Talk:List of quantum-mechanical systems with analytical solutions

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Goal of Page[edit]

I created this page to be a centralized location for links to all the quantum mechanics problems with analytical solutions on wikipedia. I have created a preliminary list, but envisage much more information. Perhaps the Hamiltonian (particularly V(r)), the eigenvalues (En), and wavefunctions ((r)) for each system could be included in the list.--Zolot 19:37, 14 February 2007 (UTC)[reply]

Thanks for starting this page. I think it will be quite useful. I started list of quantum mechanical potentials too. --HappyCamper 18:29, 14 April 2007 (UTC)[reply]
  • A similar list of analytic solutions can be found here. Would it not be nice if the lists were identical? (For instance via a link?)--P.wormer 00:38, 15 April 2007 (UTC)[reply]
I overlooked that list. I'll move them over here instead. --HappyCamper 01:37, 15 April 2007 (UTC)[reply]

hydrogen-molecule ion[edit]

The hydrogen-molecule ion, being a 3-particle system, is non-soluble. Even in the clamped nuclei approximation and use of prolate spheroidal coordinates (in which the 3D electronic equation separates into three 1D equations), one of the equations is done by series expansion (if my memory serves me right, I don't have access to any scientific literature right now). It is arguable if such an expansion may be called an analytic solution.--P.wormer 00:47, 15 April 2007 (UTC)[reply]

Table[edit]

So, going on the suggestion above, I thought about making a table that might look like this:

System Potential V(r) = Eigenfunctions Eigenvalues
Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum
Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

But this is complicated by a few things: How would we put the free particle in here? And how should we distinguish between systems solved with cartesian coordinates, and ones with spherical coordinates? --HappyCamper 19:35, 15 April 2007 (UTC)[reply]

This table would work, but you make a good point about it being excessively complicated. Expressing V(r) in various coordinate systems may be confusing to a general reader. Including eigenfunctions can be similarly complicated. For example, some sort of note would be needed to explain the Hermite polynomials in the solution to the harmonic oscillator (and the spherical harmonics when they arise, etc...). Clearly, such a comprehensive table would become rather confusing, and maybe it would be better to simply link to the main articles.
Perhaps a better solution would be to put a standardized Infobox on each of the linked pages with this information. The individual pages can then provide details of the coordinate systems, polynomial series, etc... Zolot 17:35, 23 April 2007 (UTC)[reply]
Ah yes, now the infobox idea sounds viable. I like that. --HappyCamper 18:02, 23 April 2007 (UTC)[reply]

Soluble or Solvable?[edit]

Should it be soluble or solvable?

According to the Oxford English Dictionary:

Solvable, a.

  1. Able to pay; solvent. Obs.
  2. Payable. Obs. rare.
    1. Capable of being solved.
    2. Math. Of a group: that may be regarded as the last of a finite series of groups of which the first is trivial, each being a normal subgroup of the next and each of the quotients being Abelian.
  4. Capable of being dissolved. Also absol.
  5. Capable of being resolved into something.


Soluble, a. and n.

  1. adj.
    1. Med.
      1. Of the bowels, etc.: Free from constipation or costiveness; relaxed. Now rare or Obs.
      2. Laxative; causing looseness of the bowels.
      1. Capable of being melted or dissolved.
      2. As a specific epithet with names of substances. soluble blue (also Soluble Blue), any of a class of water-soluble dyes that are di- and trisulphonic acid derivatives of aniline blue and are now used chiefly in papers and inks. In Biochem. applied to those species of RNA now usu. known as transfer RNA.
      3. Dissolving, solvent. rare.
    3. Capable of being untied or loosed. rare.
    4. Plastic, pliable. Also fig. Obs.
      1. Capable of being solved or explained; solvable.
      2. Math. = SOLVABLE a. 3b.
    6. Capable of being resolved; reducible.
  2. n. A soluble constituent, esp. of a foodstuff. —Preceding unsigned comment added by Arnob1 (talk • contribs) 14:02, 22 December 2007 (UTC)[reply]

Solutions of Schrodingers equation[edit]

Should this page be renamed Solutions of Schrodinger's equation ? I know that analytical solutions are the focus of this article, but there should be an article with the above name to collect together (or discuss) known solutions to SE. Compare Solutions of the Einstein field equations with Exact solutions in general relativity as an analogy. MP (talk•contribs) 21:18, 22 January 2008 (UTC)[reply]


The hydrogen molecular ion[edit]

It's unclear why this was included in this page, as it is not analytically solvable. The reference[1] from the cited article, Lambert W function, explicitly states that it is only solvable in that way via a limit to the one-dimensional case.

We can now make a synthesis with the results from dimensional scaling: the Schrödinger wave equation can be generalized to an arbitrary number of dimensions D which can be subsequently treated as continuous variable [14], [15], [18], [19], [20] and [21]. In the limit as D → 1+, the hydrogen molecular ion can be solved exactly [22] in terms of the Lambert W function

This item should be removed from the list.


Episanty (talk) 14:24, 9 September 2015 (UTC)[reply]

References

  1. ^ Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. doi:10.1016/j.chemphys.2005.10.031.
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