Best way to incorporate orthogonality/orthonormality proofs

[Dan Nessett]

I am not sure whether this is a topic for the physics or mathematical fora. I will start the topic here, since it arises in the context of studying quantum mechanics. However, if the mathematics forum is a better place to discuss this issue, let me know and I will request it be moved (or, figure out how to move it myself, if that is something I can do).

I am studying quantum mechanics (using the textbook by Shankar) and having arrived at the point where rotationally symmetric systems are introduced came across the mathematical assertion (without proof) that the spherical harmonics are orthonormal. Having a somewhat skeptical nature, I attempted to prove this myself without success. So, I poked around on the web trying to find a proof. No luck. This led me to ask my informal tutor whether he could provide a proof. Working together we finally developed one (based largely on the proof of orthonormality in the book by Riley, Hobson, and Bence (Mathematical methods for physics and engineering). Concurrently, my tutor used another text to show that solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues are orthogonal. We think it would be useful to other students to make these proofs available on Citizendium.

The questions is: where is the best fit for these proofs in the Citizendium physics/mathematics ecology. I did some initial investigations and noticed there is nothing in the mathematics section on Hilbert spaces. Since the question of orthonormality arises in the context of these spaces, it would seem necessary to first add some material about these spaces before providing the proofs. This question is actually a specific example of a more general one: does material on function family orthonormality belong in the vector space discussion or in the differential equations section? What's more, this particular orthogonality/orthonormality issue may be of more interest to mathematical physicists than pure mathematicians, which suggests that the material belongs in a mathematical physics context.

I have no problem writing the necessary material (I am not a physicist, but have some familiarity with the mathematical topics underlying the orthonormality question). I just need some guidance what connective material is required and where to put it.

[Dan Nessett]

There seems to be some concern on the non-members discussion forum that the proofs I am proposing might be original work. While my collaborator and I reworked the presentation, the logic comes from textbooks. The proof of orthonormality of the associated Legendre functions follows the logic of Riley, Hobson, and Bence (Mathematical methods for physics and engineering) pg. 590, (2006) 3rd Edition, Cambridge University Press, ISBN 0-521-67971-0. The proof of orthogonality of solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues follows Churchill ("Fourier Series and Boundary Value Problems", pp. 70-72, (1963) McGraw-Hill, ISBN 0-07-010841-2).

If anyone wants to look at the proofs they are in my Wikipedia user space (when I attempted to add them there, Wikipedia editors responded that proofs are not welcome on Wikipedia). The Associated Legendre Functions orthonormality proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Legendre/Associated_Legendre_Functions_Orthonormality_for_fixed_m .
The Sturm-Liouville proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Sturm-Liouville/Orthogonality_proof .

[Matt Innis]

There seems to be some concern on the non-members discussion forum that the proofs I am proposing might be original work. While my collaborator and I reworked the presentation, the logic comes from textbooks. The proof of orthonormality of the associated Legendre functions follows the logic of Riley, Hobson, and Bence (Mathematical methods for physics and engineering) pg. 590, (2006) 3rd Edition, Cambridge University Press, ISBN 0-521-67971-0. The proof of orthogonality of solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues follows Churchill ("Fourier Series and Boundary Value Problems", pp. 70-72, (1963) McGraw-Hill, ISBN 0-07-010841-2).

If anyone wants to look at the proofs they are in my Wikipedia user space (when I attempted to add them there, Wikipedia editors responded that proofs are not welcome on Wikipedia). The Associated Legendre Functions orthonormality proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Legendre/Associated_Legendre_Functions_Orthonormality_for_fixed_m .
The Sturm-Liouville proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Sturm-Liouville/Orthogonality_proof .

No problem Dan, I wouldn't recognize original research in mathematical proofs if you hit me in the face with it   You would know that better than I would.  The nice thing is that we have experts here that would be much more likely to understand what you are doing and they will help you decide where it goes and how to present it.  You are going about it in the right way, so keep going!

[Howard C. Berkowitz]

There seems to be some concern on the non-members discussion forum that the proofs I am proposing might be original work. While my collaborator and I reworked the presentation, the logic comes from textbooks. The proof of orthonormality of the associated Legendre functions follows the logic of Riley, Hobson, and Bence (Mathematical methods for physics and engineering) pg. 590, (2006) 3rd Edition, Cambridge University Press, ISBN 0-521-67971-0. The proof of orthogonality of solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues follows Churchill ("Fourier Series and Boundary Value Problems", pp. 70-72, (1963) McGraw-Hill, ISBN 0-07-010841-2).

If anyone wants to look at the proofs they are in my Wikipedia user space (when I attempted to add them there, Wikipedia editors responded that proofs are not welcome on Wikipedia). The Associated Legendre Functions orthonormality proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Legendre/Associated_Legendre_Functions_Orthonormality_for_fixed_m .
The Sturm-Liouville proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Sturm-Liouville/Orthogonality_proof .

No problem Dan, I wouldn't recognize original research in mathematical proofs if you hit me in the face with it   You would know that better than I would.  The nice thing is that we have experts here that would be much more likely to understand what you are doing and they will help you decide where it goes and how to present it.  You are going about it in the right way, so keep going!

I wouldn't know if you hit me in the back with original chiropractic. I'd just know the results.

[Hayford Peirce]

There seems to be some concern on the non-members discussion forum that the proofs I am proposing might be original work.

That really wasn't a *concern* of mine, I was just trying to cover all bases in a theoretical discussion of whether proofs could be used in CZ math. articles.  Goodness knows we haven't had very *many* of them, thank goodness, but over the years there *have* been a couple of Citizens, who, during their brief stays with us, managed to introduce long articles about Crop Circles, the Odin Brotherhood, and other weird things, so we *do* have to be alert to the possibility that something similar may come along again.  But as Matt says in his comment above, who are *we* to know the merits of a math. proof? But, of course, somewhere among the Citizenry there *is* someone who knows....

[Matt Innis]

Okay, Hayford, I have to ask why you always add the period after math.

[Matt Innis]

There seems to be some concern on the non-members discussion forum that the proofs I am proposing might be original work. While my collaborator and I reworked the presentation, the logic comes from textbooks. The proof of orthonormality of the associated Legendre functions follows the logic of Riley, Hobson, and Bence (Mathematical methods for physics and engineering) pg. 590, (2006) 3rd Edition, Cambridge University Press, ISBN 0-521-67971-0. The proof of orthogonality of solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues follows Churchill ("Fourier Series and Boundary Value Problems", pp. 70-72, (1963) McGraw-Hill, ISBN 0-07-010841-2).

If anyone wants to look at the proofs they are in my Wikipedia user space (when I attempted to add them there, Wikipedia editors responded that proofs are not welcome on Wikipedia). The Associated Legendre Functions orthonormality proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Legendre/Associated_Legendre_Functions_Orthonormality_for_fixed_m .
The Sturm-Liouville proof is at: http://en.wikipedia.org/wiki/User:Dnessett/Sturm-Liouville/Orthogonality_proof .

No problem Dan, I wouldn't recognize original research in mathematical proofs if you hit me in the face with it   You would know that better than I would.  The nice thing is that we have experts here that would be much more likely to understand what you are doing and they will help you decide where it goes and how to present it.  You are going about it in the right way, so keep going!

I wouldn't know if you hit me in the back with original chiropractic. I'd just know the results.

And that's all that counts, right 

[Hayford Peirce]

Okay, Hayford, I have to ask why you always add the period after math.

Because I'm too lazy to write out "mathematics" and by putting the period after "math" I'm indicating that this is actually an abbreviation, not the semi-word "math" as in, "he is a math teacher." A subtle distinction, but, as the lawyers say, probably without a difference.... But if I'm talking with a prof. of mathematics I don't want him to think that I'm calling him a lowly math teacher. Or, as I think they would say in Blighty, maths....

[Peter Schmitt]

to show that solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues are orthogonal. We think it would be useful to other students to make these proofs available on Citizendium.

The questions is: where is the best fit for these proofs in the Citizendium physics/mathematics ecology. I did some initial investigations and noticed there is nothing in the mathematics section on Hilbert spaces. Since the question of orthonormality arises in the context of these spaces, it would seem necessary to first add some material about these spaces before providing the proofs. This question is actually a specific example of a more general one: does material on function family orthonormality belong in the vector space discussion or in the differential equations section? What's more, this particular orthogonality/orthonormality issue may be of more interest to mathematical physicists than pure mathematicians, which suggests that the material belongs in a mathematical physics context.

I think that there should be a general article on the Sturm-Liouville equation,
then (maybe) a separate page on "Solutions of the Sturm-Liouville equation" describing these solutions and their properties.
Then the proof of orthonormality could go into a subpage (/Advanced, as long as there is no /Proofs).
If this turns out impractical, the material can be moved.
(Whether this is mathematics or physics is not an important question.)

Both these pages would be nice to have together with the solution proof. Of course, there should be an article on Hilbert space
(and many more articles), but not necessarily because of Sturm-Liouville -- there are other topics which are much more
closely related, and the Hilbert space article should be a rather basic introduction, I think.

[Dan Nessett]

I think that there should be a general article on the Sturm-Liouville equation,
then (maybe) a separate page on "Solutions of the Sturm-Liouville equation" describing these solutions and their properties.
Then the proof of orthonormality could go into a subpage (/Advanced, as long as there is no /Proofs).
If this turns out impractical, the material can be moved.
(Whether this is mathematics or physics is not an important question.)

Both these pages would be nice to have together with the solution proof. Of course, there should be an article on Hilbert space
(and many more articles), but not necessarily because of Sturm-Liouville -- there are other topics which are much more
closely related, and the Hilbert space article should be a rather basic introduction, I think.

Thanks for your response, Peter. The only reason I mentioned Hilbert Space (HS) is the whole notion of orthogonality rests on the definition of an inner product in a vector space. Orthogonality of continuous (and square integrable) functions requires the notion of an infinite dimensional vector space, which is normally framed in terms of HS. I wouldn't expect that an article on HS would go into great detail, but it should at least suggest the idea of an infinite dimensional vector space and present (perhaps without much comment) the inner product for functions meeting the necessary criteria.

[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. The eigenvalue equation of L² is of the Sturm-Liouville type. See http://en.citizendium.org/wiki/Spherical_harmonics.
Apparently your "poking on the web" did not  extend to  Citizendium.

[Peter Schmitt]

For a quantum mechanician the orthogonality of spherical harmonics is obvious: they are eigenfunctions of two commuting Hermitian operators L² and L_z. The eigenvalue equation of L² is of the Sturm-Liouville type. See http://en.citizendium.org/wiki/Spherical_harmonics.
Apparently your "poking on the web" did not  extend to  Citizendium.

Paul, you confirm my suspicion which I wanted to check first (I am not familiar with details of PDEs) before mentioning it.
Of course, such general properties belong to the main articles.
(It might still be of some interest to explicitly check orthogonality - without reference to general Hilbert space and operator theory -
as an exercise, which then would better fit into /Tutorials.)

[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. The eigenvalue equation of L² is of the Sturm-Liouville type. See http://en.citizendium.org/wiki/Spherical_harmonics.
Apparently your "poking on the web" did not  extend to  Citizendium.

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.

When you conjecture: '[a]pparently your "poking on the web" did not  extend to  Citizendium.', you are entirely correct. I didn't even know Citizendium existed until a couple of weeks ago. And it is perhaps instructive briefly to relate why. I rarely go to a site looking for an answer. I almost always use a search engine (almost always Google) to find what I am looking for. Invariably among some of the first entries on the Google search list is a Wikipedia article. I can't recall one that cited a Citizendium article. Perhaps somewhere down the list a Citizendium article appeared, but if so, I have always ended my quest before arriving there.

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.

Thanks very much for the pointer to the article on spherical harmonics. It is very well done. Also, the pointer (in the article) to the Associated Legendre Function article suggests a natural place to link to the orthonormality proof (i.e., a link somewhere in the "Orthogonality Relations" section). It looks like the text of this article is imported from Wikipedia (or perhaps it went from Citizendium to there). In any case, the proof may need to be modified to conform to the notation used in the article (e.g., in the article the sub-script variables are l and l', whereas in the proof they are l and k)

[Hayford Peirce]

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.

[Peter Schmitt]

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 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.)

In any case, the proof may need to be modified to conform to the notation used in the article (e.g., in the article the sub-script variables are l and l', whereas in the proof they are l and k)

It probably will never be possible to have uniform notation ...
I prefer l and k because (especially in subscripts) they are easier to read and to distinguish.)