text (Lorrain, Corson and Lorrain) and the section on the Laplace Equation (Chapter 12) that solves it in spherical coordinates uses the Laplacian operator as a notational convenience. There is no mention of Hermitian operators, eigenvalues, etc. The solution approach is to first separate variables and re-express the DE in terms of these. It solves the radial part and then tackles the angular part. It does this by a change of variable that turns the angular differential equation into Legendre's equation. After solving this equation, it simply asserts that the Legendre polynomials (without comment it restricts the solutions from those expressed using associated Legendre functions to those using Legendre polynomials) are orthogonal and gives the normalization information. It does not prove this orthogonality/normalization assertion.

So, a student taking  a course that uses this text (or one with a similar approach) has had no contact with operators and their properties. If he/she goes searching on the web looking for a proof of the orthonormality of Legendre polynomials (or associated Legendre functions, if the text he/she is using provides a more general result), then they will have no idea to look in the Citizendium section on operators and the properties of hermitian operators. Even if they did, some at least are unlikely to have the background in Hilbert Spaces and operator mathematics necessary to understand the proof using techniques based on those concepts.

I think this makes a case that a project like Citizendium should provide different routes to the same result, based on what are the most prominent mathematical frameworks in current use. So, I would argue, by all means Citizendium should have material on operators, hermitian operators, eigenvectors and the orthogonality of hermitian operator eigenfunctions. But, it also should have material that provides a route to results that does not depend on this way of formulating the problem.