Hydrogen atom - complete orthornormal set

[QuantumCosmo]

Hi,
I was wondering if the bound solutions to the radial part of the hydrogen atom form a complete set for the functions in L^2(0,\infty). I know that the laguerre polynomials are complete and that they only differ from the radial solutions by factors of x^l * exp, so I thought that they would have to be complete too (apparently, it is not so simple as to say: "well, you are missing the scattering states, so of course they are not complete").
But I heard that it followed from Sturm-Liouville-Theory that they are not complete, which doesent seem to make any sense to me.
Does anyone have an idea?
Thank you,
QuantumCosmo

 

[dextercioby]

As far as I know, the bound states of the H-atom form a complete orthonormal set in a Hilbert space containing a sum of spaces one of them being L^{2} \left(\left(0,\infty\right), \, r^2 dr\right), the other components of the sum being related to the spherical angles.

 

[QuantumCosmo]

So that would mean the solutions to the hydrogen atom do form a complete set. Its what I thought but somehow not what my professor seems to think...
Thanks for the answer :)

 

[The_Duck]

Can't you make a wave packet out of scattering eigenstates to get a perfectly normalizable state that has no projection onto any bound state? That would imply that the bound states do not form a complete basis, yes?

 

[Meir Achuz]

You have to include the positive energy (scattering) states.

 

[dextercioby]

Can't you make a wave packet out of scattering eigenstates to get a perfectly normalizable state that has no projection onto any bound state? That would imply that the bound states do not form a complete basis, yes?

As far as I know, the bound states of the H-atom form a complete orthonormal set in a Hilbert space containing a sum of spaces one of them being L^{2} \left(\left(0,\infty\right), \, r^2 dr\right), the other components of the sum being related to the spherical angles.

I based my reply above on the purely Hilbert space analysis of the H-atom (Gerald Teschl's notes) which is obviously incomplete, due to its failure to account for the scattering states (<=>continuous spectrum of the Hamiltonian). Indeed, if one goes to an extension of L^2(R^3), called rigged Hilbert space (RHS), the bound states no longer form a complete set, because, as you say, one can form wavepackets out of scattering states which can be orthogonal to the bound states. One, of course, has to add the generalized eigenvectors corresponding to the continuous values of the energy, which is basically what the RHS achieves.

 

[A. Neumaier]

I based my reply above on the purely Hilbert space analysis of the H-atom (Gerald Teschl's notes) which is obviously incomplete, due to its failure to account for the scattering states (<=>continuous spectrum of the Hamiltonian). Indeed, if one goes to an extension of L^2(R^3), called rigged Hilbert space (RHS), the bound states no longer form a complete set, because, as you say, one can form wavepackets out of scattering states which can be orthogonal to the bound states. One, of course, has to add the generalized eigenvectors corresponding to the continuous values of the energy, which is basically what the RHS achieves.

Even in the Hilbert space, the bound states do not form a complete set, as you can see by looking at the spectral resolution. The space spanned by the bound states contains not a single scattering state. (The full spectrum can be seen by means of the dynamical group SO(2,4) rather than O(4), which only gives the discrete spectrum. Wybourne's book ''Classical groups for physicists'' gives a readable account.

 

[dextercioby]

(The full spectrum can be seen by means of the dynamical group SO(2,4) rather than O(4), which only gives the discrete spectrum. Wybourne's book ''Classical groups for physicists'' gives a readable account.

I tried to check his 50 pages long <readable account> of the H-atom and I was dissapointed. Formal mathematics with an abuse of bracket notation, no functional analysis, 90% of his statements unproven, 50 articles in his generous bibliography (to which I don't have access anyway) probably containing the missing proofs and making your head spin between the last pages containing the bibliography and the pages containing the links to it.

Oh, I may have only skimmed through, but a mention of the continuous portion of the spectrum appears on 2 pages at most, so I guess I'll give that part a second chance.

 

[A. Neumaier]

I tried to check his 50 pages long <readable account> of the H-atom and I was dissapointed. Formal mathematics with an abuse of bracket notation, no functional analysis, 90% of his statements unproven,

Well, as the title says this is a book for physicists, not for mathematicians. So he takes all the liberties a physicist needs to get to the results quickly. Mathematicians can add the required rigor themselves.

For a more rigorous treatment, try the book by Barut and Raczka. But I am not sure they are rigorous enough, and I am also not sure whether they exhibit the continuous portion of the hydrogen spectrum in detail enough to be useful.

 

[dextercioby]

[...]
For a more rigorous treatment, try the book by Barut and Raczka. But I am not sure they are rigorous enough, and I am also not sure whether they exhibit the continuous portion of the hydrogen spectrum in detail enough to be useful.

I have checked it and there's no detailed account on the unitary irreds of SO(4,2) or their influence in the quantum mechanical description of the H-atom.

 

[A. Neumaier]

I have checked it and there's no detailed account on the unitary irreds of SO(4,2) or their influence in the quantum mechanical description of the H-atom.

I haven't the book anymore, but there was a long sequence of exercises on the hydrogen atom in the the chapter on dynamical groups.

 

[A. Neumaier]

I haven't the book anymore, but there was a long sequence of exercises on the hydrogen atom in the the chapter on dynamical groups.

Actually, I found the book, and the exercises are _not_ there. This surprises me - I am sure that I had seen them somewhere...

Another possible source is: B. Cordani, The Kepler Problem, Birkh"auser 2003.
It treats the classical and the quantum Kepler problem along similar lines.

see also
http://slac.stanford.edu/pubs/slacpubs/0000/slac-pub-0141.pdf
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085769/pdf/pnas01823-0045.pdf

 

[DrDu]

Hi,
I was wondering if the bound solutions to the radial part of the hydrogen atom form a complete set for the functions in L^2(0,\infty). I know that the laguerre polynomials are complete and that they only differ from the radial solutions by factors of x^l * exp, so I thought that they would have to be complete too (apparently, it is not so simple as to say: "well, you are missing the scattering states, so of course they are not complete").
But I heard that it followed from Sturm-Liouville-Theory that they are not complete, which doesent seem to make any sense to me.
Does anyone have an idea?
Thank you,
QuantumCosmo

Although I don't remember the details, there is a difference of how the Laguerre polynomials enter in their definition as a complete set of orthogonal polynomials and how they enter into the radial solution of the Schroedinger equation. Namely in the latter, the radial dependence is scaled by a factor containing the main quantum number n. This makes the decisive difference.

 

[dextercioby]

[...] Another possible source is: B. Cordani, The Kepler Problem, Birkh"auser 2003.
It treats the classical and the quantum Kepler problem along similar lines.

see also
http://slac.stanford.edu/pubs/slacpubs/0000/slac-pub-0141.pdf
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085769/pdf/pnas01823-0045.pdf

Haha, yes, you pulled that one again on me. I don't have that very book. I thank you very much for the <stanford> article.

 

[A. Neumaier]

I thank you very much for the <stanford> article.

Some more free related articles on the topic:
http://arxiv.org/pdf/quant-ph/0409209
http://www.ifi.unicamp.br/~aguiar/Cursos/FI264/a90110.pdf

 

[dextercioby]

Actually, I found the book, and the exercises are _not_ there. This surprises me - I am sure that I had seen them somewhere...
[...]

Hi, Arnold, I found them at the end of Chapter 12, p. 378 pp 392. :D